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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Integer multiplication exponent saving

Track exact exponent savings κ for integer multiplication in $O(n L(n)^{1-\kappa})$ time, where $L(n)=\max(\lceil\log_2 n\rceil,1)$ . Higher κ is better. Every entry uses the same public IntMul.KappaBound definition: one deterministic machine, a fixed finite alphabet and tape count, exact multiplication for every positive input length, and an eventual worst-case time bound.

Avi’s Harvey–van der Hoeven mission supplies the shared foundation. Its main goal uses the natural-logarithm formulation of the 2021 bound, so it serves as the foundation rather than a numeric entry. Avi’s positive-κ mission targets Jain’s round-six value 0.00003666565558019; it is a historical checkpoint. The reviewed community PR #62 checkpoint targets 0.000051016920170078. Open entries record goals to prove, not established records. The full multiplication theorem remains Open even when finite numerical certificates have been verified.

Community checkpoint and credits · Original framework · Harvey–van der Hoeven

NoneFormalized record≥ 0.00003666565558019Open frontier

2 provers on it0 of 2 missions formalized

3SUM Exponent

Classical algorithms solve 3SUM in $O(n^2)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic $O(n^{1.9992})$ algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.999074Formalized record

3 provers on it4 of 4 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Classical algorithms solve all-pairs shortest paths in $O(n^3)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic $O(n^{2.99942})$ algorithm. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.103205334138Formalized record≤ 2Open frontier

9 provers on it7 of 8 missions formalized

Sharp diagonal Hlawka constant

The sharp Hlawka inequality for Schatten $p$ -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real $p\ge256$ . We conjecture that the same formula holds for all $p\ge2$ .

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 80Formalized record≤ 70Open frontier

3 provers on it7 of 8 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

Schnirelmann (1930) showed some finite $k$ works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved $k = 5$ unconditionally. Helfgott (2013) proved that every odd number greater than $5$ is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: $27$ is neither prime nor $2$ + prime.

≤ 27Formalized record≤ 5Open frontier

35 provers on it13 of 15 missions formalized

Matrix multiplication exponent

Schoolbook matrix multiplication takes $n^3$ operations. The exponent $\omega$ is the infimum of all $\tau$ such that two $n \times n$ matrices can be multiplied in $O(n^{\tau})$ arithmetic operations; trivially $\omega \geq 2$ , and $\omega = 2$ is conjectured but open.

Strassen gave the first nontrivial bound, $\omega < 2.81$ , in 1969, and introduced the laser method in 1986 to reach $\omega < 2.48$ . Coppersmith and Winograd's 1990 bound of $2.376$ stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached $\omega < 2.371339$ in 2025, and the current record is $\omega < 2.371177$ , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Integer multiplication exponent saving

Community checkpoint and credits · Original framework · Harvey–van der Hoeven

NoneFormalized record≥ 0.00003666565558019Open frontier

2 provers on it0 of 2 missions formalized

3SUM Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.999074Formalized record

3 provers on it4 of 4 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.103205334138Formalized record≤ 2Open frontier

9 provers on it7 of 8 missions formalized

Sharp diagonal Hlawka constant

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 80Formalized record≤ 70Open frontier

3 provers on it7 of 8 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

≤ 27Formalized record≤ 5Open frontier

35 provers on it13 of 15 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

On a Problem of Optimal Transport Under Marginal Martingale Constraints 7: For c = |y − x| and Continuous µ the Optimizer Is Unique, Keeps µ ∧ ν in Place, Splits the Rest in TwoResearch Paper

Motivation

A martingale transport plan between two laws $\mu$ and $\nu$ on $\mathbb R$ is a joint law of a pair $(X,Y)$ with $X\sim\mu$ , $Y\sim\nu$ and $E[Y\mid X]=X$ . Minimizing or maximizing $E[c(X,Y)]$ over such plans gives the model-independent price bounds of an option with payoff $c(X,Y)$ when the market quotes vanilla options at two maturities, and so fixes the marginal laws $\mu$ and $\nu$ of the asset price. For the forward-starting straddle, with payoff $|Y-X|$ , the two bounds are the problems with costs $-|y-x|$ and $|y-x|$ .

1965: Strassen (doi:10.1214/aoms/1177700153) shows that martingale plans between $\mu$ and $\nu$ exist if and only if $\mu$ and $\nu$ are in convex order.
2012: Hobson and Neuberger (doi:10.1111/j.1467-9965.2010.00473.x) identify the optimizer for $-|y-x|$ through a construction of dual maximizers, under conditions on the marginals.
2012: Hobson and Klimmek communicate a description of the optimizer for $+|y-x|$ to Beiglböck and Juillet, cited there as private communication (arXiv:1208.1509v2, §7.4 and reference 15).
2016: Beiglböck and Juillet (arXiv:1208.1509v2) prove existence, uniqueness and the shape of the optimizer for $|y-x|$ for every continuous starting law, using only the primal problem and their variational lemma.

Setting

$\mu$ and $\nu$ are Borel probability measures on $\mathbb R$ with finite first moments, in convex order: $\int\varphi\,d\mu\le\int\varphi\,d\nu$ for every convex $\varphi:\mathbb R\to\mathbb R$ (Definition 2.1). $\Pi_M(\mu,\nu)$ is the set of measures $\pi$ on $\mathbb R^2$ with marginals $\mu$ and $\nu$ such that $y-x$ is $\pi$ -integrable and

\int\rho(x)\,(y-x)\,d\pi(x,y)=0\qquad\text{for every bounded Borel }\rho,

which is the paper's characterization (4) of "the disintegration $\pi_x$ has barycentre $x$ ". For the cost $c(x,y)=|y-x|$ the cost of a plan is $\int|y-x|\,d\pi\in[0,\infty)$ , and $\pi$ is optimal if it minimizes this over $\Pi_M(\mu,\nu)$ . $\mu$ is continuous if $\mu(\{x\})=0$ for every $x$ .

$\mu\wedge\nu$ is the largest measure below both $\mu$ and $\nu$ (Example 2.5); $(\mathrm{Id}\otimes\mathrm{Id})_\#\eta$ is the image of $\eta$ under $x\mapsto(x,x)$ , a measure on the diagonal $\Delta=\{(x,x)\}$ . For $\Gamma\subseteq\mathbb R^2$ , $\Gamma_x=\{y:(x,y)\in\Gamma\}$ , and $\operatorname{graph}(T)=\{(x,T(x))\}$ .

Formalization targets

Goal: Theorem 7.4 (p. 43)

If $\mu\preceq_C\nu$ and $\mu$ is continuous, there is a unique optimal $\pi_{\mathrm{abs}}\in\Pi_M(\mu,\nu)$ for $c(x,y)=|y-x|$ ; it is concentrated on a set $\Gamma$ with $|\Gamma_x|\le3$ for every $x$ ; and

\pi_{\mathrm{abs}}=(\mathrm{Id}\otimes\mathrm{Id})_\#(\mu\wedge\nu)+\pi_{\mathrm{go}},\qquad \pi_{\mathrm{go}}\ \text{concentrated on}\ \operatorname{graph}(T_1)\cup\operatorname{graph}(T_2)

for some functions $T_1,T_2:\mathbb R\to\mathbb R$ .

Milestones, in attack order

Attainment of the minimum (§2.1, pp. 10–11).
Lemma 1.11, the variational lemma (p. 8).
Lemma 7.5, the sign of a three-point cost difference (pp. 43–44).
The forbidden configurations (24) on a finitely optimal set (p. 45).
The static part: $\pi|_\Delta=(\mathrm{Id}\otimes\mathrm{Id})_\#(\mu\wedge\nu)$ for every optimal $\pi$ (pp. 45–46).
Lemma 3.2, accumulation of uncountably many large fibres (p. 19).
At most two off-diagonal points per fibre (p. 46).
The reduced problem between $\mu-\mu\wedge\nu$ and $\nu-\mu\wedge\nu$ (p. 46).
Lemma 5.5, two Borel graphs (p. 35), and Lemma 5.6, uniqueness from at most two points per fibre (p. 36).

Significance

The theorem identifies the lower model-independent bound for the forward-starting straddle: the extremal model keeps the mass that $\mu$ and $\nu$ share in place and splits every other starting point between at most two destinations. With Theorem 7.3 for $-|y-x|$ it settles both bounds for continuous $\mu$ without any dual attainment, which is known to fail in general. The decomposition into a static part $\mu\wedge\nu$ and a part between marginals with $\bar\mu\wedge\bar\nu=0$ is a reduction that applies to other costs vanishing on the diagonal.

The result is proved in the paper; nothing in it has been machine-checked. The mission produces a formal statement of Theorem 7.4 and of each step of its proof, a Lean vocabulary for martingale transport on $\mathbb R$ shared with the other missions of this series, and the general measure-theoretic lemmas (3.2, 5.5, 5.6) that the structure theorems of the paper all use.

Difficulty

Lemma 7.5 and (24) are elementary; the work lies in passing from them to statements about measures. The variational lemma needs Kellerer's duality-type result for finitely optimal sets. The static part requires showing that a positive defect $\kappa=\mu\wedge\nu-\pi(\Delta\text{-projection})$ produces, at $\kappa$ -almost every point, a forbidden configuration, which uses the disintegration of the moving part. The cardinality bound requires the accumulation argument of Lemma 3.2, and the forbidden configurations only hold for points that lie strictly inside the range of their own fibre, which the martingale condition supplies only almost everywhere. Uniqueness does not follow from strict convexity of a cost functional, since the problem is linear: it comes from the two-graph structure through Lemma 5.6, and it fails if $\mu$ has atoms (Remark 7.7).

Formalization scope

Measures are Mathlib Measure ℝ and Measure (ℝ × ℝ) with Borel σ-algebras. Costs are integrals in the extended reals through the published ModelRiskOT.Duality.extIntegral; the cost $|y-x|$ is nonnegative and, under finite first moments, finite. Martingale plans are encoded by condition (4), competitors by the same device. $\mu\wedge\nu$ is the infimum in Mathlib's complete lattice of measures, measure subtraction is Mathlib's truncated subtraction, cardinalities of fibres are Set.encard in $\mathbb N\cup\{\infty\}$ , and "concentrated on $A$ " is $\pi(A^c)=0$ . Continuity of $\mu$ is $\mu(\{x\})=0$ for all $x$ . As on the page, $\Gamma$ , $T_1$ and $T_2$ in the goal carry no measurability requirement.

Two choices differ from a literal reading and are disclosed in the items: (24) carries the hypothesis $y^-<x<y^+$ of Lemma 7.5, which the page invokes and without which (24) is false; and the steps that do not need continuity of $\mu$ (the static part, the reduced problem) are stated without it.

A trivializing formalization is ruled out: $\mu\wedge\nu$ is the lattice infimum of measures, not a product and not a pointwise minimum of set values, and all four conjuncts of the goal (existence, uniqueness, the bound 3, the decomposition) are asserted together.

Reusable beyond this mission: the Setting layer (convex order, $\Pi_M$ , competitors), Lemma 3.2, Lemma 5.5 (a Lusin–Novikov consequence) and Lemma 5.6. Proofs of any milestone, and infrastructure on disintegrations of plans on $\mathbb R^2$ , are welcome.

Selected references

M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints, Ann. Probab. 44(1), 42–106, 2016. arXiv:1208.1509v2, doi:10.1214/14-AOP966
D. Hobson, A. Neuberger, Robust bounds for forward start options, Mathematical Finance 22(1), 31–56, 2012. doi:10.1111/j.1467-9965.2010.00473.x
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36, 423–439, 1965. doi:10.1214/aoms/1177700153
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics 156, Springer, 1995 (Theorem 18.11). doi:10.1007/978-1-4612-4190-4

13 thms1 active userReviewed

Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

On a Problem of Optimal Transport Under Marginal Martingale Constraints 3: Between Probability Measures in Convex Order There Is Exactly One Left-Monotone Martingale PlanResearch Paper

Motivation

In classical optimal transport on the real line, one coupling plays a distinguished role: the monotone (Hoeffding–Fréchet) coupling, which sends the $q$ -quantile of the first marginal to the $q$ -quantile of the second. It is canonical (every initial segment of the first marginal goes as far left as possible) and it is optimal for a whole family of costs at once.

Martingale optimal transport adds the constraint that the coupling be the law of a one-step martingale $(X, Y)$ , $\mathbb E[Y\mid X]=X$ . The problem arises in robust (model-independent) mathematical finance: prices of vanilla options fix the marginal laws $\mu$ of $X$ and $\nu$ of $Y$ , and bounds on the price of an exotic option $c(X,Y)$ that hold under every arbitrage-free model are values of the martingale transport problem (Beiglböck, Henry-Labordère, Penkner 2013; Galichon, Henry-Labordère, Touzi 2014). Martingale couplings exist exactly when $\mu$ and $\nu$ are in convex order (Strassen 1965).

Beiglböck and Juillet (arXiv:1208.1509, Ann. Probab. 44(1), 2016) asked for the martingale counterpart of the monotone coupling, and found it: the left-curtain coupling $\pi_{\mathrm{lc}}$ . This mission formalizes its existence and uniqueness (Theorem 1.5), which is the foundation for the paper's later optimality results (Theorems 1.7, 6.1 and 6.3, other missions of this series).

Setting

All measures are Borel measures on $\mathbb R$ or $\mathbb R\times\mathbb R$ . $\mathcal M$ is the set of finite measures $\mu$ with $\int|x|\,d\mu<\infty$ . For $\mu,\nu\in\mathcal M$ :

Convex order $\mu\preceq_C\nu$ : $\int\varphi\,d\mu\le\int\varphi\,d\nu$ for every convex $\varphi:\mathbb R\to\mathbb R$ (Definition 2.1). It forces equal mass and equal mean. Extended convex order $\mu\preceq_E\nu$ : the same inequality for nonnegative convex $\varphi$ only (Definition 4.3); it allows $\mu(\mathbb R)<\nu(\mathbb R)$ .
Martingale transport plans $\Pi_M(\mu,\nu)$ : measures $\pi$ on $\mathbb R^2$ with marginals $\mu$ and $\nu$ and $\int\rho(x)(y-x)\,d\pi(x,y)=0$ for every bounded Borel $\rho$ , that is, the conditional barycentre of $\pi_x$ is $x$ for $\mu$ -a.e. $x$ .
Left-monotone plans (Definition 1.4): $\pi$ is concentrated on a Borel set $\Gamma$ that contains no three points $(x,y^-),(x,y^+),(x',y')$ with $x<x'$ and $y^-<y'<y^+$ . Mass leaving a point $x$ to both sides of $y'$ forbids any later point $x'>x$ from sending mass to $y'$ .
Shadow (Lemma 4.6): for $\mu\preceq_E\nu$ , $S^\nu(\mu)$ is the measure $\eta\le\nu$ with $\mu\preceq_C\eta$ that is least in the convex order among all such $\eta$ : the least spread-out part of $\nu$ into which $\mu$ can be embedded by a martingale.
Left-curtain coupling (Theorem 4.18): the plan $\pi_{\mathrm{lc}}$ with $\operatorname{proj}^x_\#(\pi_{\mathrm{lc}}|_{]-\infty,x]\times\mathbb R})=\mu|_{]-\infty,x]}$ and $\nu^{\pi_{\mathrm{lc}}}_x:=\operatorname{proj}^y_\#(\pi_{\mathrm{lc}}|_{]-\infty,x]\times\mathbb R})=S^\nu(\mu|_{]-\infty,x]})$ for every $x$ .

In Lean these are InM, ConvexLE, ExtConvexLE, IsMartingalePlan, IsLeftMonotone, IsShadow, IsLeftCurtain and targetUpTo in the shared namespace MartOT.Var. The mission's own theorems are in MartOT.Curtain.

Formalization targets

Goal: Theorem 1.5 (p. 6)

For probability measures $\mu\preceq_C\nu$ on $\mathbb R$ with finite first moments,

\exists!\,\pi\quad\pi\in\Pi_M(\mu,\nu)\ \text{ and }\ \pi\text{ is left-monotone}.

Existence alone, or uniqueness only among left-curtain plans, is not the goal: the statement says that monotonicity alone singles out one martingale coupling.

Milestones

Lemma 4.6: shadows exist and are unique, and satisfy (iii′) for the extended order.
Monotonicity of shadows (§4.4, p. 31): $\mu\le\mu'\preceq_E\nu$ implies $S^\nu(\mu)\le S^\nu(\mu')$ .
Theorem 4.18: $\pi_{\mathrm{lc}}$ exists, is unique, is a probability measure and lies in $\Pi_M(\mu,\nu)$ .
Theorem 1.8: $\nu^{\pi_{\mathrm{lc}}}_t\preceq_C\nu^\pi_t$ for every $t$ and every $\pi\in\Pi_M(\mu,\nu)$ .
Lemma 1.11 (variational lemma): an optimal plan of finite cost lives on a Borel set on which no finitely supported measure has a cheaper competitor.
Proof of Theorem 4.21: $\pi_{\mathrm{lc}}$ is optimal for every cost $c_{s,t}(x,y)=\mathbf 1_{]-\infty,s]}(x)|y-t|$ .
Theorem 4.21: $\pi_{\mathrm{lc}}$ is left-monotone (the existence half of the goal).
Lemma 5.1: end-point behaviour of $\mu\preceq_C\nu$ at $\sup\operatorname{spt}\mu$ and $\inf\operatorname{spt}\mu$ .
Lemma 5.2: a nonzero signed measure of mass $0$ is detected by a test function $g_{a,b}$ anchored in the support of its positive part.
Theorem 5.3: every left-monotone martingale plan is the left-curtain coupling of its marginals (the uniqueness half).

Significance

Theorem 1.5 identifies a canonical martingale coupling defined by a geometric property of its support alone. The paper then shows that $\pi_{\mathrm{lc}}$ is the unique optimizer of the martingale transport problem for costs $h(y-x)$ with $h'$ strictly convex (Theorem 1.7) and that it is characterized by the convex-order minimality of Theorem 1.8. The left-curtain coupling has since been studied and generalized in a series of works (for instance Henry-Labordère and Touzi 2016).

The result is proved in the paper. It has, as far as the platform's catalogue shows, no machine-checked proof: no statement about martingale transport plans, shadows or the left-curtain coupling exists on Prove2Me. A formal development would produce reusable infrastructure: the convex order on finite measures of arbitrary mass, shadows, and the passage between the barycentre characterization (4) of martingale plans and their disintegrations.

Difficulty

Existence is not the hard part in the abstract: a left-monotone plan can be obtained as an optimizer for a suitable cost via the variational lemma. The construction through shadows is more delicate: one must show that the shadows of the initial segments $\mu|_{]-\infty,x]}$ increase with $x$ (this rests on Theorem 4.8, the shadow of a sum) and that the resulting plan satisfies the martingale property.

Uniqueness is the main obstacle. The classical argument for uniqueness of optimal plans (averaging two candidates and using strict convexity) requires a continuous first marginal and does not apply to arbitrary $\mu$ with atoms. The paper's argument is specific to the problem: it compares $\nu^\pi_x$ with the shadow $\nu^{\pi_{\mathrm{lc}}}_x$ through the test functions $g_{u,v}$ and a case analysis at the support end-points, which needs the measure-theoretic Lemmas 5.1 and 5.2.

Formalization scope

Measures are MeasureTheory.Measure ℝ and Measure (ℝ × ℝ) with Borel σ-algebras. The goal and Theorems 1.8, 4.18, 4.21 take $\mu,\nu$ probability measures (IsProbabilityMeasure) in convex order; Lemma 4.6, the shadow monotonicity, Lemma 5.1 and Theorem 5.3 work in $\mathcal M$ , as the paper's Sections 4–5 do.
Integrals of convex functions and costs are extended-real valued (EReal, through the published ModelRiskOT.Duality.extIntegral), so no Bochner integral of a non-integrable function enters a statement. $\Pi_M$ is encoded by characterization (4) of the paper, not by disintegrations.
Shadows and the left-curtain coupling are predicates (IsShadow, IsLeftCurtain), never chosen functions: their existence and uniqueness are milestones, not definitions. A formalization that defined $\pi_{\mathrm{lc}}$ by a choice and the left-monotone plan as "the left-curtain plan" would make the goal circular; the goal mentions only $\Pi_M$ and left-monotonicity.
" $\pi(\Gamma)=1$ " is written $\pi(\Gamma^c)=0$ ; the support of a measure is Mathlib's Measure.support.
No hypothesis is added to any statement of the page. In Lemma 5.1, $\sup\operatorname{spt}\mu$ is the real supremum of the support under BddAbove; at $\mu=0$ Lean's convention $\sup\emptyset=0$ applies.
Reading: Theorem 1.8's "minimal" is formalized as "least" (below every member of the family), which is how the paper uses it.
Strassen's theorem is not restated; it is already posed on the platform as PalmQueueing.Ordering.strassen_cx.

Contributions are welcome at every level: proofs of the milestones, auxiliary lemmas on the convex order of finite measures (potential functions $u_\mu$ , equality of mass and mean), and the equivalence between (4) and the disintegration form of the martingale property.

Selected references

M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints, Ann. Probab. 44(1), 42–106, 2016. arXiv:1208.1509, doi:10.1214/14-AOP966
M. Beiglböck, P. Henry-Labordère, F. Penkner, Model-independent bounds for option prices — a mass transport approach, Finance Stoch. 17(3), 477–501, 2013. arXiv:1106.5929
A. Galichon, P. Henry-Labordère, N. Touzi, A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options, Ann. Appl. Probab. 24(1), 312–336, 2014. doi:10.1214/13-AAP925
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36(2), 423–439, 1965. doi:10.1214/aoms/1177700153
P. Henry-Labordère, N. Touzi, An explicit martingale version of the one-dimensional Brenier theorem, Finance Stoch. 20(3), 635–668, 2016. arXiv:1302.4854

14 thms1 active userReviewed

Dynamical Systems Machine Learning Optimization+1·Captain: mikedeng1

Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization 5: Decreasing-Step Adam Converges Almost Surely to the Critical Points of FResearch Paper

Motivation

Adam (Kingma and Ba, 2015) is the default optimizer for training neural networks: a stochastic gradient method that keeps an exponential moving average $m_n$ of past gradients and an exponential moving average $v_n$ of their coordinatewise squares, and divides the first by the square root of the second. Its practical success came well before any convergence theory for nonconvex objectives. Reddi, Kale and Kumar (2018) showed that Adam with constant momentum parameters can fail to converge even on convex problems, which made the question of under which parameter schedules Adam provably converges a live one.

Barakat and Bianchi (arXiv:1810.02263, SIAM J. Math. Data Sci. 2021) answer it through the ODE method of stochastic approximation (Benaïm, 1999). This mission takes the paper's almost-sure convergence theorem for Adam with decreasing stepsizes, Theorem 5.2. Under a stepsize schedule in which $1 - \alpha_n$ and $1 - \beta_n$ shrink in proportion to the stepsize $\gamma_n$ , the iterates converge almost surely to the critical points of the objective, provided they stay bounded.

Setting

Let $\xi$ be a random variable on a measurable space $\Xi$ with law $\mu$ , let $f : \mathbb R^d \times \Xi \to \mathbb R$ be an integrand with gradient $\nabla f(x, \xi)$ in $x$ , and set

F(x) = \mathbb E f(x, \xi), \qquad S(x) = \mathbb E\big(\nabla f(x,\xi)^{\odot 2}\big), \qquad \mathcal S = \nabla F^{-1}(\{0\}),

where $\odot 2$ is the coordinatewise square and $\mathcal S$ is the critical set. Let $\xi_1, \xi_2, \dots$ be iid copies of $\xi$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ .

Algorithm 5.1 takes stepsizes $\gamma_n > 0$ , weights $\alpha_n, \beta_n \in [0,1]$ and a constant $\varepsilon > 0$ . Starting from $x_0 \in \mathbb R^d$ , $m_0 = v_0 = 0$ and $r_0 = \bar r_0 = 0$ , it computes for $n \ge 1$ , with all vector operations coordinatewise:

\begin{aligned} m_n &= \alpha_n m_{n-1} + (1-\alpha_n)\nabla f(x_{n-1}, \xi_n), & v_n &= \beta_n v_{n-1} + (1-\beta_n)\nabla f(x_{n-1}, \xi_n)^{\odot 2},\\ r_n &= \alpha_n r_{n-1} + (1 - \alpha_n), & \bar r_n &= \beta_n \bar r_{n-1} + (1 - \beta_n),\\ x_n &= x_{n-1} - \gamma_n \frac{\hat m_n}{\varepsilon + \sqrt{\hat v_n}}, & \hat m_n &= m_n / r_n,\quad \hat v_n = v_n / \bar r_n . \end{aligned}

The divisions by $r_n$ , $\bar r_n$ are the bias correction.

The stepsizes satisfy Assumption 5.1: $\gamma_{n+1}/\gamma_n \to 1$ , $\sum_n \gamma_n = +\infty$ , $\sum_n \gamma_n^2 < +\infty$ , and there are $a, b$ with $0 < b < 4a$ , $(1-\alpha_n)/\gamma_n \to a$ and $(1-\beta_n)/\gamma_n \to b$ . The limiting dynamics is the autonomous ODE $\dot z = h_\infty(z)$ on $\mathcal Z_+ = \mathbb R^d \times \mathbb R^d \times [0,\infty)^d$ , written $(\mathrm{ODE}_\infty)$ , with

h_\infty(x, m, v) = \Big( -\frac{m}{\varepsilon + \sqrt v},\ a(\nabla F(x) - m),\ b(S(x) - v) \Big).

Formalization targets

Goal: Theorem 5.2

Assume Assumption 2.2 (regularity and moments of $f$ ), $F$ coercive (2.3), $S(x) > 0$ coordinatewise (2.4), iid samples (4.1), Assumption 5.1, $\sup_{x\in K}\mathbb E\|\nabla f(x,\xi)\|^4 < \infty$ on compacts (4.2 i) with $p = 4$ ), that $F(\mathcal S)$ has empty interior, and that $(x_n, m_n, v_n)$ is bounded with probability one. Then, almost surely,

d(x_n, \mathcal S) \to 0, \qquad m_n \to 0, \qquad S(x_n) - v_n \to 0,

and if $\mathcal S$ is countable, almost surely $(x_n, m_n, v_n) \to (x^*, 0, S(x^*))$ for some $x^* \in \mathcal S$ .

Milestones

Lemma 9.1 i)–ii): $r_n = 1 - \prod_{i=1}^n \alpha_i$ , and $r_n$ increases to $1$ .
§9.1: in the decomposition $\bar z_{n+1} = \bar z_n + \gamma_{n+1} h_\infty(\bar z_n) + \gamma_{n+1}\chi_{n+1} + \gamma_{n+1}\varsigma_{n+1}$ of $\bar z_n = (x_{n-1}, m_n, v_n)$ , the remainder satisfies $\varsigma_n \to 0$ almost surely.
§9.1: the piecewise-affine interpolation of $(\bar z_n)$ on the times $\tau_n = \sum_{k\le n}\gamma_k$ is almost surely a bounded asymptotic pseudotrajectory of the semiflow of $(\mathrm{ODE}_\infty)$ .
Proposition 7.13: $(\mathrm{ODE}_\infty)$ is well posed on $\mathcal Z_+$ and defines a semiflow $\Phi$ .
Proposition 7.14 (after Benaïm): the limit set of a relatively compact asymptotic pseudotrajectory of a semiflow with a strict Lyapunov function lies in the set of equilibria.
Proposition 7.15: on the closed hull of the orbits of a compact set, $W_\delta = V_\infty - \delta\langle\nabla F(x), m\rangle + \delta\|S(x) - v\|^2$ is a strict Lyapunov function for some $\delta > 0$ .

Significance

The theorem identifies a regime of Adam's hyperparameters, $1 - \alpha_n \sim a\gamma_n$ and $1 - \beta_n \sim b\gamma_n$ with $b < 4a$ , in which the algorithm behaves like a gradient method with vanishing noise: the iterates approach critical points, the momentum dies out, and the second-moment estimate converges to $S(x_n)$ . The condition $b < 4a$ ties the two averaging rates together. It is the condition under which the paper's Lyapunov function for the continuous dynamics decreases (Lemma 7.5). The argument structure is also a template for other adaptive methods: once the limiting ODE has a strict Lyapunov function, almost-sure convergence of the iterates follows from the same steps.

The paper proves the result on paper; none of it is machine-checked. A formal proof would also be the first formal instance on Prove2Me of the full ODE method for a concrete stochastic algorithm. That pipeline runs from the iterates to a perturbed Euler scheme, then to an asymptotic pseudotrajectory and a limit-set theorem, and finally to convergence. Benaïm's general results are being formalized separately on the platform (the asymptotic-pseudotrajectory definition this mission reuses comes from that effort), so the two developments meet in milestone 3.

Difficulty

The obvious approach is to treat Adam as stochastic gradient descent with a preconditioner and run a descent-lemma argument on $F(x_n)$ . This fails because the update direction $\hat m_n/(\varepsilon + \sqrt{\hat v_n})$ is not a descent direction for $F$ : $m_n$ lags behind $\nabla F(x_n)$ . Any Lyapunov function has to involve the momentum, and $F$ alone does not decrease. The continuous-time energy $V_\infty$ decreases only weakly, and only its perturbation $W_\delta$ is strict, and only on compact sets. A second difficulty is that the stepsize-dependent coefficients $(1-\alpha_{n+1})/\gamma_{n+1}$ and the bias corrections $r_n$ , $\bar r_n$ make the recursion a non-autonomous perturbation of the Euler scheme of $h_\infty$ . The remainder must be shown to vanish along almost every path before the general theory applies. Finally, the abstract limit-set theorem needs the semiflow to be well defined on the closed set $\mathcal Z_+$ , where $h_\infty$ involves $\sqrt v$ and is not Lipschitz at $v = 0$ .

Formalization scope

$\mathbb R^d$ is EuclideanSpace ℝ (Fin d). The state space $\mathcal Z$ is the product of three copies with Mathlib's max-of-factors norm, which is equivalent to the Euclidean norm of $\mathbb R^{3d}$ and leaves boundedness and limits unchanged. $F$ and $S$ are Bochner integrals against the law $\mu$ . Assumption 2.2 makes the integrands integrable, so they are the true expectations. Moments are lower Lebesgue integrals. Algorithm 5.1 is a pathwise recursion adamIter, driven by a random sequence $\xi : \mathbb N \to \Omega \to \Xi$ whose index $0$ is unused. The iid hypothesis is iIndepFun with every $\xi_{n+1}$ of law $\mu$ .

Two hypotheses are left implicit on the page and are added explicitly. The first is $\varepsilon > 0$ . The second is $\alpha_1 < 1$ and $\beta_1 < 1$ : Assumption 5.1 iii) permits $\alpha_1 = 1$ , which gives $r_1 = 0$ , and Lean's division by zero returns $0$ . Lemma 9.1 is stated with $\gamma_n > 0$ and $\sum\gamma_n = \infty$ , which its convergence claim needs. Semiflows are Flow ℝ≥0 on the subtype $\mathcal Z_+$ , and the asymptotic pseudotrajectory and limit set are the published StochApproxDyn.LimitSet definitions. The ODE milestones 4–6 hold for abstract $F$ , $S$ under Assumptions 7.1–7.2 with $0 < b \le 4a$ , as in the paper's §7.

The goal concerns the iterates of Algorithm 5.1 itself. A statement about trajectories of $(\mathrm{ODE}_\infty)$ , or one that replaces almost-sure boundedness by a bound uniform in $\omega$ together with deterministic gradients, is a different and much weaker theorem and is ruled out. The interior condition on $F(\mathcal S)$ and the countability of $\mathcal S$ enter exactly as on the page.

A complete development needs the following:

well-posedness of the ODE on a closed set with a non-Lipschitz field;
the Benaïm machinery: martingale-noise control via Doob's convergence theorem, and the passage from vanishing perturbations to asymptotic pseudotrajectories;
the limit-set theorem for strict Lyapunov functions.

The last two are reusable for any stochastic approximation scheme, and contributions to them are welcome independently of Adam.

Selected references

A. Barakat, P. Bianchi, Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization, SIAM J. Math. Data Sci. 3(1), 2021; arXiv:1810.02263v4. https://arxiv.org/abs/1810.02263
M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités XXXIII, LNM 1709, Springer, 1999. https://doi.org/10.1007/BFb0096509
D. P. Kingma, J. Ba, Adam: A Method for Stochastic Optimization, ICLR 2015. https://arxiv.org/abs/1412.6980
S. J. Reddi, S. Kale, S. Kumar, On the Convergence of Adam and Beyond, ICLR 2018. https://arxiv.org/abs/1904.09237

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Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

On a Problem of Optimal Transport Under Marginal Martingale Constraints 5: For c = ϕ(x)ψ(y), ψ Strictly Convex, ϕ Strictly Decreasing, the Left-Curtain Coupling Is the Unique OptimizerResearch Paper

Motivation

Martingale optimal transport asks for the cheapest way to couple two given laws $\mu$ and $\nu$ on $\mathbb R$ by a one-step martingale: a pair $(X,Y)$ with $X\sim\mu$ , $Y\sim\nu$ and $E[Y\mid X]=X$ . The problem arises in mathematical finance, where $\mu$ and $\nu$ are the risk-neutral laws of an asset at two maturities, implied by quoted vanilla option prices, and the extremal values of $E[c(X,Y)]$ over all such couplings are model-independent price bounds for the exotic payoff $c$ (Beiglböck, Henry-Labordère, Penkner 2013; Galichon, Henry-Labordère, Touzi 2014). In classical optimal transport on the line, a single coupling, the monotone (quantile) coupling, is optimal for a whole family of costs. Beiglböck and Juillet (arXiv:1208.1509, Ann. Probab. 2016) identify its martingale counterpart, the left-curtain coupling $\pi_{lc}$ , and determine costs for which it is optimal.

This mission formalizes one of those optimality results, Theorem 6.3: for product costs $c(x,y)=\varphi(x)\psi(y)$ with $\psi$ strictly convex and $\varphi$ decreasing, $\pi_{lc}$ is the unique optimizer.

Setting

Let $\mathcal M$ be the set of finite Borel measures on $\mathbb R$ with finite first moment. For $\mu,\nu\in\mathcal M$ the convex order $\mu\preceq_C\nu$ means $\int\phi\,d\mu\le\int\phi\,d\nu$ for every convex $\phi:\mathbb R\to\mathbb R$ ; it forces equal mass and equal mean. The extended convex order $\mu\preceq_E\nu$ asks this only for nonnegative convex $\phi$ .

A transport plan from $\mu$ to $\nu$ is a measure $\pi$ on $\mathbb R\times\mathbb R$ with marginals $\mu$ and $\nu$ ; the set of them is $\Pi(\mu,\nu)$ . It is a martingale transport plan, $\pi\in\Pi_M(\mu,\nu)$ , if $y-x$ is $\pi$ -integrable and

\int\rho(x)\,(y-x)\,d\pi(x,y)=0\quad\text{for every bounded Borel }\rho,

that is, the conditional law $\pi_x$ of $y$ given $x$ has barycentre $x$ for $\mu$ -almost every $x$ . For a cost $c:\mathbb R^2\to\mathbb R$ the cost of a plan is $E_\pi[c]=\int c\,d\pi\in(-\infty,+\infty]$ , the value is $C_M(\mu,\nu)=\inf_{\pi\in\Pi_M(\mu,\nu)}E_\pi[c]$ , and $\pi$ is optimal if it belongs to $\Pi_M(\mu,\nu)$ and attains this infimum.

For a plan $\pi$ and $t\in\mathbb R$ , the measure $\nu^\pi_t=\mathrm{proj}^y_\#(\pi|_{(-\infty,t]\times\mathbb R})$ is where $\pi$ sends the left part $\mu|_{(-\infty,t]}$ of $\mu$ . Given $\mu\preceq_E\nu$ , the shadow $S^\nu(\mu)$ is the measure $\eta$ with $\eta\le\nu$ and $\mu\preceq_C\eta$ that is $\preceq_C$ -below every other such measure. The left-curtain coupling of $\mu\preceq_C\nu$ is the plan $\pi_{lc}$ with

\mathrm{proj}^x_\#(\pi_{lc}|_{(-\infty,x]\times\mathbb R})=\mu|_{(-\infty,x]},\qquad \nu^{\pi_{lc}}_x=S^\nu(\mu|_{(-\infty,x]})\qquad(x\in\mathbb R).

It sends every left part of $\mu$ to the most concentrated part of $\nu$ that can receive it by a martingale.

Formalization targets

Goal: Theorem 6.3

Let $\mu\preceq_C\nu$ be finite measures, $\psi\ge0$ strictly convex, $\varphi\ge0$ strictly decreasing, $c(x,y)=\varphi(x)\psi(y)$ , and assume $C_M(\mu,\nu)<+\infty$ . Then

\pi_{lc}\ \text{is optimal, and every optimal }\pi\in\Pi_M(\mu,\nu)\text{ equals }\pi_{lc}.

Milestones

Lemma 4.6: for $\mu\preceq_E\nu$ in $\mathcal M$ the shadow exists and is unique.
Theorem 4.18: for $\mu\preceq_C\nu$ the left-curtain coupling exists, is unique, and lies in $\Pi_M(\mu,\nu)$ .
Proof of Theorem 4.21: for every $\pi\in\Pi_M(\mu,\nu)$ and $s$ , $\nu^\pi_s\le\nu$ , $\mu|_{(-\infty,s]}\preceq_C\nu^\pi_s$ , and $\nu^{\pi_{lc}}_s\preceq_C\nu^\pi_s$ .
Equation (16): $\displaystyle\int\varphi(x)\psi(y)\,d\pi=\int_0^\infty\Big(\int\psi\,d\nu^\pi_{\{\varphi\ge t\}}\Big)dt$ .
Equality case: for $\eta\preceq_C\eta'$ and $\psi$ strictly convex with $\int\psi\,d\eta'<\infty$ , $\int\psi\,d\eta=\int\psi\,d\eta'$ iff $\eta=\eta'$ .
§1.3: a plan in $\Pi(\mu,\nu)$ is determined by the family $(\nu^\pi_t)_{t\in\mathbb R}$ .

Significance

Theorem 6.3 gives a class of costs for which the martingale transport problem has an explicit, cost-independent solution, and does so with uniqueness and without any regularity of $\varphi$ beyond monotonicity. In the financial reading, every payoff of the form $\varphi(X)\psi(Y)$ with these shape properties has its model-independent lower price bound attained by one and the same model, the left-curtain martingale, which depends only on the marginals. The result complements Theorem 6.1 of the same paper (costs $h(y-x)$ with $h'$ strictly convex), proved by a different, variational route.

The theorem is proved in the paper. No machine-checked version is known: neither the left-curtain coupling nor shadows nor martingale optimal transport on $\mathbb R$ appears in Mathlib or on the platform. A formalization produces the shadow construction, the left-curtain coupling as a Lean object with its characterization, and the comparison principle " $\pi_{lc}$ has the $\preceq_C$ -least targets", all of which are reused by the other results of the series (uniqueness of the left-monotone plan, Theorem 6.1, the support bounds of §7).

Difficulty

The obvious argument compares costs plan by plan, but the convex order of two measures alone does not order the integrals of a strictly convex $\psi$ strictly: that $\int\psi\,d\eta=\int\psi\,d\eta'$ forces $\eta=\eta'$ requires representing $\eta'$ as a martingale image of $\eta$ (Strassen's theorem on $\mathbb R$ ) and the equality case of Jensen's inequality, with care about infinite integrals. The second obstacle is the passage from "equal targets for almost every level $t$ of $\varphi$ " to "equal plans": the superlevel sets of a strictly decreasing $\varphi$ are half-lines that may be open or closed at the jumps of $\varphi$ , so the targets must be controlled for both kinds of half-line, and a null set of levels must be bridged by monotonicity of $t\mapsto\nu^\pi_t$ . Existence of shadows (Lemma 4.6) is itself a nontrivial construction through potential functions $u_\mu(x)=\int|y-x|\,d\mu(y)$ .

Formalization scope

Measures are Mathlib Measure ℝ and Measure (ℝ × ℝ) with the Borel σ-algebras. $\mu$ and $\nu$ are finite, not necessarily probability, measures, as the theorem states; membership in $\mathcal M$ is part of ConvexLE. Integrals that may be infinite take values in EReal, through the published ModelRiskOT.Duality.extIntegral ( $\int f^+-\int f^-$ ). $\Pi_M$ is encoded by the characterization above with bounded test functions $\rho$ . The shadow and the left-curtain coupling are predicates (IsShadow, IsLeftCurtain); their existence and uniqueness are milestones, never built into definitions. "Optimal" is minimality of cost over martingale plans; attainment is not assumed.

Two hypotheses are read into the printed statement. "Decreasing" is strict: with $\varphi\equiv1$ all martingale plans have the same cost $\int\psi\,d\nu$ , and for $\mu$ uniform on $\{-1,1\}$ , $\nu$ uniform on $\{-2,0,2\}$ there are infinitely many of them. $C_M(\mu,\nu)<+\infty$ is added, as the proof assumes a plan of finite cost; otherwise every plan is optimal. A formalization in which $\pi_{lc}$ is merely some plan, the cost is a Bochner integral (which is $0$ for non-integrable integrands), or optimality is taken among all plans rather than martingale plans would not be Theorem 6.3. Theorem 4.18 is stated for finite measures of equal mass, the generality Theorem 6.3 needs. Strassen's theorem is already posed on the platform (PalmQueueing.Ordering.strassen_cx) and is not restated.

A complete development needs the convex order on finite measures, shadows, the left-curtain coupling, a layer-cake identity for product integrands and the equality case of the convex order for strictly convex test functions. The shadow and left-curtain layers are reusable across the whole series; contributions to any milestone, and to general lemmas on the convex order of finite (not only probability) measures, are welcome.

Selected references

M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints, Ann. Probab. 44(1), 42–106, 2016. arXiv:1208.1509v2, doi:10.1214/14-AOP966
M. Beiglböck, P. Henry-Labordère, F. Penkner, Model-independent bounds for option prices — a mass transport approach, Finance Stoch. 17, 477–501, 2013. doi:10.1007/s00780-013-0205-8
A. Galichon, P. Henry-Labordère, N. Touzi, A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options, Ann. Appl. Probab. 24(1), 312–336, 2014. doi:10.1214/13-AAP925
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36, 423–439, 1965. doi:10.1214/aoms/1177700153

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Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

On a Problem of Optimal Transport Under Marginal Martingale Constraints 6: For c = −|y − x| and Continuous µ the Optimizer Is Unique and Lives on Two Monotone GraphsResearch Paper

Motivation

A forward-start straddle pays $|S_2 - S_1|$ , the absolute change of an asset price between two future dates. A trader who knows the market prices of all vanilla options at both dates knows the laws $\mu$ of $S_1$ and $\nu$ of $S_2$ , but not their joint law. Under a pricing measure the price is a martingale, so the admissible joint laws are exactly the martingale couplings of $\mu$ and $\nu$ . The highest price consistent with the data is therefore the value of an optimization problem over martingale couplings: maximize $\mathbb E|S_2-S_1|$ . Hobson and Neuberger (Robust bounds for forward start options, 2012) studied this problem and identified the maximizing coupling through the dual problem. Beiglböck, Henry-Labordère and Penkner (arXiv:1106.5929) showed that the dual maximizers need not exist in general.

Beiglböck and Juillet (arXiv:1208.1509v2, Ann. Probab. 2016) approach martingale transport through the structure of the support of optimal plans. Their Theorem 7.3 recovers the Hobson–Neuberger optimizer for every continuous starting law $\mu$ : it is unique and splits each starting point into at most two destinations along two monotone maps.

Timeline.

1965: Strassen (doi:10.1214/aoms/1177700153) shows that martingale couplings of $\mu,\nu$ exist iff $\mu$ and $\nu$ are in convex order.
2012: Hobson and Neuberger identify the optimizer for $c=-|y-x|$ via a construction of dual maximizers, under conditions on the marginals.
2013: Beiglböck, Henry-Labordère and Penkner give an example where the dual maximizers do not exist.
2016: Beiglböck and Juillet prove existence, uniqueness and the two-graph structure for every continuous $\mu$ , using only the primal problem and a variational lemma.

Setting

A martingale transport plan between probability measures $\mu,\nu$ on $\mathbb R$ with finite first moments is a probability measure $\pi$ on $\mathbb R^2$ with first marginal $\mu$ , second marginal $\nu$ , and

\int \rho(x)\,(y-x)\,d\pi(x,y)=0\qquad\text{for every bounded Borel }\rho,

that is, the conditional mean of $y$ given $x$ is $x$ . Their set is $\Pi_M(\mu,\nu)$ . It is nonempty exactly when $\mu$ and $\nu$ are in convex order, $\mu\preceq_C\nu$ : $\int\varphi\,d\mu\le\int\varphi\,d\nu$ for every convex $\varphi$ .

For a cost $c:\mathbb R^2\to\mathbb R$ the problem is

C_M(\mu,\nu)=\inf\Big\{\int c\,d\pi:\ \pi\in\Pi_M(\mu,\nu)\Big\},

and $\pi$ is optimal if it attains this infimum. This mission concerns the Hobson–Neuberger cost $c(x,y)=-|y-x|$ , so optimal plans maximize $\int|y-x|\,d\pi$ . The measure $\mu$ is continuous: $\mu(\{x\})=0$ for every $x$ .

For $\Gamma\subseteq\mathbb R^2$ write $\Gamma_x=\{y:(x,y)\in\Gamma\}$ . A competitor of a finitely supported measure $\alpha$ on $\mathbb R^2$ is a measure $\alpha'$ with the same two marginals and the same conditional barycentres $\int y\,d\alpha_x(y)$ .

Formalization targets

Goal: Theorem 7.3 (p. 43)

If $\mu\preceq_C\nu$ are probability measures and $\mu$ is continuous, there is a unique optimal $\pi_{\mathrm{HN}}\in\Pi_M(\mu,\nu)$ for $c(x,y)=-|y-x|$ . Moreover there are a Borel set $S$ with $\mu(S)=1$ and functions $T_1,T_2$ , nondecreasing on $S$ , with

T_1(x)\le x\le T_2(x)\ (x\in S),\qquad \pi_{\mathrm{HN}}\big(\{(x,y):x\in S,\ y\in\{T_1(x),T_2(x)\}\}\big)=1.

Milestones, in attack order

Attainment (§2.1, pp. 10–11): for a lower semicontinuous, sufficiently integrable cost the infimum is attained when $\Pi_M(\mu,\nu)\neq\emptyset$ .
Lemma 1.11 (p. 8): an optimal plan of finite cost is concentrated on a Borel $\Gamma$ such that no finitely supported $\alpha$ on $\Gamma$ has a cheaper competitor.
Lemma 7.5 (pp. 43–44): the sign of $A-B$ , an explicit difference of absolute values, as a function of one variable.
(23) (p. 44): on such a $\Gamma$ , for $(x,y^-),(x,y^+),(x',y')\in\Gamma$ with $y^-<y'<y^+$ , neither $y'\le x'<x$ nor $x<x'\le y'$ .
Lemma 3.2 (p. 19) and the countability of $\{a:|\Gamma_a|>2\}$ (p. 45).
Lemma 5.5 (p. 35): a Borel set with at most two points per fibre is the union of two Borel graphs.
Monotonicity of $T_1,T_2$ (p. 45).
Convexity of the set of optimizers (p. 45) and Lemma 5.6 (p. 36): a nonempty convex set of martingale plans each living on a set with two-point fibres is a single plan.

Significance

The result. Theorem 7.3 gives the extremal martingale coupling for the forward-start straddle in closed structural form: a plan that sends each $x$ to one point below and one point above it, both chosen monotonically in $x$ . Combined with the martingale constraint, this determines the transition probabilities at each $x$ , so the plan, and the model-independent upper price bound, are pinned down by $\mu$ and $\nu$ . Remark 7.6 of the paper shows that continuity of $\mu$ cannot be dropped: for atomic $\mu$ uniqueness can fail.

Formalizing it. The result is proved in the paper; to our knowledge none of it is machine-checked. The mission produces a formal account of a support-based method (variational lemma, forbidden configurations, fibre counting, selection of Borel graphs, uniqueness from convexity). Lemma 3.2, Lemma 5.5 and Lemma 5.6 are general statements reused by the paper's other structure theorems (Corollary 1.6, Theorems 7.1, 7.4) and are of independent use.

Difficulty

The forbidden configurations (23) are pointwise statements about a set $\Gamma$ of full measure, while the conclusion is about functions. Passing from one to the other needs three ingredients that do not follow from (23) alone: a counting argument showing that only countably many fibres can have three or more points (this is where continuity of $\mu$ enters, through Lemma 3.2), a measurable selection of the two branches (Lemma 5.5, a Lusin–Novikov type theorem not in Mathlib), and a uniqueness argument comparing two optimizers through their average. The naive idea of reading uniqueness off from (23) directly fails: (23) constrains each optimizer separately, and two optimizers could a priori live on different pairs of graphs.

Lemma 1.11 itself rests on a duality theorem of Kellerer type for product sets and is the subject of a separate mission of this series.

Formalization scope

The Lean development works with measures on ℝ and ℝ × ℝ and their Borel σ-algebras. Integrals that may be infinite are taken in EReal through the published ModelRiskOT.Duality.extIntegral (positive minus negative part, with $(+\infty)-(+\infty)=-\infty$ ); the cost of a plan, $C_M$ , and the convex order are defined through it. $\Pi_M(\mu,\nu)$ is encoded by the paper's own characterization (4) with bounded test functions, and competitors by the same device; no disintegrations appear in the statements. Cardinalities of fibres are Set.encard, valued in $\mathbb N\cup\{\infty\}$ . Continuity of $\mu$ is the hypothesis $\mu(\{x\})=0$ for every $x$ . "Concentrated on $A$ " is $\pi(A^c)=0$ .

The attainment milestone uses the finite-measure class $\mathcal M$ introduced in §2.1: probability normalization is required for Lemma 1.11 and the goal, but not for attainment. The first moments of both marginals are explicit hypotheses in that milestone.

The goal states $T_1,T_2$ as nondecreasing on a Borel set $S$ of full $\mu$ -measure, with $T_1\le\mathrm{id}\le T_2$ on $S$ . The page writes $T_1,T_2:\mathbb R\to\mathbb R$ nondecreasing everywhere; read literally that version fails when $\mu$ has bounded support and $\nu$ does not, and the paper's proof produces exactly the full-measure form (as in its Corollary 1.6).

Statements (23), the fibre count and the monotonicity of $T_1,T_2$ are stated for an arbitrary set $\Gamma$ with the finite-optimality property of Lemma 1.11 for $c=-|y-x|$ , so they do not depend on a plan. A formalization proving only existence of an optimizer, or uniqueness without the two-graph structure, does not settle the goal: both parts are required.

Needed infrastructure: weak compactness of $\Pi_M(\mu,\nu)$ and lower semicontinuity of the cost functional; a Kellerer-type duality for Lemma 1.11; a Lusin–Novikov selection theorem for Lemma 5.5. Contributions to any of these, and alternative proofs of Lemma 5.6, are welcome.

Selected references

M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints, Ann. Probab. 44(1), 42–106, 2016. arXiv:1208.1509v2, doi:10.1214/14-AOP966
D. Hobson, A. Neuberger, Robust bounds for forward start options, Mathematical Finance 22(1), 31–56, 2012. doi:10.1111/j.1467-9965.2010.00473.x
M. Beiglböck, P. Henry-Labordère, F. Penkner, Model-independent bounds for option prices — a mass transport approach, Finance Stoch. 17, 477–501, 2013. arXiv:1106.5929
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36, 423–439, 1965. doi:10.1214/aoms/1177700153
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics 156, Springer, 1995 (Theorem 18.11). doi:10.1007/978-1-4612-4190-4

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Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

On a Problem of Optimal Transport Under Marginal Martingale Constraints 4: For c = h(y − x) with h′ Strictly Convex the Left-Curtain Coupling Is the Unique OptimizerResearch Paper

Motivation

Martingale optimal transport asks for the cheapest way to couple two probability laws $\mu$ and $\nu$ on $\mathbb R$ under the extra constraint that the coupling is the law of a one-step martingale. The problem arose in robust mathematical finance: when the prices of all call options at two maturities are observed, the marginal laws of the underlying asset at those maturities are known, and the no-arbitrage bounds on the price of a path-dependent option are the values of a martingale transport problem (Beiglböck, Henry-Labordère, Penkner 2013; Galichon, Henry-Labordère, Touzi 2014). The paper of Beiglböck and Juillet (arXiv:1208.1509v2, Ann. Probab. 44 (2016)) introduced the structural theory of the one-dimensional problem: a variational principle for optimal plans, a canonical coupling called the left-curtain coupling, and the identification of that coupling as the optimizer for a family of costs.

This mission formalizes that last identification. In classical optimal transport on the line, the monotone (quantile) coupling is the unique optimizer for every cost $h(y-x)$ with $h$ strictly convex; the result here is the martingale counterpart, with a different coupling and a different convexity condition.

Setting

Let $\mu,\nu$ be Borel probability measures on $\mathbb R$ with finite first moment. They are in convex order, $\mu\preceq_C\nu$ , if $\int\varphi\,d\mu\le\int\varphi\,d\nu$ for every convex $\varphi:\mathbb R\to\mathbb R$ (Definition 2.1). A transport plan is a measure $\pi$ on $\mathbb R\times\mathbb R$ with first marginal $\mu$ and second marginal $\nu$ ; it is a martingale transport plan, $\pi\in\Pi_M(\mu,\nu)$ , if under $\pi$ the conditional mean of $y$ given $x$ equals $x$ , equivalently

\int\rho(x)\,(y-x)\,d\pi(x,y)=0\quad\text{for every bounded Borel }\rho .

Strassen's theorem says $\Pi_M(\mu,\nu)\ne\emptyset$ exactly when $\mu\preceq_C\nu$ .

A cost is a function $c:\mathbb R^2\to\mathbb R$ . It satisfies the sufficient integrability condition if $c(x,y)\ge a(x)+b(y)$ for all $x,y$ with $a\in L^1(\mu)$ , $b\in L^1(\nu)$ ; then $\int c\,d\pi\in(-\infty,+\infty]$ is well defined for every plan. The martingale transport problem is

C_M(\mu,\nu)=\inf_{\pi\in\Pi_M(\mu,\nu)}\int c\,d\pi ,

and a plan attaining the infimum is optimal.

A plan $\pi$ is (left-)monotone (Definition 1.4) if it is concentrated on a Borel set $\Gamma$ containing no three points $(x,y^-),(x,y^+),(x',y')$ with $x<x'$ and $y^-<y'<y^+$ . For a finite measure $\eta\le\mu$ , its shadow $S^\nu(\eta)$ is the least measure in the convex order among the measures $\eta'\le\nu$ with $\eta\preceq_C\eta'$ (Lemma 4.6). The left-curtain coupling $\pi_{lc}$ is the plan that transports $\mu|_{]-\infty,x]}$ onto $S^\nu(\mu|_{]-\infty,x]})$ for every $x$ (Theorem 4.18); writing $\nu^\pi_t$ for the second marginal of $\pi|_{]-\infty,t]\times\mathbb R}$ , this says $\nu^{\pi_{lc}}_t=S^\nu(\mu|_{]-\infty,t]})$ .

Formalization targets

Goal: Theorem 1.7 (= Theorem 6.1)

Let $\mu\preceq_C\nu$ be probability measures, let $h:\mathbb R\to\mathbb R$ be differentiable with strictly convex derivative $h'$ , let $c(x,y)=h(y-x)$ satisfy the sufficient integrability condition, and assume $C_M(\mu,\nu)<\infty$ . Then

\pi_{lc}\ \text{is optimal, and every optimal}\ \pi\in\Pi_M(\mu,\nu)\ \text{equals}\ \pi_{lc}.

Milestones

Attainment (§2.1): for a lower semicontinuous cost with the integrability condition, the infimum $C_M(\mu,\nu)$ is attained whenever $\Pi_M(\mu,\nu)\ne\emptyset$ .
Lemma 1.11 (variational lemma): an optimal plan of finite cost is concentrated on a Borel $\Gamma$ such that no finitely supported $\alpha$ with support in $\Gamma$ has a strictly cheaper competitor (same marginals, same conditional barycentres).
The rerouting (proof of Theorem 6.1): for $\lambda y^+ + (1-\lambda)y^- = y'$ , the measure $\alpha'$ with masses $1-\lambda,\lambda,1$ at $(x',y^-),(x',y^+),(x,y')$ is a competitor of $\alpha$ with masses $\lambda,1-\lambda,1$ at $(x,y^+),(x,y^-),(x',y')$ .
Display (15): $d(t)=\lambda h(y^+-t)+(1-\lambda)h(y^--t)-h(y'-t)$ is strictly decreasing.
The rerouting is strictly cheaper: for $x<x'$ ,

\lambda c(x,y^+)+(1-\lambda)c(x,y^-)+c(x',y')>\lambda c(x',y^+)+(1-\lambda)c(x',y^-)+c(x,y').

Every finite optimizer is monotone.
Theorem 4.18: $\pi_{lc}$ exists, is unique and lies in $\Pi_M(\mu,\nu)$ .
Theorem 5.3: a monotone martingale transport plan is the left-curtain coupling of its marginals.
Theorem 1.9 (companion): under the goal's hypotheses, for $\pi\in\Pi_M(\mu,\nu)$ , "monotone", "optimal" and " $\nu^\pi_t\preceq_C\nu^{\pi'}_t$ for all $\pi'\in\Pi_M(\mu,\nu)$ and $t$ " are equivalent.

Significance

The theorem shows that, for every cost $h(y-x)$ with $h'$ strictly convex (for instance $h=\exp$ ), the martingale transport problem has the same optimizer, determined by $\mu$ and $\nu$ alone. In the financial reading, the model-independent bound on the price of the corresponding forward-start payoff is attained by one explicit martingale, the same for the whole family. Remark 6.2 of the paper derives from it that $\pi_{lc}$ also minimizes the essential supremum of $y-x$ . Theorem 1.9 packages the result as an equivalence of a geometric property (monotonicity), an optimality property and an order property of the curtain; later work on the left-curtain coupling, its explicit construction, and its extensions to higher dimensions and to multiple marginals takes these characterizations as its starting point.

The result is proved in the paper. As far as is known, none of it is formalized: Mathlib has the convex order neither for finite measures nor in the form used here, no martingale couplings, and no theory of shadows. The mission produces the statement layer (convex order on finite measures, martingale plans, extended-real costs, shadows, left-monotone plans, competitors) and asks for a machine-checked proof of the goal and of each milestone. The real-variable milestones (displays around (15)) are self-contained and elementary; the measure-theoretic ones carry the weight.

Difficulty

Optimality is a global property of a measure on $\mathbb R^2$ , while monotonicity is a pointwise property of its support. Passing from the first to the second requires the variational lemma, whose proof uses a duality theorem for measures with prescribed marginal bounds and a measurable-selection argument; a direct perturbation of $\pi$ along a single bad triple fails because a triple of points carries no mass. The second obstacle is uniqueness of the monotone plan (Theorem 5.3), which needs the shadow construction and the associativity of shadows; monotonicity alone does not visibly pin down the plan. Existence of an optimizer needs weak compactness of $\Pi_M(\mu,\nu)$ , which is not immediate because the martingale condition is tested against bounded measurable, not continuous, functions.

Formalization scope

All measures are Borel measures on $\mathbb R$ and $\mathbb R\times\mathbb R$ ; $\mu,\nu$ in the goal are probability measures. Convex order requires finite mass and finite first moment of both sides. Integrals that may be infinite ( $\int c\,d\pi$ , $\int\varphi\,d\mu$ for convex $\varphi$ ) are computed in the extended reals as $\int f^+-\int f^-$ (the published ModelRiskOT.Duality.extIntegral), never as Bochner integrals. $\Pi_M$ is encoded by the test-function condition displayed above (the paper's (4)). Shadows and the left-curtain coupling are predicates (IsShadow, IsLeftCurtain); their existence and uniqueness are theorems, and the goal asserts them rather than assuming them. Strict convexity of $h'$ is strict convexity of deriv h on $\mathbb R$ , with $h$ differentiable everywhere. Finiteness of costs is $C_M(\mu,\nu)<+\infty$ , which is the same as the existence of a martingale plan of finite cost (Theorem 6.1's wording).

A formalization that assumes an optimizer exists, or that asserts uniqueness only among left-curtain plans, is trivial and is not the goal: the statement asserts a plan with the left-curtain property, its optimality, and that every optimal plan equals it.

The development needs: weak compactness of plan sets on $\mathbb R^2$ ; lower semicontinuity of extended-real cost functionals; the variational lemma (shared with the companion mission on Lemma 1.11); shadows and their order properties; and the uniqueness of monotone martingale plans. The shadow theory and the variational lemma are reusable across the whole series on this paper. Contributions on any milestone are welcome, including proofs of the elementary displays (15) and the rerouting inequality.

Selected references

M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints, Ann. Probab. 44(1), 42–106, 2016. https://arxiv.org/abs/1208.1509v2
M. Beiglböck, P. Henry-Labordère, F. Penkner, Model-independent bounds for option prices: a mass transport approach, Finance Stoch. 17, 477–501, 2013. https://doi.org/10.1007/s00780-013-0205-8
A. Galichon, P. Henry-Labordère, N. Touzi, A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options, Ann. Appl. Probab. 24(1), 312–336, 2014. https://doi.org/10.1214/13-AAP925
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36, 423–439, 1965. https://doi.org/10.1214/aoms/1177700153
C. Villani, Optimal Transport: Old and New, Springer, 2009. https://doi.org/10.1007/978-3-540-71050-9

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Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

On a Problem of Optimal Transport Under Marginal Martingale Constraints 8: If Affine Lines Meet h′ in at Most k Points, Optimal Plans Split Each Non-Atom Into at Most k PointsResearch Paper

Motivation

Martingale optimal transport asks for the cheapest way to couple two given laws $\mu$ and $\nu$ of a price at two dates under the constraint that the coupling is a martingale: the conditional mean of the later price given the earlier one equals the earlier one. It is the model-independent pricing problem of mathematical finance: given the marginal laws implied by vanilla option prices, the extreme values of $E[c(X,Y)]$ over all martingale couplings bound the price of the exotic payoff $c$ (Beiglböck, Henry-Labordère, Penkner 2013; Galichon, Henry-Labordère, Touzi 2014). In the classical (non-martingale) transport problem, the shape of the cost determines the shape of the optimal coupling: for strictly convex costs of $y-x$ on the line the optimizer is a monotone map. The question this mission addresses is the martingale analogue: how much can an optimal martingale coupling split a single starting point, as a function of the cost?

Beiglböck and Juillet (arXiv:1208.1509, Ann. Probab. 2016) introduced the variational lemma for this problem and used it to prove, among other results, that the left-curtain coupling is optimal for a family of costs and that, for costs $h(y-x)$ , the number of points into which an optimal plan sends a non-atom of $\mu$ is bounded by a geometric property of $h'$ alone (their Theorem 7.1). This mission formalizes that bound.

Setting

All measures are Borel measures on $\mathbb R$ or $\mathbb R^2$ . A martingale transport plan between probability measures $\mu,\nu$ on $\mathbb R$ is a measure $\pi$ on $\mathbb R\times\mathbb R$ with marginals $\mu$ and $\nu$ such that $\int\rho(x)(y-x)\,d\pi(x,y)=0$ for every bounded Borel $\rho$ ; equivalently, the disintegration $(\pi_x)$ of $\pi$ along its first coordinate satisfies $\int y\,d\pi_x(y)=x$ for $\mu$ -almost every $x$ . The set of such plans is $\Pi_M(\mu,\nu)$ ; it is nonempty exactly when $\mu$ and $\nu$ are in convex order, $\mu\preceq_C\nu$ , meaning $\int\varphi\,d\mu\le\int\varphi\,d\nu$ for every convex $\varphi$ .

A cost is a function $c:\mathbb R^2\to\mathbb R$ . It satisfies the sufficient integrability condition if $c(x,y)\ge a(x)+b(y)$ for some $a\in L^1(\mu)$ , $b\in L^1(\nu)$ ; then $E_\pi[c]=\int c\,d\pi\in(-\infty,+\infty]$ is well defined for every plan. A plan $\pi\in\Pi_M(\mu,\nu)$ is optimal if $E_\pi[c]\le E_{\pi'}[c]$ for every $\pi'\in\Pi_M(\mu,\nu)$ .

A competitor of a finitely supported measure $\alpha$ on $\mathbb R^2$ is a measure $\alpha'$ with the same two marginals and the same conditional barycentres $\int y\,d\alpha_x(y)$ . For a set $\Gamma\subseteq\mathbb R^2$ , $\Gamma_x=\{y:(x,y)\in\Gamma\}$ is its fibre over $x$ .

This mission concerns costs of the form $c(x,y)=h(y-x)$ with $h:\mathbb R\to\mathbb R$ twice continuously differentiable, and the hypothesis that affine functions meet $h'$ in at most $k$ points: for all $s,t\in\mathbb R$ , $|\{x:h'(x)=sx+t\}|\le k$ . For instance $h(t)=t^4$ satisfies it with $k=3$ .

Formalization targets

Goal: Theorem 7.1 (p. 38)

Under the hypotheses above, with $\pi$ an optimal martingale plan of finite cost, there is a disintegration $(\pi_x)_{x\in\mathbb R}$ of $\pi$ such that for every $x\in\mathbb R$

\mu(\{x\})>0\qquad\text{or}\qquad\operatorname{card}(\operatorname{spt}\pi_x)\le k,

and if $\mu$ has no atoms, $\operatorname{card}(\operatorname{spt}\pi_x)\le k$ holds $\mu$ -almost surely for every disintegration of $\pi$ .

Milestones

Lemma 1.11 (variational lemma, p. 8): an optimal plan of finite cost is concentrated on a Borel set $\Gamma$ such that no finitely supported $\alpha$ with $\operatorname{spt}\alpha\subseteq\Gamma$ has a strictly cheaper competitor.
Lemma 3.2 (p. 19): if uncountably many fibres $\Gamma_a$ have at least $k$ points, some $(a,b_1<\dots<b_k)$ is a limit of such configurations from the right and from the left.
An interior point off the chord (proof of Theorem 7.1, p. 39): for $b_0<\dots<b_k$ , some interior $b_i$ has $h'(b_i-a)$ off the chord of $h'(\cdot-a)$ through $b_0,b_k$ .
The local sign (p. 39): near such $(a,b_i)$ , the cost difference (17) − (18) of a three-point rerouting is nonzero with sign fixed by $a-a'$ .
Countably many exceptional points (pp. 38–39): on such a $\Gamma$ , only countably many continuity points $x$ of $\mu$ have $|\Gamma_x|\ge k+1$ .

Significance

The result itself. Theorem 7.1 converts a one-dimensional condition on $h'$ into a sparsity statement for every optimal martingale coupling: non-atoms of $\mu$ are sent to at most $k$ points. For $h(t)=t^4$ this gives at most three points, and Section 7.3 of the paper gives an example with continuous $\mu$ where the optimizer splits into more than two points, so for this cost the bound cannot be lowered to two. For costs where $h'$ is strictly convex (so $k=2$ ), it gives the two-point structure also enjoyed by the left-curtain coupling, which is optimal for those costs (Theorem 1.9). Such support bounds are what make the optimizers computable and the corresponding robust price bounds explicit.

Formalizing it. The theorem is proved in the paper; to our knowledge none of it is formalized. A formal proof requires machine-checked versions of the variational lemma (itself resting on a duality theorem of Kellerer type), the countability argument of Lemma 3.2, and the real-analysis estimate behind the three-point rerouting. Each of these is reusable: Lemma 1.11 drives every structural result of the paper, and Lemma 3.2 is used again for the left-monotonicity theorems.

Difficulty

The natural first attempt is to argue pointwise: if a fibre $\Gamma_a$ has $k+1$ points, reroute mass within that fibre to lower the cost. This fails, because a competitor must preserve the barycentre of each fibre, and within a single fibre no rerouting both preserves the barycentre and the marginals. Any improvement must move mass between two different starting points $a$ and $a'$ , and this requires the two fibres to be close to each other and arranged consistently, which a single fibre cannot guarantee. Producing such pairs of fibres needs uncountably many exceptional points (Lemma 3.2) and the set $\Gamma$ of the variational lemma, whose existence is the deep part (it rests on a duality theorem for measures with given marginals). A second difficulty is that the conclusion is required at every non-atom $x$ , not only almost everywhere, so the exceptional set must be controlled exactly.

Formalization scope

Lean represents measures with Mathlib's Measure ℝ and Measure (ℝ × ℝ). $\Pi_M$ is encoded through the test-function characterization $\int\rho(x)(y-x)\,d\pi=0$ for bounded Borel $\rho$ ; costs are extended-real integrals $\int c^+-\int c^-\in[-\infty,+\infty]$ (the published ModelRiskOT.Duality.extIntegral); competitors of a finitely supported $\alpha=\sum_{s\in S}w_s\delta_s$ are encoded through the same test-function device. A disintegration is a Markov kernel $\kappa$ with $\pi=\mu\otimes\kappa$ , and $\operatorname{card}(\operatorname{spt}\pi_x)\le k$ is written as " $\kappa(x)$ is concentrated on a set of at most $k$ points", which is equivalent for probability measures on $\mathbb R$ . Cardinalities take values in $\mathbb N\cup\{\infty\}$ ; $h'$ is deriv h.

Standing and added hypotheses of the goal: $\mu,\nu$ are probability measures in convex order; "optimal transport plan" means optimal martingale plan, as throughout Section 7; and, because the proof starts from Lemma 1.11, the hypotheses of that lemma are added: the sufficient integrability condition for $c$ and finite cost of $\pi$ . "Continuous $\mu$ " is $\mu(\{x\})=0$ for every $x$ .

The conclusion is not trivialized: the first part holds for every $x$ with a single chosen kernel, not almost everywhere, and the second part quantifies over all kernels, not one. The hypotheses are jointly satisfiable (for example $h(t)=t^4$ , $k=3$ , $\mu=\nu=\delta_0$ ).

A complete development needs disintegration of measures on $\mathbb R^2$ (available in Mathlib as condKernel), the variational lemma with its duality ingredient, and elementary real analysis. Contributions to the variational lemma are reusable across the whole series of missions on this paper.

Selected references

M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints, Ann. Probab. 44(1), 42–106, 2016. arXiv:1208.1509
M. Beiglböck, P. Henry-Labordère, F. Penkner, Model-independent bounds for option prices — a mass transport approach, Finance Stoch. 17, 477–501, 2013. doi:10.1007/s00780-013-0205-8
A. Galichon, P. Henry-Labordère, N. Touzi, A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options, Ann. Appl. Probab. 24(1), 312–336, 2014. doi:10.1214/13-AAP925
M. Beiglböck, M. Goldstern, G. Maresch, W. Schachermayer, Optimal and better transport plans, J. Funct. Anal. 256(6), 1907–1927, 2009. doi:10.1016/j.jfa.2009.01.013

8 thms1 active userReviewed

Convex Optimization Dynamical Systems Functional Analysis+2·Captain: mikedeng1

Tikhonov Regularization of a Second Order Dynamical System with Hessian Driven Damping 1: For α > 3 and ∫ tε(t) dt < +∞ the Trajectory Converges Weakly to a Minimizer of gResearch Paper

Motivation

Minimizing a convex function $g$ on a real Hilbert space $\mathcal H$ by first-order methods has a continuous-time counterpart: second-order differential equations whose trajectories descend $g$ . Su, Boyd and Candès (20) showed that Nesterov's accelerated gradient method is a discretization of $\ddot x+\frac{\alpha}{t}\dot x+\nabla g(x)=0$ with $\alpha=3$ , along which $g(x(t))-\min g=O(1/t^2)$ . Two modifications of this equation have been studied separately because each adds a property the plain system lacks.

A Hessian-driven damping term $\beta\nabla^2g(x(t))\dot x(t)$ (Attouch, Peypouquet, Redont, J. Differential Equations 2016) damps oscillations and, for $\beta>0$ , makes $t^2\|\nabla g(x(t))\|^2$ integrable, while keeping fast values and weak convergence of trajectories.
A Tikhonov regularization term $\epsilon(t)x(t)$ with $\epsilon(t)\downarrow0$ (Attouch, Chbani, Riahi, J. Math. Anal. Appl. 2018) can steer trajectories to the minimum-norm minimizer.

Boţ, Csetnek and László (arXiv:1911.12845v2, 2020) combine both terms and ask which properties of the two parent systems survive. This mission covers the first of their answers: when $\epsilon$ decays fast enough, the fast rates and the weak convergence of trajectories are preserved.

Timeline. 2016 (published 2018): weak convergence of trajectories of $\ddot x+\frac\alpha t\dot x+\nabla g(x)=0$ for $\alpha>3$ (Attouch, Chbani, Peypouquet, Redont, Math. Program. 168). 2016: the same with Hessian-driven damping, $\beta>0$ (Attouch, Peypouquet, Redont). 2018: Tikhonov regularization of that system, with the $O(1/t^2)$ rate kept and strong convergence to the minimum-norm minimizer under slow decay of $\epsilon$ (Attouch, Chbani, Riahi). 2020: both terms together (Boţ, Csetnek, László, this paper).

Setting

Fix a real Hilbert space $\mathcal H$ , a starting time $t_0>0$ , parameters $\alpha\ge3$ and $\beta\ge0$ , and initial data $u_0,v_0\in\mathcal H$ . The General assumption of the paper is:

$g:\mathcal H\to\mathbb R$ is convex and twice Fréchet differentiable, $\nabla g$ is Lipschitz continuous on bounded sets, and $\operatorname{argmin} g\ne\emptyset$ ; $\min g$ denotes its minimal value;
$\epsilon:[t_0,+\infty)\to[0,+\infty)$ is nonincreasing, of class $C^1$ , and $\epsilon(t)\to0$ as $t\to+\infty$ .

The dynamical system is

\ddot x(t)+\frac{\alpha}{t}\dot x(t)+\beta\nabla^2g(x(t))\dot x(t)+\nabla g(x(t))+\epsilon(t)x(t)=0,\quad t\ge t_0,\quad x(t_0)=u_0,\ \dot x(t_0)=v_0. \tag{5}

A global $C^2$ -solution is a twice continuously differentiable $x:[t_0,+\infty)\to\mathcal H$ satisfying (5) for every $t\ge t_0$ . The decay of $\epsilon$ is controlled by $\int_{t_0}^{+\infty}t\epsilon(t)\,dt<+\infty$ together with one of two conditions:

(a) there exist $a>1$ and $t_1\ge t_0$ with $\dot\epsilon(t)\le-\frac{a\beta}{2}\epsilon^2(t)$ for all $t\ge t_1$ ;
(b) there exist $a>0$ and $t_1\ge t_0$ with $\epsilon(t)\le\frac at$ for all $t\ge t_1$ .

A curve $x(t)$ converges weakly to $\bar x$ if $\langle x(t),v\rangle\to\langle\bar x,v\rangle$ for every $v\in\mathcal H$ . The Lean development lives in the namespace TikhonovHDD.Weak, with IsSolution, GenAssumptionG, GenAssumptionEps, CondA, CondB, argmin, minVal and WeakTendstoAtTop for these objects.

Formalization targets

Goal: Theorem 3.5 (p. 15)

Under the General assumption, $\int_{t_0}^{+\infty}t\epsilon(t)\,dt<+\infty$ and (a) or (b), if $\alpha>3$ then every global $C^2$ -solution of (5) satisfies

\exists\,\bar x\in\operatorname{argmin} g:\qquad x(t)\rightharpoonup\bar x\quad(t\to+\infty).

Milestones

In the order the argument uses them:

Theorem 2.1 (p. 4): existence and uniqueness of a global $C^2$ -solution of (5).
Lemma A.2 (p. 29): a bounded-below, locally absolutely continuous $F$ with $\dot F\le G\in L^1$ a.e. has a finite limit.
Theorem 3.3 (p. 11), split three ways: for $\alpha\ge3$ , $g(x(t))-\min g=O(1/t^2)$ ; for $\alpha>3$ , $x$ is bounded and $t(g(x(t))-\min g)$ , $t\|\dot x\|^2$ , $t\epsilon\|x-x^*\|^2$ , $t\epsilon\|x\|^2\in L^1$ ; and separately $t^2\|\nabla g(x(t))\|^2\in L^1$ .
Lemma A.3 (p. 30): $u(t)+\frac t\alpha\dot u(t)\to\bar u$ with $\alpha>0$ implies $u(t)\to\bar u$ .
Theorem 3.4 (p. 12): for $\alpha>3$ and every $x^*\in\operatorname{argmin} g$ , $t\langle\nabla g(x),x-x^*\rangle\in L^1$ , $\lim\|x(t)-x^*\|$ and $\lim t\langle\dot x+\beta\nabla g(x),x-x^*\rangle$ exist, $g(x(t))-\min g=o(1/t^2)$ , $\|\dot x+\beta\nabla g(x)\|=o(1/t)$ , and $t^2\epsilon(t)\|x(t)\|^2\to0$ .

Significance

The result. Theorem 3.5 shows that the Tikhonov term, when it decays at least as fast as $\int t\epsilon<\infty$ requires, does not disturb the asymptotics of the Hessian-damped system: values converge at rate $o(1/t^2)$ and trajectories converge weakly to a minimizer. With $\beta=0$ it recovers the corresponding results for the regularized system without Hessian damping, and with $\epsilon\equiv0$ those for the Hessian-damped system. It is also the boundary case of the paper's second half, where slower decay of $\epsilon$ is shown to give strong convergence to the minimum-norm minimizer instead.

Formalizing it. The results are proved in the paper; none of them is machine-checked. A formal development would produce reusable pieces: a Lyapunov-energy analysis of second-order systems with time-dependent coefficients, the asymptotic Lemmas A.2 and A.3, and the bridge from "the distance to every minimizer converges and weak cluster points are minimizers" to weak convergence of a curve (the continuous Opial lemma).

Difficulty

The obvious approach, a single Lyapunov function that decreases along trajectories, fails here: the Tikhonov term makes the natural energy $\mathcal E_b$ only almost nonincreasing, with an error $\propto t\epsilon(t)\|x^*\|^2$ that must be integrated, and the Hessian term produces a cross term $-\beta\epsilon(t)t^2\langle\nabla g(x),x\rangle$ that is controlled only by condition (a) or (b). The step from $O(1/t^2)$ to $o(1/t^2)$ and to the existence of $\lim\|x(t)-x^*\|$ cannot be read off a single bounded energy: boundedness of $\mathcal E_b$ gives rates in $O$ , not in $o$ , and says nothing directly about $\|x(t)-x^*\|$ . Weak convergence then requires Opial's lemma, which is not in Mathlib in continuous form. Existence of solutions is itself nontrivial in infinite dimension: the right-hand side involves $\nabla^2g$ , so the system has to be rewritten as a first-order system before the Cauchy–Lipschitz theorem applies.

Formalization scope

$\mathcal H$ is any real Hilbert space (InnerProductSpace ℝ H, CompleteSpace H); nothing is finite-dimensional, so weak convergence is genuinely weaker than norm convergence. It is stated through inner products with every $v\in\mathcal H$ ; stating it as norm convergence, or testing only some $v$ , would change the theorem.
Trajectories are maps $\mathbb R\to\mathcal H$ with velocity and acceleration maps; derivatives are one-sided at $t_0$ and values before $t_0$ play no role. Every theorem is stated for every $C^2$ -solution; Theorem 2.1 shows this class is a singleton, so the hypothesis is not vacuous. A sorry-free check that the hypotheses of the goal can be met ( $g=0$ , $\epsilon=0$ , $x=0$ ) accompanies the drafts.
$\nabla^2g(x)v$ is the derivative of $\nabla g$ at $x$ applied to $v$ ; $\dot\epsilon$ is a derivative map tied to $\epsilon$ on $[t_0,+\infty)$ ; $\min g=\inf g$ , attained since $\operatorname{argmin} g\ne\emptyset$ ; $L^1$ membership is Lebesgue integrability on $[t_0,+\infty)$ (never a comparison of a Bochner integral with a number); $O$ , $o$ are Asymptotics relations at $+\infty$ .
Repair. Theorem 2.1 carries the extra hypothesis " $\beta=0$ or $g\in C^2$ ". As printed (only twice differentiable $g$ ) it is false for $\beta>0$ : on $\mathbb R$ , $g'(x)=2x+x^2\sin(1/x)$ gives a strongly convex $g$ with bounded but discontinuous $g''$ , and the solution from $u_0=0$ , $v_0=1$ is not $C^2$ . The paper's own proof uses continuity of $\nabla^2 g$ .
Flagged, not repaired. The last claim of Theorem 3.3, $t^2\|\nabla g(x(t))\|^2\in L^1$ , is its own item: the paper's argument for it uses a term proportional to $\beta$ , so for $\beta=0$ the paper gives no proof. No counterexample is known, and the statement is drafted as printed.
Referenced, not re-posed. Lemma A.4 (continuous Opial lemma, p. 30), the last step of the goal's proof, is the published InertialAVD.Traj.lemma_A_2 (same hypotheses: $S$ nonempty, $\lim\|x(t)-z\|$ exists for every $z\in S$ , every weak sequential limit point lies in $S$ ; same conclusion). It is a reference milestone of this mission; its weak convergence InertialAVD.Traj.WeakTendstoAtTop has the same body as TikhonovHDD.Weak.WeakTendstoAtTop.
The model is defined from scratch: the published HessianDamping.DINAVD.Setting concerns a different equation (no $\epsilon x$ term) and is not reused.
Welcome contributions: a general Cauchy–Lipschitz existence theorem for locally Lipschitz first-order systems in Banach spaces on $[t_0,+\infty)$ with a blow-up alternative, the continuous Opial lemma, and Lemma A.2 for absolutely continuous functions.

Selected references

R.I. Boţ, E.R. Csetnek, S.C. László, Tikhonov regularization of a second order dynamical system with Hessian driven damping, arXiv:1911.12845v2, 2020; published in Math. Program., https://doi.org/10.1007/s10107-020-01528-8. https://arxiv.org/abs/1911.12845v2
H. Attouch, J. Peypouquet, P. Redont, Fast convex optimization via inertial dynamics with Hessian driven damping, J. Differential Equations 261(10), 5734–5783, 2016. https://doi.org/10.1016/j.jde.2016.08.020
H. Attouch, Z. Chbani, H. Riahi, Combining fast inertial dynamics for convex optimization with Tikhonov regularization, J. Math. Anal. Appl. 457(2), 1065–1094, 2018. https://doi.org/10.1016/j.jmaa.2016.12.017
H. Attouch, Z. Chbani, J. Peypouquet, P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math. Program. 168, 123–175, 2018. https://doi.org/10.1007/s10107-016-0992-8
W. Su, S. Boyd, E.J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: theory and insights, J. Mach. Learn. Res. 17(153), 1–43, 2016. https://jmlr.org/papers/v17/15-084.html

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Convex Optimization Dynamical Systems Functional Analysis+2·Captain: mikedeng1

Tikhonov Regularization of a Second Order Dynamical System with Hessian Driven Damping 2: lim inf ‖x(t) − x‖ = 0 at the Minimum-Norm Minimizer, and x(t) → x If x(t) Ends In or Out of B(0, ‖x*‖)Research Paper

Motivation

Many methods for minimizing a smooth convex function $g$ on a Hilbert space can be read as discretizations of second order ordinary differential equations, and the asymptotic behaviour of the continuous trajectories predicts the behaviour of the iterates. The inertial system with vanishing damping $\ddot x+\frac{\alpha}{t}\dot x+\nabla g(x)=0$ is the continuous counterpart of Nesterov's accelerated gradient method (Su, Boyd, Candès 2016); for $\alpha>3$ its trajectories converge weakly to some minimizer of $g$ (Attouch, Chbani, Peypouquet, Redont 2018). Two modifications of this system have been studied separately. A Hessian driven damping term $\beta\nabla^2g(x)\dot x$ reduces oscillations while keeping the fast rate of the values (Attouch, Peypouquet, Redont 2016). A Tikhonov regularization term $\epsilon(t)x$ with $\epsilon(t)\to 0$ selects a particular minimizer: under suitable decay of $\epsilon$ , trajectories approach the minimizer of minimum norm in the strong (norm) topology (Attouch, Chbani, Riahi 2018).

Boţ, Csetnek and László (arXiv:1911.12845v2, published in Mathematical Programming 2021) combine both terms and ask which properties of the two separate systems survive. This mission formalizes their strong convergence result, Theorem 4.4.

Setting

Let $\mathcal H$ be a real Hilbert space, $t_0>0$ , $\alpha\ge 3$ , $\beta\ge0$ , and $u_0,v_0\in\mathcal H$ . The General assumption of the paper (p. 2) is:

$g:\mathcal H\to\mathbb R$ is convex and twice Fréchet differentiable, $\nabla g$ is Lipschitz continuous on bounded sets, and $\operatorname{argmin} g\neq\emptyset$ ;
$\epsilon:[t_0,+\infty)\to[0,+\infty)$ is nonincreasing, of class $C^1$ , and $\lim_{t\to+\infty}\epsilon(t)=0$ .

A global $C^2$ -solution of the system

\ddot x(t)+\frac{\alpha}{t}\dot x(t)+\beta\nabla^2g(x(t))\dot x(t)+\nabla g(x(t))+\epsilon(t)x(t)=0,\quad t\ge t_0,\qquad x(t_0)=u_0,\ \dot x(t_0)=v_0, \tag{5}

is a twice continuously differentiable curve $x:[t_0,+\infty)\to\mathcal H$ satisfying (5) at every $t\ge t_0$ . The set $\operatorname{argmin} g$ is nonempty, closed and convex, so it has a unique element of minimum norm, denoted $x^*$ . The open ball $B(0,\|x^*\|)$ is $\{y:\|y\|<\|x^*\|\}$ . In the Lean development these objects are GeneralAssumption g t₀ ε ε', IsSolution g α β t₀ ε u₀ v₀ x xd xdd and IsMinNormMinimizer g xstar in the namespace TikhonovHDD.Strong, with $\min g$ written minValue g.

Formalization targets

Goal: Theorem 4.4 (p. 21)

Assume, in addition to the above,

\int_{t_0}^{+\infty}\frac{\epsilon(t)}{t}\,dt<+\infty,\qquad \lim_{t\to+\infty}\frac{\beta}{\epsilon(t)t^{\frac{\alpha}{3}+1}}\int_{t_0}^{t}\epsilon^2(s)s^{\frac{\alpha}{3}+1}\,ds=0,

that $\dot\epsilon(t)\le-\frac{a\beta}{2}\epsilon^2(t)$ for all $t\ge t_1$ , for some $a>1$ and $t_1\ge t_0$ , and that $\lim_{t\to+\infty}t^2\epsilon(t)=+\infty$ if $\alpha=3$ , while $t^2\epsilon(t)\ge\frac23\alpha\left(\frac13\alpha-1+\beta c^2\right)$ for large $t$ , for some $c>0$ , if $\alpha>3$ . Then

\liminf_{t\to+\infty}\|x(t)-x^*\|=0,

and

\lim_{t\to+\infty}\|x(t)-x^*\|=0

if there exists $T\ge t_0$ such that $\{x(t):t\ge T\}$ stays either in $B(0,\|x^*\|)$ or in its complement.

Milestones

Lemma A.1 (p. 29): for nonnegative continuous integrable $f$ on $(\delta,+\infty)$ and nondecreasing $\varphi\ge 0$ with $\varphi(t)\to+\infty$ , $\frac1{\varphi(t)}\int_\delta^t\varphi(s)f(s)\,ds\to0$ .
Theorem 3.1 (pp. 5–6): for $\alpha\ge3$ , $g(x(t))\to\min g$ under either (a) $\int_{t_0}^{+\infty}\epsilon(t)/t\,dt<+\infty$ together with $\dot\epsilon\le-\frac{a\beta}{2}\epsilon^2$ eventually, or (b) $\epsilon(t)\le a/t$ eventually.
Case I of the proof of Theorem 4.4 (pp. 21–25): the trajectory eventually stays outside the open ball, and $x(t)\to x^*$ .
Case II (pp. 25–26): the trajectory eventually stays inside the open ball, and $x(t)\to x^*$ .
Case III (p. 26): the trajectory is inside and outside the ball for arbitrarily large times, and $\liminf\|x(t)-x^*\|=0$ .

The three cases are exhaustive, so Cases I–III together give the goal.

Significance

Theorem 4.4 shows that adding Hessian driven damping does not destroy the selection property of Tikhonov regularization: the trajectory comes arbitrarily close to the minimum-norm minimizer, and converges to it in norm whenever it does not oscillate across the sphere $\|y\|=\|x^*\|$ forever. Since $\epsilon(t)\to 0$ , the system still approaches $\operatorname{argmin} g$ , but the limit is a canonical minimizer determined by $g$ alone, not by the initial data. In infinite dimensions this is the difference between weak and strong convergence, a distinction that matters for ill-posed problems and inverse problems, where the minimum-norm solution is the one usually sought. Remark 4.5 of the paper compares the hypotheses with those of Attouch, Chbani and Riahi for the system without Hessian damping: for $\beta=0$ the lower bound required of $t^2\epsilon(t)$ when $\alpha>3$ is, in the authors' words, "less tight" than theirs.

The result is proved in the paper; no part of it has a machine-checked proof. Formalizing it requires a Lyapunov-energy argument for a second order ODE in a Hilbert space, the weak lower semicontinuity of convex functions and of the norm, and the passage from weak to strong convergence via convergence of norms. Theorem 3.1 and Lemma A.1 are standalone results reusable for other Tikhonov-regularized inertial systems.

Difficulty

The weak convergence argument for the unregularized system (Opial's lemma) does not identify the limit, and it does not give strong convergence. Tikhonov regularization supplies the identification only through the Tikhonov curve $x_{\epsilon}$ , the minimizer of $g+\frac{\epsilon}{2}\|\cdot\|^2$ , and comparing $x(t)$ with $x_{\epsilon(t)}$ requires an energy whose derivative is controlled with weights tuned to $\alpha$ . The Hessian term produces cross terms of the form $\beta\epsilon(t)t^2\langle\nabla g(x(t)),x(t)\rangle$ that are absent from the system without Hessian damping; the integral hypothesis on $\epsilon^2(s)s^{\alpha/3+1}$ and the slow-decay condition $\dot\epsilon\le-\frac{a\beta}{2}\epsilon^2$ exist to absorb them. A direct energy estimate only works when $\|x(t)\|\ge\|x^*\|$ ; inside the ball a different argument is required, and when the trajectory crosses the sphere infinitely often neither argument yields convergence of the whole trajectory, only a liminf statement.

Formalization scope

Conventions committed to in Lean:

Curves are maps $\mathbb R\to\mathcal H$ ; only their values on $[t_0,+\infty)$ matter. Derivatives of $x$ , $\dot x$ and $\epsilon$ are taken within $[t_0,+\infty)$ (one-sided at $t_0$ ) and passed as explicit maps xd, xdd, ε'; $\ddot x$ and $\dot\epsilon$ are continuous there.
$\nabla g$ is gradient g; $\nabla^2g(x)v$ is the Fréchet derivative of gradient g at $x$ applied to $v$ . "Lipschitz on bounded sets" means Lipschitz on every closed ball centred at $0$ .
$\min g$ is the infimum of the range of $g$ , attained because $\operatorname{argmin} g\neq\emptyset$ .
$\int_{t_0}^{+\infty}\epsilon(t)/t\,dt<+\infty$ is Lebesgue integrability of $\epsilon(t)/t$ on $[t_0,+\infty)$ , never a comparison of a Bochner integral with a number. $t^{\alpha/3+1}$ is the real power.
$\liminf_{t\to+\infty}\|x(t)-x^*\|=0$ is written as: for every $\delta>0$ , $\|x(t)-x^*\|<\delta$ for arbitrarily large $t$ . Lean's Filter.liminf is not used, since it returns $0$ for a function tending to $+\infty$ .
The alternative "in the ball or in its complement" uses one $T$ for both branches, with the disjunction outside the universal quantifier. Placing the disjunction inside ( $\forall t\ge T$ , $\|x(t)\|<\|x^*\|$ or $\|x(t)\|\ge\|x^*\|$ ) would make the hypothesis always true and the strong convergence claim unconditional, which the paper does not prove.
Every statement is made for every global $C^2$ -solution of (5). Existence and uniqueness of that solution (Theorem 2.1 of the paper) is a separate result, formalized in the companion mission on weak convergence; the hypothesis is not vacuous.
No positivity of $\epsilon$ is assumed: it follows from the conditions for $\alpha=3$ and $\alpha>3$ .
Cases I–III carry the full hypothesis list of Theorem 4.4, although Case II uses only part of it.

No statement of the mission has been repaired against the printed text.

A complete development needs: weak lower semicontinuity of convex continuous functions and of the norm on Hilbert spaces; weak sequential compactness of bounded sets; existence of the minimum-norm element of a closed convex set (Mathlib's projection onto closed convex sets); the strongly convex Tikhonov curve and its convergence to $x^*$ ; and calculus for energy functionals along $C^2$ curves on half-lines. Lemma A.1 and the Tikhonov-curve facts are reusable for every Tikhonov-regularized dynamics. Contributions of proofs of individual milestones, of supporting lemmas, and of alternative arguments are welcome.

Selected references

R.I. Boţ, E.R. Csetnek, S.C. László, Tikhonov regularization of a second order dynamical system with Hessian driven damping, arXiv:1911.12845v2, 2020; Mathematical Programming 189, 151–186, 2021. https://arxiv.org/abs/1911.12845v2, https://doi.org/10.1007/s10107-020-01528-8
H. Attouch, Z. Chbani, H. Riahi, Combining fast inertial dynamics for convex optimization with Tikhonov regularization, Journal of Mathematical Analysis and Applications 457, 1065–1094, 2018. https://doi.org/10.1016/j.jmaa.2016.12.017
H. Attouch, J. Peypouquet, P. Redont, Fast convex optimization via inertial dynamics with Hessian driven damping, Journal of Differential Equations 261(10), 5734–5783, 2016. https://doi.org/10.1016/j.jde.2016.08.020
H. Attouch, Z. Chbani, J. Peypouquet, P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Mathematical Programming 168, 123–175, 2018. https://doi.org/10.1007/s10107-016-0992-8
W. Su, S. Boyd, E.J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: theory and insights, Journal of Machine Learning Research 17(153), 1–43, 2016. https://jmlr.org/papers/v17/15-084.html

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Operations Research Optimization Probability·Captain: mikedeng1

Approximation Algorithms for Product Framing and Pricing 2: With Type-Dependent Choice and IFR Page Views, the TRUNC Algorithm Earns at Least 1/3 of the Optimal RevenueResearch Paper

Motivation

Online retailers, search engines and streaming services show products on a sequence of pages, and most consumers look at only the first few. Which products are seen therefore depends on where they are placed. Gallego, Li, Truong and Wang (Operations Research, 2020) call the problem of placing products on pages to maximize expected revenue product framing. They show that it is NP-hard even with two pages and the multinomial logit model, and they give simple algorithms with constant-factor guarantees.

This mission concerns their second guarantee. In their basic model every consumer chooses with the same choice model, whatever the number of pages she views. In practice the number of pages viewed is itself informative: consumers who browse many pages may be keener buyers. Section 6 of the paper lets the choice model depend on the number of pages viewed and proposes the TRUNC algorithm for this setting. The guarantee is that TRUNC earns at least one third of the optimal expected revenue.

Setting

There are $n$ products with revenues $r_i\ge 0$ and $m$ virtual pages, each holding at most $p$ products; write $[k]=\{1,\dots,k\}$ . A framing places each product on at most one page, with at most $p$ products per page. A consumer views a random number $X\in[m]$ of pages, with law $\lambda(x)=\mathbb P[X=x]$ , tail $\Lambda(x)=\mathbb P[X\ge x]$ and failure rate $h(x)=\lambda(x)/\Lambda(x)$ . Her consideration set is the set of products on pages $1,\dots,X$ .

A consumer who views $x$ pages is of type $x$ and buys product $i$ from consideration set $S$ with probability $P_x(i,S)$ , where $P_x(i,S)\ge 0$ , $P_x(i,S)=0$ for $i\notin S$ , and $\sum_{i\in S}P_x(i,S)\le 1$ . The revenue of $S$ for type $x$ and the best capacitated revenue are

R_x(S)=\sum_{i\in S}r_iP_x(i,S),\qquad U(x)=\max_{S\subseteq[n],\ |S|\le xp}R_x(S).

A framing $f$ earns $\sum_{x\in[m]}\lambda(x)R_x(C_f(x))$ , where $C_f(x)$ is the set of products on pages $1,\dots,x$ . Its maximum over all framings is $V^{OPT}$ . The assumptions are:

B1: $R_x(S)\le R_y(S)$ whenever $x\le y$ and $|S|\le xp$ ;
B2: $U(x)/x$ is nonincreasing in $x$ ;
B3: problem (9) defining $U$ can be solved, here exactly;
B4: $X$ has an increasing failure rate (IFR), i.e. $h$ is nondecreasing (Shaked and Shanthikumar 2007).

TRUNC( $y$ ), for $y\in[m]$ , picks an optimal assortment $S(y)$ for $U(y)$ and places it, in any arrangement, on pages $1,\dots,y$ , leaving the remaining pages blank. TRUNC takes the best of these $m$ framings; its revenue is $V^{TRUNC}$ .

The analysis passes through the bound-revealing program (10): minimize $\max_{x\in[m]}U(x)\Lambda(x)$ over nondecreasing $U\ge 0$ with $U(x)/x$ nonincreasing and over tails $1=\Lambda(1)\ge\cdots\ge\Lambda(m)\ge 0$ with $\Lambda(x+1)\Lambda(x-1)\le\Lambda(x)^2$ , subject to $\sum_x\lambda(x)U(x)=1$ .

Formalization targets

Goal: Theorem 4

Under Assumptions B1–B4, for every choice of optimal assortments and every arrangement on the pages,

V^{TRUNC}\;\ge\;\frac13\,V^{OPT}.

Milestones

In the order the proof uses them:

Upper bound (§6, p. 13): $V^{OPT}\le\mathbf E[U(X)]=\sum_x\lambda(x)U(x)$ .
Lower bound (§6.1, p. 14): every run of TRUNC( $y$ ) earns at least $U(y)\Lambda(y)$ .
Lemma 8: for an IFR tail with $\Lambda>0$ , $g(x)=x\Lambda(x)$ is weakly unimodal, and its largest maximizer is $\min\{x: h(x)>1/(x+1)\}$ .
Proposition 5, Lemma 9, Proposition 6, Proposition 7: the structure of every optimal solution of (10) with $\Lambda>0$ $Λ > 0$ . With $y$ $y$ the largest maximizer of $U(x)\Lambda(x)$ $U (x) Λ (x)$ :
- $U(x)/x$ is constant for $x\ge y$ ;
- $y$ also maximizes $x\Lambda(x)$ ;
- $U(x)\Lambda(x)$ is constant for $x\le y$ ;
- $h$ is constant on $[y,m-1]$ .
Value of (10) (A.4, p. 42): every feasible point of (10) has $\max_xU(x)\Lambda(x)\ge 1/3$ .

Significance

The theorem shows that framing under type-dependent choice admits a constant-factor approximation using only $m$ calls to an assortment oracle, although the framing problem itself is NP-hard. The constant does not depend on the number of products, pages or page capacity. The bound-revealing program (10) is a self-contained extremal problem about log-concave tails and is reusable wherever a revenue curve with decreasing average returns meets an IFR horizon. The numerical value of (10) decreases with $m$ ( $2/3$ at $m=2$ , about $0.36$ at $m=40$ ), so the constant $1/3$ is approached only as the number of pages grows.

The result is proved in the paper. No part of it has a machine-checked proof: the platform has no item on product framing, discrete failure rates or program (10). A complete development would give a machine-checked proof of the approximation ratio. It would also give a checked treatment of the perturbation arguments of Appendix A.4, which the paper presents briefly.

Difficulty

Two steps are elementary: the upper bound (a consumer of type $x$ never sees more than $xp$ products) and the lower bound (every consumer of type $x\ge y$ sees all of $S(y)$ ).

The difficulty lies in the extremal problem (10). It is not convex: the IFR constraint is a log-concavity condition, and the objective is a maximum of products $U(x)\Lambda(x)$ . A direct bound such as $\max_xU(x)\Lambda(x)\ge\sum_x\lambda(x)U(x)/m$ gives a constant that decays with $m$ . The uniform constant comes from showing that the extremal tail is geometric beyond the maximizing page and that the extremal $U$ is pinned down on both sides of it. The paper reaches this structure through optimality conditions, so a formal proof must also establish that (10) attains its minimum, or must avoid that step.

Formalization scope

Products are Fin n. Pages are the natural numbers $1,\dots,m$ , as in the paper, with $m\ge 1$ and $p\ge 1$ . A framing is f : Fin n → ℕ: product $i$ is on page f i when $1\le f(i)\le m$ , and f i = 0 means it is not displayed. The law $\lambda$ and the functions of (10) are functions on $\mathbb N$ of which only the values on $[m]$ are used, and $\Lambda(m+1)=0$ .

$V^{OPT}$ and $U$ are maxima over finite nonempty families, and $V^{TRUNC}$ is a Finset.sup' over $y\in[m]$ . TRUNC is a predicate on its outputs, not a function, and the goal is quantified over every family of runs. Assumption B3 is used with $\epsilon=0$ , as everywhere in the paper. IFR is required only where the failure rate is defined ( $\Lambda>0$ ), so laws whose support ends before page $m$ are allowed. Nonnegative revenues $r_i\ge 0$ are assumed; the lower bound of milestone 2 needs them.

No trivializing reading. The goal is not stated for one particular filling heuristic, or for some run only, and $V^{TRUNC}$ is the revenue of the framings themselves, not the lower bound $U(y)\Lambda(y)$ .

Page numbers refer to the authors' accepted manuscript of the Operations Research article, which differ from the journal typesetting. The development needs only Mathlib (finite sums and maxima over Finset). Program (10) and Lemma 8 are independent of the framing model. Contributions are welcome at every milestone: Lemma 8 and the bound for (10) are the most self-contained.

Selected references

G. Gallego, A. Li, V.-A. Truong, X. Wang, Approximation Algorithms for Product Framing and Pricing, Operations Research, 2020. https://doi.org/10.1287/opre.2019.1875
M. Shaked, J. G. Shanthikumar, Stochastic Orders, Springer, 2007. https://doi.org/10.1007/978-0-387-34675-5

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Convex Optimization Operations Research Optimization+1·Captain: mikedeng1

Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization 1: The Perspective Interval Relaxation Is a Least Squares Problem with a Separable Reverse-Huber or ℓ1 PenaltyResearch Paper

Motivation

Best subset selection with ridge shrinkage asks for a sparse linear model: given a design matrix $X\in\mathbb R^{n\times p}$ and responses $y\in\mathbb R^n$ , minimize

\tfrac12\|y-X\beta\|_2^2+\lambda_0\|\beta\|_0+\lambda_2\|\beta\|_2^2 ,

where $\|\beta\|_0$ counts nonzero coefficients. The problem is NP-hard, and exact methods solve it as a mixed integer program by branch-and-bound (Bertsimas, King, Mazumder 2016). The speed of branch-and-bound is governed by two things: how tight the continuous relaxation at each node is, and how cheaply that relaxation can be solved.

Hazimeh, Mazumder and Saab (arXiv:2004.06152v2; Mathematical Programming 2022) build a branch-and-bound solver whose node relaxations are solved by first-order methods rather than by a conic interior-point solver. Their starting point is the perspective formulation, a standard device for strengthening relaxations of mixed integer programs with indicator variables (Frangioni, Gentile 2006; Günlük, Linderoth 2010). Theorem 1 of the paper shows that the interval relaxation of this formulation is a box-constrained least squares problem with a separable penalty in $\beta$ alone. This mission formalizes Theorem 1.

Timeline. Owen (2007) introduced the reverse Huber penalty as a hybrid of lasso and ridge. Dong, Chen, Linderoth (2015) showed that the interval relaxation of the perspective formulation without a Big-M bound reduces to least squares with a reverse Huber penalty. Hazimeh, Mazumder and Saab (2020–2021) added the Big-M constraint and obtained Theorem 1, with a second regime in which the penalty becomes $\ell_1$ .

Setting

Fix $X\in\mathbb R^{n\times p}$ , $y\in\mathbb R^n$ , $\lambda_0>0$ , $\lambda_2>0$ and $M>0$ , and write $[p]=\{1,\dots,p\}$ . The perspective formulation $\mathrm{PR}(M)$ is problem (3) of the paper:

\min_{\beta,z,s}\ \tfrac12\|y-X\beta\|_2^2+\lambda_0\sum_{i\in[p]}z_i+\lambda_2\sum_{i\in[p]}s_i \quad\text{s.t.}\quad \beta_i^2\le s_iz_i,\ \ -Mz_i\le\beta_i\le Mz_i,\ \ z_i\in\{0,1\},\ \ s_i\ge0 .

Here $z_i$ indicates whether $\beta_i$ may be nonzero, $s_i$ stands in for $\beta_i^2$ through the rotated second-order cone constraint $\beta_i^2\le s_iz_i$ , and $M$ bounds $|\beta_i|$ . The interval relaxation replaces $z_i\in\{0,1\}$ with $z_i\in[0,1]$ .

The reverse Huber penalty (4) is $\mathcal B(t)=|t|$ for $|t|\le1$ and $\mathcal B(t)=(t^2+1)/2$ for $|t|\ge1$ . Put

\psi_1(b;\lambda_0,\lambda_2)=2\lambda_0\,\mathcal B\big(b\sqrt{\lambda_2/\lambda_0}\big),\qquad \psi_2(b;\lambda_0,\lambda_2,M)=\Big(\frac{\lambda_0}{M}+\lambda_2M\Big)|b|,

and let $\psi=\psi_1$ if $\sqrt{\lambda_0/\lambda_2}\le M$ and $\psi=\psi_2$ if $\sqrt{\lambda_0/\lambda_2}>M$ . The reduced relaxation (5) is

\min_{\beta\in\mathbb R^p}\ F(\beta):=\tfrac12\|y-X\beta\|_2^2+\sum_{i\in[p]}\psi(\beta_i;\lambda_0,\lambda_2,M)\quad\text{s.t.}\quad\|\beta\|_\infty\le M ,

with optimal value $V_{\mathrm{PR}(M)}$ . For one coordinate $b$ , $\omega(b;\lambda_0,\lambda_2,M)$ is the minimum of $\lambda_0z+\lambda_2s$ over $b^2\le sz$ , $-Mz\le b\le Mz$ , $z\in[0,1]$ , $s\ge0$ .

Formalization targets

Goal: Theorem 1 (Reduced Relaxation), p. 6

The interval relaxation of (3) is equivalent to (5):

\exists\,(z,s):\ (\beta,z,s)\ \text{feasible}\iff\|\beta\|_\infty\le M,\qquad \min_{z,s}\Big\{\tfrac12\|y-X\beta\|_2^2+\lambda_0\textstyle\sum_i z_i+\lambda_2\sum_i s_i\Big\}=F(\beta)\ \ (\|\beta\|_\infty\le M),

and both problems attain the common optimal value $V_{\mathrm{PR}(M)}$ .

Milestones (proof of Theorem 1, p. 28)

Reduction to (36). For $|b|\le M$ , the smallest feasible $z$ is $\hat z=\max\{b^2/s,|b|/M\}$ , so

\omega(b;\lambda_0,\lambda_2,M)=\min_{s\ge b^2}\max\Big\{\lambda_0\tfrac{b^2}{s}+\lambda_2s,\ \lambda_0\tfrac{|b|}{M}+\lambda_2s\Big\}.

Case I. If $\sqrt{\lambda_0/\lambda_2}\le M$ and $|b|\le M$ , then $\omega(b;\lambda_0,\lambda_2,M)=2\lambda_0\mathcal B(b\sqrt{\lambda_2/\lambda_0})$ .
Case II. If $\sqrt{\lambda_0/\lambda_2}>M$ and $|b|\le M$ , then $\omega(b;\lambda_0,\lambda_2,M)=(\lambda_0/M+\lambda_2M)|b|$ .

Significance

The result. Theorem 1 removes the $2p$ auxiliary variables $z,s$ and all conic constraints from the node relaxation. What remains is a composite problem, a smooth least squares loss plus a separable, convex penalty under a box constraint, to which coordinate descent and proximal methods apply directly. The paper's solver, its active-set strategy, and the dual bounds of its Section 3 are all built on (5). The theorem also exposes a regime change: when $\sqrt{\lambda_0/\lambda_2}>M$ , the ridge term of the perspective formulation disappears from the relaxation and the penalty is pure $\ell_1$ . Propositions 1 and 2 of the paper, which compare the strengths of the perspective, Big-M and $\mathrm{PR}(\infty)$ relaxations, use this reduced form.

Formalizing it. The result is proved in the paper (Appendix A) by a coordinatewise argument. To our knowledge it has no machine-checked proof. Formalizing it gives a verified bridge from the conic mixed integer formulation to the penalized form that later results of the paper use, and a checked case analysis at the regime boundary $\sqrt{\lambda_0/\lambda_2}=M$ , where the two formulas must agree with the stated tie-breaking.

Difficulty

The obvious approach, "minimize over $z$ and $s$ coordinate by coordinate", is correct in outline, but each step hides a boundary case. The rotated cone constraint $\beta_i^2\le s_iz_i$ degenerates at $s_i=0$ or $z_i=0$ , where $\beta_i^2/s_i$ is undefined, and the paper's convention for $\beta_i=s_i=0$ must be handled. The one-variable problem (36) is a minimum of a pointwise maximum of two functions of $s$ , one convex and decreasing then increasing, the other increasing. Its minimizer lies either in the interior of the region where the first term dominates or on the crossover $s=|b|M$ , depending on the regime. The paper's proof describes the cases loosely: it optimizes the Case I term without imposing the side condition that Case I attains the maximum. A complete argument must prove the lower bound over all feasible $s$ , not just evaluate at a candidate. Finally, the minimum over $(z,s)\in\mathbb R^{2p}$ must be shown to split into a sum of coordinate minima, each attained.

Formalization scope

Data are X : Matrix (Fin n) (Fin p) ℝ, y : Fin n → ℝ, and reals lam0 lam2 M with 0 < lam0, 0 < lam2, 0 < M; the paper's $[p]$ is Fin p. The squared norm $\|y-X\beta\|_2^2$ is the explicit sum $\sum_r(y_r-(X\beta)_r)^2$ , and $\|\beta\|_\infty\le M$ is $|\beta_i|\le M$ for all $i$ . The cone constraint is kept in product form $\beta_i^2\le s_iz_i$ . In $\hat z$ and (36), Lean's $b^2/0=0$ coincides with the paper's convention $\beta_i^2/s_i=0$ for $\beta_i=s_i=0$ . Every "min" is stated with IsLeast (membership plus lower bound), so attainment is part of each statement. $V_{\mathrm{PR}(M)}$ is defined as a real infimum over the box, and Theorem 1 asserts that this infimum is attained by both problems. No normalization of $X$ or $y$ is assumed; the paper's unit-norm assumption starts in Section 3. The paper writes no $O(\cdot)$ in these results, so there are no constants to instantiate.

A goal stating only one inequality (that $F(\beta)$ is achieved by some $(z,s)$ , or that it bounds the relaxation from below), or a goal over $z_i\in\{0,1\}$ , would not be Theorem 1. The IsLeast form rules out both. The goal uses $\psi$ with both regimes.

A complete development needs: a two-variable convexity argument for $\lambda_0b^2/s+\lambda_2s$ on $s>0$ ; a lemma that the infimum of a separable objective over a product of feasible sets is the sum of the coordinate infima; and continuity of $F$ with compactness of the box to obtain an optimal $\beta$ . The coordinate lemmas and the definition of the reverse Huber penalty are reusable in the companion missions on relaxation strength and Lagrangian duality. Contributions to any milestone, or an alternative proof of the goal, are welcome.

Selected references

H. Hazimeh, R. Mazumder, A. Saab, Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization, arXiv:2004.06152v2 (2021); Mathematical Programming (2022). https://arxiv.org/abs/2004.06152v2
A. Frangioni, C. Gentile, Perspective cuts for a class of convex 0–1 mixed integer programs, Mathematical Programming 106(2), 225–236, 2006. https://doi.org/10.1007/s10107-005-0594-3
O. Günlük, J. Linderoth, Perspective reformulations of mixed integer nonlinear programs with indicator variables, Mathematical Programming 124(1–2), 183–205, 2010. https://doi.org/10.1007/s10107-010-0360-z
H. Dong, K. Chen, J. Linderoth, Regularization vs. Relaxation: A conic optimization perspective of statistical variable selection, arXiv:1510.06083, 2015. https://arxiv.org/abs/1510.06083
A. B. Owen, A robust hybrid of lasso and ridge regression, Contemporary Mathematics 443, 59–72, 2007. https://doi.org/10.1090/conm/443/08555
D. Bertsimas, A. King, R. Mazumder, Best subset selection via a modern optimization lens, Annals of Statistics 44(2), 813–852, 2016. https://doi.org/10.1214/15-AOS1388

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Algorithmic Game Theory Control Theory Probability·Captain: mikedeng1

A Probabilistic Weak Formulation of Mean Field Games and Applications 1: When the Hamiltonian's Maximizer Sets Are Convex, the Mean Field Game in Weak Formulation Has a SolutionResearch Paper

Motivation

Mean field games (MFGs) describe Nash equilibria of stochastic differential games with a very large number of symmetric players, each of whom interacts with the others only through the empirical distribution of their states (and, in this paper, of their controls). They were introduced independently by Lasry and Lions and by Huang, Malhamé and Caines around 2006, and are used for models of systemic risk, price impact in optimal execution, flocking and crowd motion, and large-population economics. The basic mathematical question is existence of an equilibrium: a flow of measures that is reproduced by the optimal behaviour of a representative player who takes that flow as given.

Two families of methods dominate the literature. The analytic approach of Lasry and Lions solves a coupled Hamilton–Jacobi–Bellman / Fokker–Planck system. The probabilistic approach of Carmona and Delarue (arXiv:1210.5780) solves a forward–backward SDE of McKean–Vlasov type in the strong formulation, on a fixed probability space with the controlled state as a strong solution. Both require the coefficients to be regular (Lipschitz or continuous) in the state variable. Carmona and Lacker (arXiv:1307.1152, Ann. Appl. Probab. 2015) introduce a weak formulation, in which controls act through a Girsanov change of measure on a fixed driftless state process. This lets the drift, running reward and terminal reward be merely measurable and path-dependent in the state. The interaction can enter through the law of the state path and through the law of the control. This mission formalizes their existence theorem.

Setting

Fix a dimension $d$ , a horizon $T>0$ and the path space $\mathcal C=C([0,T];\mathbb R^d)$ with the sup norm. A measurable $\psi:\mathcal C\to[1,\infty)$ fixes the admissible population laws, $\mathcal P_\psi(\mathcal C)=\{\mu:\int\psi\,d\mu<\infty\}$ . The topology $\tau_\psi(\mathcal C)$ is the weakest one making $\mu\mapsto\int f\,d\mu$ continuous for every measurable $f$ with $|f|\le c\,\psi$ . A compact convex control set $A$ sits in a normed space, and $\mathcal P(A)$ carries the weak topology.

A probability space carries an initial state $\xi$ and an independent Brownian motion $W$ , with filtration $\mathbb F$ generated by $(\xi,W)$ . Admissible controls $\mathbb A$ are the progressively measurable $A$ -valued processes. The driftless state $X$ solves $dX_t=\sigma(t,X)\,dW_t$ , $X_0=\xi$ , and its law is $\mathcal X=P\circ X^{-1}$ . For a population law $\mu$ and a control $\alpha$ , the measure $P^{\mu,\alpha}$ has density $\mathcal E\big(\int_0^\cdot\sigma^{-1}b(t,X,\mu,\alpha_t)\,dW_t\big)_T$ with respect to $P$ . Under $P^{\mu,\alpha}$ , $X$ is a weak solution of $dX_t=b(t,X,\mu,\alpha_t)\,dt+\sigma(t,X)\,dW^{\mu,\alpha}_t$ . Given also a measurable flow $q:[0,T]\to\mathcal P(A)$ of control laws, the reward is

J^{\mu,q}(\alpha)=\mathbb E^{\mu,\alpha}\Big[\int_0^Tf(t,X,\mu,q_t,\alpha_t)\,dt+g(X,\mu)\Big],\qquad V^{\mu,q}=\sup_{\alpha\in\mathbb A}J^{\mu,q}(\alpha).

The standing assumptions (S) ask for:

measurability of the coefficients;
continuity in the control;
a unique strong solution $X$ with $\mathbb E[\psi^2(X)]<\infty$ ;
$\sigma(t,X)$ nonsingular, with $\sigma^{-1}b$ bounded;
growth of $f$ and $g$ controlled by $\psi$ ;
a separated running reward $f=f_1(t,x,\mu,a)+f_2(t,x,\mu,q)$ .

The Hamiltonian is $h=f+z\cdot\sigma^{-1}b$ . Its maximum over $A$ is $H$ , and the set of maximizers is $A(t,x,\mu,z)$ .

Formalization targets

Goal: Theorem 3.5

Assume (S), (E) (for each $(t,x)$ , the maps $b$ , $f$ , $g$ are sequentially continuous in the measure and control arguments, on the laws $\mu\sim\mathcal X$ with $\tau_\psi$ ) and (C) (each $A(t,x,\mu,z)$ is convex). Then there exist $\mu\in\mathcal P_\psi(\mathcal C)$ , a measurable $q:[0,T]\to\mathcal P(A)$ and $\alpha\in\mathbb A$ with

V^{\mu,q}=J^{\mu,q}(\alpha),\qquad P^{\mu,\alpha}\circ X^{-1}=\mu,\qquad P^{\mu,\alpha}\circ\alpha_t^{-1}=q_t\ \text{ for a.e. } t.

This is Definition 3.4 of a solution of the MFG. No constants or rates appear; the goal asserts existence only.

Milestones

The milestones are the statements the paper's proof (§7) names, in attack order:

the BSDE characterization of optimal controls, (7.1)–(7.4) and Remark 7.2;
Lemma 7.6 (densities of image measures);
Lemma 7.7 (uniform moment bounds on Girsanov densities);
Proposition 7.8 (compactness and metrizability of the set $\mathcal Q$ of candidate laws);
Berge's maximum theorem (Theorem 7.5);
Lemma 7.9 (continuity of the Hamiltonian on $\mathcal Q$ );
Lemma 7.10 (stability of the BSDE adjoint process);
Lemma 7.11 (upper hemicontinuity of the optimal-control correspondence);
the set-valued fixed point theorem, Proposition 7.4.

Significance

The theorem yields equilibria for interactions that are discontinuous in the state and nonlocal in time. Examples are rank-based rewards, path-dependent payoffs, and dependence on the law of the controls ("extended" MFGs, needed for price impact models). Strong-formulation methods do not cover these. It is also the input to the companion results: uniqueness under Lasry–Lions monotonicity (Theorem 3.8) and the construction of approximate Nash equilibria for the $n$ -player game (Theorem 4.2), formalized as missions 2 and 3 of this series.

The result is proved in the paper. What this mission adds is a machine-checked development of the weak formulation:

Girsanov densities with uniform moment bounds;
the topology $\tau_\psi$ and the compactness of $\mathcal Q$ ;
BSDE comparison for control problems;
upper hemicontinuity of argmax correspondences;
a Kakutani-type fixed point theorem in locally convex spaces.

None of these is in Mathlib, and no mean field game existence theorem in the weak formulation has been formalized. Formalization also checks the proof. The fixed point theorem as printed (Proposition 7.4, with closed rather than compact values) is false, and the milestone states the corrected version. Connecting it to Lemma 7.11, whose values are closed but not compact in the norm $\mathbb E\int_0^T\|\alpha_t\|\,dt$ , is part of the work.

Difficulty

The fixed point is set-valued. Without uniqueness of maximizers, the best response to $(\mu,\nu)$ is a set of controls. Kakutani's theorem in finite dimensions does not apply: the laws live in $\mathcal P_\psi(\mathcal C)$ , which with $\tau_\psi$ is neither metrizable nor separable.

The obvious argument would apply Schauder's theorem to $\mu\mapsto P^{\hat\alpha}\circ X^{-1}$ for a measurable selection $\hat\alpha$ of the maximizers. It fails twice. A selection need not depend continuously on $\mu$ . And the adjoint process $Z^{\mu,\nu}$ , which enters the maximizer, depends on $(\mu,\nu)$ only through a BSDE, with no pointwise regularity.

Compactness has to come from somewhere other than the coefficients. All laws of the form $P^{\mu,\alpha}\circ X^{-1}$ lie in a set $\mathcal Q$ of laws with bounded densities relative to $\mathcal X$ . Turning that into compactness for $\tau_\psi$ needs the $\psi^2$ -moment of $X$ to upgrade $\tau_1$ -convergence to $\tau_\psi$ -convergence. Continuity in the control law goes through the stable topology on measures over $[0,T]\times\mathcal P(A)$ .

Formalization scope

The Lean development (namespace WeakMFG.Existence) builds on the published Brownian/Itô/BSDE layer Peng1990.SMP.Stochastic. It commits to these conventions:

Base space. It is an arbitrary probability space carrying $(\xi,W)$ , with the filtration of $(\xi,W)$ augmented by null sets. The canonical space $\mathbb R^d\times\mathcal C$ is one instance, and every object of the theorem is a law.
State equation. It is read through the L² Itô integral, so solutions are square integrable. " $\sigma>0$ " is nonsingularity, and the growth function $\rho$ is monotone.
Types. $\mathbb R^d$ is Fin d → ℝ, time is ℝ≥0, and paths are continuous maps on $[0,T]$ .
Topologies. $\mathcal P_\psi(\mathcal C)$ is its own type carrying $\tau_\psi$ . $\mathcal P(A)$ has the weak topology and its Borel $\sigma$ -field.
Densities. Girsanov densities and BSDE solutions are relations. The solution concept asks that a density version exists and that the identities hold for all versions.
Optimality. $V^{\mu,q}=J^{\mu,q}(\alpha)$ is stated as optimality of $\alpha$ against every admissible control; no real supremum is formed.
Assumption (E) is sequential continuity, as printed.
Assumption (C) is required wherever $\sigma(t,x)$ is nonsingular.
Control set. $A$ is assumed nonempty.

Some formalizations would be trivial and are ruled out. Every clause of Definition 3.4 is stated. No density version may be missing. Optimality is over the whole admissible class, not a subclass. Replacing the goal by a fixed-point statement about the map $\Phi$ is not acceptable, since that is a step of the proof.

Milestone conventions:

Milestone (7.4). It adds joint measurability of $f$ in the control-law argument. The page presupposes it when it integrates $f$ against $\nu_t(dq)$ .
Berge's theorem. It assumes a nonempty $K$ .
Proposition 7.4. It assumes nonempty compact values (see Significance).

Infrastructure that is reusable beyond this mission:

Doléans exponentials of bounded integrands and their $L^q$ bounds;
weak compactness in $L^1$ (Dunford–Pettis) for density sets;
the stable topology;
Cellina's approximate selection theorem;
Schauder–Tychonoff fixed points in locally convex spaces.

Contributions to any of these, and alternative proofs that avoid the non-compact-valued fixed point step, are welcome.

Selected references

R. Carmona, D. Lacker, A probabilistic weak formulation of mean field games and applications, Ann. Appl. Probab. 25(3), 2015; arXiv:1307.1152v2 (2014). https://arxiv.org/abs/1307.1152
R. Carmona, F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim. 51(4), 2013. https://arxiv.org/abs/1210.5780
J.-M. Lasry, P.-L. Lions, Mean field games, Jpn. J. Math. 2(1), 2007. https://doi.org/10.1007/s11537-007-0657-8
M. Huang, R. Malhamé, P. Caines, Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst. 6(3), 2006. https://doi.org/10.4310/CIS.2006.v6.n3.a5
E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14(1), 1990. https://doi.org/10.1016/0167-6911(90)90082-6
A. Cellina, Approximation of set valued functions and fixed point theorems, Ann. Mat. Pura Appl. 82, 1969. https://doi.org/10.1007/BF02410784
C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis, 3rd ed., Springer, 2006. https://doi.org/10.1007/3-540-29587-9

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Differential Geometry Numerical Analysis Optimization·Captain: mikedeng1

Riemannian Proximal Gradient Methods I: Every Accumulation Point of the Riemannian Proximal Gradient Method Is Stationary, with an Explicit Iteration BoundResearch Paper

Motivation

Many problems in statistics, signal processing and machine learning minimize a smooth loss plus a nonsmooth regularizer over a set that is a manifold rather than a vector space: sparse principal component analysis over the Stiefel manifold, sparse blind deconvolution over the sphere, low-rank recovery over fixed-rank matrices. In Euclidean space the standard tool for $\min f + g$ with $f$ smooth and $g$ nonsmooth is the proximal gradient method, whose convergence theory (stationarity of limit points, an $O(1/\epsilon^2)$ bound on the number of iterations to reach an $\epsilon$ -small step, an $O(1/k)$ rate under convexity) is classical (Beck–Teboulle 2009).

Huang and Wei (arXiv:1909.06065, Mathematical Programming 2021) proposed the Riemannian proximal gradient method (RPG): at each iterate the proximal subproblem is posed on the tangent space, the nonsmooth term is pulled back through a retraction, and the next iterate is obtained by retracting the computed step. Unlike the earlier ManPG method of Chen, Ma, So and Zhang (2020), which uses the Euclidean proximal mapping of an embedding space, RPG is defined intrinsically on any manifold with a retraction. Its global convergence theorem is the subject of this mission.

Setting

Let $\mathcal M$ be a finite-dimensional manifold with a smooth Riemannian metric $\langle\cdot,\cdot\rangle_x$ on each tangent space $T_x\mathcal M$ , with norm $\|\cdot\|_x$ . A retraction is a smooth map $R : T\mathcal M\to\mathcal M$ , $(x,\eta)\mapsto R_x(\eta)$ , with $R_x(0_x) = x$ and $DR_x(0_x) = \mathrm{id}$ . The vector transport by differentiated retraction is $\mathcal T_{\eta_x} = DR_x(\eta_x) : T_x\mathcal M\to T_{R_x(\eta_x)}\mathcal M$ ; $\mathcal T^\sharp_{\eta_x}$ denotes its adjoint for the two metrics and $\mathcal T^{-\sharp}_{\eta_x}$ the adjoint of its inverse.

The problem is

\min_{x\in\mathcal M} F(x) = f(x) + g(x),

with $f$ continuously differentiable, Riemannian gradient $\operatorname{grad} f$ , and $g$ continuous, possibly nonsmooth. For $h : \mathcal M\to\mathbb R$ the Riemannian Clarke subdifferential is defined through the pull-back $\hat h_x = h\circ R_x$ on the Hilbert space $T_x\mathcal M$ :

\hat h_x^\circ(\eta; v) = \limsup_{\xi\to\eta,\ t\downarrow 0}\frac{\hat h_x(\xi+tv)-\hat h_x(\xi)}{t},\qquad \partial h(x) = \{\zeta\in T_x\mathcal M \mid \langle\zeta,v\rangle_x\le\hat h_x^\circ(0_x;v)\ \forall v\}.

A point $x$ is stationary if $0\in\partial F(x)$ .

Algorithm 1 (RPG). Given $x_0$ and a constant $\tilde L > L$ , for $k = 0,1,\dots$ let

\ell_{x_k}(\eta) = \langle\operatorname{grad} f(x_k),\eta\rangle_{x_k} + \frac{\tilde L}{2}\|\eta\|_{x_k}^2 + g(R_{x_k}(\eta)),

choose $\eta^*_{x_k}\in T_{x_k}\mathcal M$ that is a stationary point of $\ell_{x_k}$ with $\ell_{x_k}(0)\ge\ell_{x_k}(\eta^*_{x_k})$ (condition (3.1)), and set $x_{k+1} = R_{x_k}(\eta^*_{x_k})$ .

Assumption 3.1. $F$ is bounded below and $\Omega_{x_0} = \{x \mid F(x)\le F(x_0)\}$ is compact. Assumption 3.2. $f$ is $L$ -retraction-smooth on $\Omega_{x_0}$ : $f(R_x(\eta))\le f(x) + \langle\operatorname{grad} f(x),\eta\rangle_x + \frac L2\|\eta\|_x^2$ (inequality (3.2)).

Formalization targets

Goal: Theorem 3.1

For every run of Algorithm 1: if $\eta^*_{x_k} = 0$ then $x_k$ is stationary. Under Assumptions 3.1 and 3.2, $\{x_k\}$ has an accumulation point, every accumulation point $x_*$ is stationary, and for every $\epsilon > 0$ some iterate with index at most

\frac{F(x_0)-F(x_*)}{\beta\epsilon^2},\qquad\beta = \frac{\tilde L - L}{2},

satisfies $\|\eta^*_{x_k}\|_{x_k}\le\epsilon$ .

Milestones, in the order the proof uses them

Lemma 3.1 (descent): $F(x_k) - F(x_{k+1})\ge\beta\|\eta^*_{x_k}\|_{x_k}^2$ .
(3.6): $\lim_k\|\eta^*_{x_k}\|_{x_k} = 0$ .
Lemma 3.2: for a continuous vector field $\xi$ , $\|\xi_y - \mathcal T^{-\sharp}_{\eta_x}\xi_x\|_y\to 0$ as $\eta_x\to 0$ on the tangent bundle, $y = R_x(\eta_x)$ .
(3.8): at every iteration some $\zeta\in\partial g(x_{k+1})$ satisfies $\operatorname{grad} f(x_k)+\tilde L\eta^*_{x_k}+\mathcal T^\sharp_{R_{\eta^*_{x_k}}}\zeta = 0$ .
(3.9): where the transport is invertible, $\operatorname{grad} f(x_{k+1}) - \mathcal T^{-\sharp}(\operatorname{grad} f(x_k)+\tilde L\eta^*_{x_k}) = \operatorname{grad} f(x_{k+1})+\zeta\in\partial F(x_{k+1})$ .

Significance

Theorem 3.1 is the basic convergence guarantee of RPG: the method is a descent method without any convexity of $g$ , its limit points satisfy the first-order necessary condition $0\in\partial F$ , and the number of iterations needed to make the step small is explicit, with the same $O(1/\epsilon^2)$ dependence as in the Euclidean case. It underlies the further analyses of the same paper (the $O(1/k)$ rate under retraction convexity and the Kurdyka–Łojasiewicz analysis) and of later Riemannian proximal methods.

The theorem is proved on paper; no machine-checked version exists. Formalizing it requires a Clarke subdifferential on tangent spaces, its calculus (sum rule with a $C^1$ function, chain rule through a local diffeomorphism) and its closedness along convergent sequences on a manifold. None of these is in Mathlib. They are reusable well beyond this paper by any Riemannian nonsmooth optimization method.

Difficulty

The descent inequality and the iteration bound are elementary once Lemma 3.1 is available. The difficulty is the stationarity of accumulation points. The subproblem's optimality condition (3.8) lives in the tangent space at $x_k$ and involves a subgradient of $g$ at $x_{k+1}$ ; to pass to the limit, the gradients and subgradients must be compared across different tangent spaces, which is what the inverse-adjoint transport and Lemma 3.2 do. This needs joint continuity of the retraction and its differential on the tangent bundle and continuity of the metric, and then a closedness property of the Riemannian Clarke subdifferential that the paper cites from Hosseini, Huang and Yousefpour (2018, Theorem 2.2(c)) and that a complete development must prove. A pointwise argument at a fixed base point does not suffice, because the base points $x_{k_j}$ move.

Formalization scope

Theorem numbers and pages are those of arXiv:1909.06065v4. The manifold is modelled on a finite-dimensional real inner-product space $E$ and carries a RiemannianBundle with a smooth metric (IsContMDiffRiemannianBundle). Retraction, differentiated-retraction transport and Riemannian gradient are the published definitions RiemOpt.BFGS.Setting and RiemOpt.FR.Setting; joint smoothness of $R$ on the tangent bundle is added, as on p. 4. The Clarke limit superior is taken in EReal along $\mathcal N(\eta)\times\mathcal N_{>0}(0)$ , so it never takes a default value. Accumulation points are MapClusterPt, and "in at most $N$ iterations" is an index $k\le N$ . A run of Algorithm 1 is any sequence of iterates and steps satisfying (3.1); the step is a selection, not a function of the iterate.

Two hypotheses are explicit in the Lean statements. Each pull-back $g\circ R_x$ is locally Lipschitz, which p. 5 requires for the subdifferential to be defined. $f$ is $C^1$ , because the proof applies Lemma 3.2 to $\operatorname{grad} f$ . Assumption 3.2 is stated in the form the proof of Lemma 3.1 uses: (3.2) at every $x\in\Omega_{x_0}$ for every tangent vector. Definition 3.1 read literally only constrains $\eta$ with $R_x(\eta)\in\Omega_{x_0}$ , and under that reading Lemma 3.1 fails: a nonconvex $g$ can send $x_1$ far outside $\Omega_{x_0}$ . The paper's own sufficient condition gives the strengthened form.

The formalization must not trivialize stationarity. The subdifferential is defined from the Clarke derivative of the pull-back alone. It does not involve $\mathcal T^{-\sharp}$ or ContinuousLinearMap.inverse, so $0\in\partial F(x)$ is a genuine condition. The run hypotheses are satisfiable, for instance on $\mathcal M = \mathbb R$ with $R_x(\eta) = x+\eta$ , $f(x) = x^2$ , $g = 0$ , $x_k = 0$ , $\eta_k = 0$ .

Welcome contributions: the Clarke calculus on a real Hilbert space (sum rule, chain rule through a $C^1$ map, upper semicontinuity), local invertibility of $DR_x(\eta)$ near the zero section, and the closedness of the Riemannian subdifferential on the tangent bundle.

Selected references

W. Huang, K. Wei, Riemannian proximal gradient methods, Mathematical Programming, 2021. arXiv:1909.06065v4. https://arxiv.org/abs/1909.06065, https://doi.org/10.1007/s10107-021-01632-3
S. Hosseini, W. Huang, R. Yousefpour, Line search algorithms for locally Lipschitz functions on Riemannian manifolds, SIAM Journal on Optimization 28(1):596–619, 2018 (reference [29] of Huang–Wei; source of the Riemannian Clarke subdifferential and of its closedness, Theorem 2.2(c)). https://epubs.siam.org/toc/sjope8/28/1
P.-A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008. https://doi.org/10.1515/9781400830244
S. Chen, S. Ma, A. M.-C. So, T. Zhang, Proximal gradient method for nonsmooth optimization over the Stiefel manifold, SIAM Journal on Optimization 30 (2020). https://doi.org/10.1137/18M122457X
A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences 2 (2009). https://doi.org/10.1137/080716542
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, 1983. https://doi.org/10.1137/1.9781611971309

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Dynamical Systems Machine Learning Optimization+1·Captain: mikedeng1

Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization 4: Constant-Step Adam Iterates Converge in Probability to the ODE Solution as γ → 0Research Paper

Motivation

Adam (Kingma and Ba, 2015, arXiv:1412.6980) is one of the most widely used optimizers for training neural networks. It is a stochastic gradient method that keeps exponential moving averages of the stochastic gradients and of their coordinatewise squares, rescales both by bias-correction factors, and divides the first by the square root of the second. Its original convergence analysis was shown to be flawed (Reddi, Kale and Kumar, 2018, arXiv:1904.09237), and in the regime in which practitioners actually run it, with a constant step size, the iterates do not converge.

Barakat and Bianchi (arXiv:1810.02263, SIAM J. Math. Data Sci. 3(1), 2021) describe what constant-step Adam does instead: as the step size $\gamma$ tends to zero, with the averaging parameters tending to one at rates proportional to $\gamma$ , the interpolated iterates follow a deterministic non-autonomous ordinary differential equation. This mission formalizes that statement, Theorem 4.3 of the paper, together with the lemmas of its proof (§8.1).

Setting

Let $\mathcal Z = \mathbb R^d\times\mathbb R^d\times\mathbb R^d$ with points $z=(x,m,v)$ , and $\mathcal Z_+$ its subset with $v\ge0$ coordinatewise. A random sample $\xi$ with law $\mu$ on a measurable space $\Xi$ enters through an integrand $f(x,\xi)$ with gradient $\nabla f(x,\xi)$ in $x$ . The objective is $F(x)=\mathbb E f(x,\xi)$ and the second-moment map is $S(x)=\mathbb E\nabla f(x,\xi)^{\odot2}$ .

Given a step size $\gamma>0$ , parameters $\alpha=\bar\alpha(\gamma)$ , $\beta=\bar\beta(\gamma)$ in $[0,1)$ and $\varepsilon>0$ , the Adam map $T_{\gamma,\alpha,\beta}(n,z,\xi)$ sends $(x,m,v)$ to

\Big(x-\frac{\gamma(1-\alpha^n)^{-1}(\alpha m+(1-\alpha)\nabla f(x,\xi))}{\varepsilon+(1-\beta^n)^{-1/2}(\beta v+(1-\beta)\nabla f(x,\xi)^{\odot2})^{1/2}},\ \alpha m+(1-\alpha)\nabla f(x,\xi),\ \beta v+(1-\beta)\nabla f(x,\xi)^{\odot2}\Big),

coordinatewise. The constant-step iterates are $z^\gamma_0=(x_0,0,0)$ and $z^\gamma_n=T_{\gamma,\bar\alpha(\gamma),\bar\beta(\gamma)}(n,z^\gamma_{n-1},\xi_n)$ , driven by iid samples $\xi_1,\xi_2,\dots$ with law $\mu$ . The interpolated process $\mathsf z^\gamma$ is the piecewise linear function with $\mathsf z^\gamma(n\gamma)=z^\gamma_n$ .

The parameters satisfy $(1-\bar\alpha(\gamma))/\gamma\to a>0$ and $(1-\bar\beta(\gamma))/\gamma\to b>0$ as $\gamma\downarrow0$ , with $b\le4a$ . The limiting Adam ODE is $\dot z(t)=h(t,z(t))$ , $z(0)=(x_0,0,0)$ , where for $t>0$

h(t,z)=\Big(-\frac{(1-e^{-at})^{-1}m}{\varepsilon+\sqrt{(1-e^{-bt})^{-1}v}},\ a(\nabla F(x)-m),\ b(S(x)-v)\Big).

The vector field is singular at $t=0$ . A global solution is a continuous map $z:[0,\infty)\to\mathcal Z_+$ , continuously differentiable on $(0,\infty)$ , that satisfies the equation for every $t>0$ .

Formalization targets

Goal: Theorem 4.3

Under the paper's assumptions (Assumptions 2.2 to 2.5, 4.1, and 4.2 ii) with $p=2$ ),

\forall T>0,\ \forall\delta>0,\qquad \lim_{\gamma\downarrow0}\mathbb P\Big(\sup_{t\in[0,T]}\|\mathsf z^\gamma(t)-z(t)\|>\delta\Big)=0,

where $z$ is the global solution of the Adam ODE from $(x_0,0,0)$ .

Milestones, in the order of the proof

Lemma 8.1. A uniform $(1+s)$ -moment bound for the normalised increment $H_\gamma(n+1,z,\xi)$ over states whose debiased version $e_\gamma(n,z)$ lies in a ball.
Lemma 8.2. Uniform integrability of the increments of the truncated iterates $z^{\gamma,R}$ , which are stopped when the debiased iterate leaves the ball of radius $R$ .
Lemma 8.3. Tightness of the truncated interpolated processes $\mathsf z^{\gamma,R}$ in $C([0,\infty),\mathcal Z)$ , and vanishing in probability of the accumulated martingale noise.
Proposition 7.6. The debiased trajectories $\bar e(t+\eta,z(t))$ of the ODE stay in one compact set, and $F(x(t))\le F(x_0)$ .
Lemma 8.4. The mean field $h_\gamma(\varphi_n,z_n)$ converges to $h(t,z)$ when $\gamma\varphi_n\to t$ .
Proposition 7.12. The ODE has exactly one global solution.
Eq. (8.4). For $R\ge R_0+1$ , with $R_0=\sup_{t>0}\|\bar e(t,z(t))\|$ , the truncated process converges in probability to $z$ , uniformly on $[0,T]$ .

Significance

The theorem justifies the continuous-time model of Adam that the rest of the paper studies. The convergence of the ODE to critical points of $F$ (Theorem 3.2) and its rates (Theorem 3.4) say something about the algorithm only through this approximation. It also isolates the role of the bias correction: the initial condition $(x_0,0,0)$ and the singular factors $(1-e^{-at})^{-1}$ are exactly the trace of the debiasing step. The result is also the first step towards the long-run statement, Theorem 4.5, which bounds the time the iterates spend away from the critical points as $\gamma\to0$ .

The result is proved on paper. To the knowledge of the mission, none of it is machine-checked: there is no formal statement of Adam's dynamics on Prove2Me. A complete development would give a checked instance of the ODE method for a non-autonomous, non-Lipschitz limit field and a non-homogeneous Markov chain. The surrounding stochastic-approximation library (truncation by stopping, uniform integrability, tightness of interpolated processes, identification of limits by uniqueness) applies to other adaptive methods, such as RMSProp and AdaGrad variants, whose limits have the same structure.

Difficulty

The standard argument for constant-step stochastic approximation assumes a time-homogeneous mean field that is locally Lipschitz. Neither holds here. The mean field $h_\gamma(n,z)$ depends on the iteration index through the bias-correction factors $(1-\bar\alpha(\gamma)^n)^{-1}$ . Its limit $h(t,z)$ blows up as $t\downarrow0$ , and it is not Lipschitz in $v$ near $v=0$ . A direct Gronwall comparison between the iterates and the ODE therefore fails at the origin of time, which is exactly where the iterates start.

The iterates are also not known to be bounded. Moment bounds are only available while the debiased iterate stays in a ball. This forces the truncation $B_R$ and an argument that the truncation is asymptotically never triggered. That argument needs uniform-in-time bounds on the ODE solution (Proposition 7.6) and uniqueness of the solution (Proposition 7.12), and both rest on a Lyapunov function specific to the case $b\le4a$ .

Formalization scope

Spaces and norm. $\mathcal Z$ is E d × E d × E d with E d = EuclideanSpace ℝ (Fin d). The paper's $\|z\|$ is the Euclidean norm of $\mathbb R^{3d}$ , written zNorm, not Mathlib's max norm on products.
Randomness. The probability space is explicit, with a probability measure $\mathbb P$ . Independence of $(\xi_{n+1})_n$ is stated with iIndepFun. Every $\xi_n$ with $n\ge1$ has law $\mu$ , and $\xi_0$ is unused.
Gradient. $\nabla f$ is a given function, measurable in $\xi$ , that equals the gradient of $f(\cdot,\xi)$ for $\mu$ -almost every $\xi$ .
Expectations. Moments, the supremum in Lemma 8.1 and the uniform-integrability expectations are lower Lebesgue integrals in $[0,\infty]$ . $F$ , $S$ and $h_\gamma$ are Bochner integrals, which the hypotheses make integrable.
Solutions of the ODE. Solutions are maps $\mathbb R\to\mathcal Z$ constrained only on $[0,\infty)$ , and uniqueness means agreement on $[0,\infty)$ . The deterministic Propositions 7.6 and 7.12 are stated for any $F$ , $S$ satisfying the paper's §7 hypotheses.
Limit and supremum. The limit is $\gamma\downarrow0$ (𝓝[>] 0). The event $\sup_{t\in[0,T]}\|\cdot\|>\delta$ is written as $\exists t\in[0,T],\ \|\cdot\|>\delta$ , which is the same event.
Tightness. Tightness uses the paper's definition with the compact-open topology on $C([0,\infty),\mathcal Z)$ .
Corrections. Two corrections to the page are recorded in the items. Lemma 8.1 restricts its $n=0$ case to $m=0$ , because as printed that case is false. Lemma 8.4 states the implicit regime $\gamma_n\to0$ .

A trivializing formalization is ruled out. $T$ and $\delta$ are universally quantified before the limit and do not depend on $\gamma$ . The limit is in $\gamma$ , not in $n$ at fixed $\gamma$ ; constant-step Adam does not converge in $n$ . The solution $z$ is tied to the $F$ , $S$ , $a$ , $b$ of the stochastic model. All hypotheses are satisfiable, for example by $f(x,\xi)=\tfrac12\|x\|^2+\xi\langle\mathbf 1,x\rangle$ with $\xi=\pm1$ equiprobable.

A complete development needs:

the deterministic ODE theory of the companion missions (Lyapunov function, boundedness, uniqueness);
moment and uniform-integrability bounds;
a tightness criterion in $C([0,\infty),\mathcal Z)$ (Arzelà–Ascoli);
Prokhorov's and Skorokhod's theorems, or a direct substitute;
dominated convergence for $h_\gamma$ .

Contributions to any of these, including reusable statements about interpolated processes, are welcome.

Selected references

A. Barakat and P. Bianchi, Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization, SIAM J. Math. Data Sci. 3(1), 2021; arXiv:1810.02263v4. https://arxiv.org/abs/1810.02263
D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, ICLR, 2015. https://arxiv.org/abs/1412.6980
S. J. Reddi, S. Kale and S. Kumar, On the Convergence of Adam and Beyond, ICLR, 2018. https://arxiv.org/abs/1904.09237
M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités XXXIII, LNM 1709, Springer, 1999. https://doi.org/10.1007/BFb0096509
P. Bianchi, W. Hachem and A. Salim, Constant step stochastic approximations involving differential inclusions: stability, long-run convergence and applications, Stochastics 91, 2019, pp. 288–320 (reference [6] of the paper; its Lemma 6.2 gives Lemma 8.3). https://arxiv.org/abs/1802.07981

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Bandit Algorithms Machine Learning Probability·Captain: mikedeng1

A Tutorial on Thompson Sampling III: In the Known-Standard Problem, Every Stationary Deterministic Strategy Has Linearly Growing Expected Regret Under Some PriorResearch Paper

Why randomize actions

Thompson sampling (TS) selects each action by sampling a parameter from the current posterior and playing the action that is optimal for the sample. It is a stationary randomized strategy: the distribution of the action is a fixed function of the posterior, the same in every period, and the action is drawn at random from it. Its Bayesian regret bounds, surveyed in §8.1 of the tutorial by Russo, Van Roy, Kazerouni, Osband and Wen (Found. Trends Mach. Learn. 11(1), 2018), grow sublinearly in the horizon $T$ .

A natural question is whether the randomization is essential. A strategy that maps the posterior to an action deterministically, for example "play the action with the highest posterior mean", is equally stationary and conceptually simpler. §8.1.3 of the tutorial answers the question with Example 8.1 (A Known Standard), taken from Russo and Van Roy, Learning to optimize via information-directed sampling (Operations Research 66(1), 2018): in a two-action problem with a known baseline action, every stationary deterministic strategy suffers linearly growing expected regret under some prior. The example is the standard justification for why posterior sampling, and not a deterministic function of the posterior, is the right stationary rule.

Setting

There are two actions $\mathcal X=\{1,2\}$ and a binary unknown parameter $\theta\in\{0,1\}$ with prior

\theta\sim\mathrm{Bernoulli}(p_0),\qquad \mathbb P(\theta=1)=p_0 .

In each period $t=1,2,\dots$ the agent selects an action $x_t$ and observes a binary reward $r_t$ . Rewards are conditionally independent across periods given $\theta$ and the actions:

action $1$ (the known standard) pays $\mathrm{Bernoulli}(1/2)$ , whatever $\theta$ is;
action $2$ pays $\mathrm{Bernoulli}(3/4)$ if $\theta=1$ and $\mathrm{Bernoulli}(1/4)$ if $\theta=0$ .

The mean reward $\mu(x,\theta)$ of action $x$ is its success probability, and the optimal action $x^*$ is action $2$ if $\theta=1$ and action $1$ if $\theta=0$ , so $\mu(x^*,\theta)=3/4$ or $1/2$ .

The history $\mathbb H_t$ lists the actions and rewards of the first $t$ periods. The posterior $p_t=\mathbb P(\theta=1\mid\mathbb H_t)$ is given by Bayes' rule (4.2) of the tutorial, applied period by period:

p_t=\frac{p_0\,L_1(\mathbb H_t)}{p_0\,L_1(\mathbb H_t)+(1-p_0)\,L_0(\mathbb H_t)},\qquad L_\theta(\mathbb H_t)=\prod_{s=1}^{t}\mathbb P_\theta(r_s\mid x_s),

with $p_0$ the prior on the empty history.

A stationary deterministic strategy is one function $f$ from posterior probabilities to actions, used in every period: $x_t=f(p_{t-1})$ . The expected cumulative regret (Bayesian regret, p. 71) of running $f$ from the prior $p_0$ for $T$ periods is

\mathbb E[\mathrm{Regret}(T)]=\mathbb E\Big[\sum_{t=1}^T\big(\mu(x^*,\theta)-\mu(x_t,\theta)\big)\Big],

the expectation taken over $\theta$ and the rewards. The Lean development names these objects succProb ( $\mu$ ), optMean ( $\mu(x^*,\cdot)$ ), post ( $p_t$ ), Strategy ( $f$ ), lawGiven (the probability of a history given $\theta$ ) and expRegret.

Formalization targets

Goal: some prior forces linear regret

\forall f\ \ \exists\,p_0\in(0,1]\ \ \exists\,c>0\ \ \forall T\in\mathbb N:\qquad \mathbb E_{p_0}[\mathrm{Regret}(T)]\ \ge\ c\,T .

This is the sentence "for any deterministic stationary strategy, there exists a prior probability $p_0$ such that expected cumulative regret grows linearly with time" (p. 79). The prior is chosen after the strategy. No constant is fixed; only the linear shape is asserted.

Milestones: the two cases of the example

$x_1=1$ for some $p_0>0$ . If $f(p_0)=1$ for some $p_0\in(0,1]$ , then on every history of positive probability $p_t=p_0$ and $x_t=1$ for all $t$ , and

\mathbb E_{p_0}[\mathrm{Regret}(T)]=\tfrac{p_0}{4}\,T .

$x_1=2$ for all $p_0>0$ . If $f(p)=2$ for every $p\in(0,1]$ , then for every $p_0\in(0,1]$ , on every history of positive probability $x_t=2$ for all $t$ , and

\mathbb E_{p_0}[\mathrm{Regret}(T)]=\tfrac{1-p_0}{4}\,T .

Significance

The example separates two properties that regret analyses often conflate: being a function of the posterior alone (stationarity) and achieving sublinear Bayesian regret. It shows that a stationary strategy with sublinear regret on all priors must randomize, which is the structural reason TS samples rather than maximizes. It also explains why deterministic algorithms with good regret, such as upper confidence bound methods, are necessarily nonstationary: their confidence radii depend on the period and on counts, not only on the posterior.

The tutorial gives the argument in one paragraph. A formal version makes precise what is left implicit there: that the posterior is the actual conditional probability computed from the history, that "uninformative" means the posterior is invariant under action $1$ , and that under action $2$ the posterior never leaves $(0,1]$ . The result is a closed-form regret identity in each case, not only an asymptotic statement. As far as is known, neither the example nor its source in Russo and Van Roy (2018) has been machine-checked.

Difficulty

The mathematics is elementary; the work lies in the bookkeeping of a history-dependent strategy. The strategy's action in period $t$ depends on the whole history through $p_{t-1}$ , so the law of $\mathbb H_T$ is a product of indicator factors, each evaluating $f$ at a posterior computed from a prefix of the same history. A direct computation of the regret must show that every history of positive probability follows a single action path, and that is a statement about the posterior along every prefix, not only at the first period. A first attempt that defines the posterior by a recursion in which "the posterior stays at $p_0$ " is built in proves nothing about the example; the posterior must be Bayes' rule applied to the history.

Formalization scope

Paper action $1$ is 0 : Fin 2, action $2$ is 1 : Fin 2; $\theta=1$ is true; a reward of $1$ is true. Lean period s : Fin T is the paper's period $s+1$ .
Histories are Fin t → Fin 2 × Bool; every probability and expectation is a finite sum, so no measure theory is needed.
The posterior post is the closed form of iterated Bayes' rule (4.2). The probabilities with which the strategy picks its actions are $0$ or $1$ and do not depend on $\theta$ ; on histories of positive probability they cancel, so post is the true conditional probability there. For $0<p_0\le1$ its denominator is positive, so Lean's convention $x/0=0$ is never used.
A strategy is a function ℝ → Fin 2; its values outside $[0,1]$ are never used. The same function acts in every period, which is what "stationary" means.
The prior ranges over $p_0\in(0,1]$ : the text requires $p_0>0$ , and $p_0\le1$ because it is a probability.
"Grows linearly with time" is read as a lower bound $cT$ with $c>0$ for all $T$ . The constant $c>0$ is part of the claim; $c=0$ would make the goal trivial. The quantifiers are in the order of the text, every strategy first and then a prior; the reverse order (one prior for all strategies) is a different statement that the tutorial does not claim.
Contributions welcome: proofs of the two milestones and of the goal from them, and lemmas on the posterior (invariance under an uninformative action, positivity) that are reusable for other Bayesian bandit examples.

The tutorial's argument for the example is on p. 79; the tutorial attributes the example to Russo and Van Roy (2018).

Selected references

D. J. Russo, B. Van Roy, A. Kazerouni, I. Osband, Z. Wen, A Tutorial on Thompson Sampling, Foundations and Trends in Machine Learning 11(1):1–96, 2018. https://doi.org/10.1561/2200000070 (Example 8.1, p. 79; Bayes' rule (4.2), p. 19; Bayesian regret, p. 71)
D. Russo, B. Van Roy, Learning to Optimize via Information-Directed Sampling, Operations Research 66(1):230–252, 2018. https://doi.org/10.1287/opre.2017.1663 (arXiv: https://arxiv.org/abs/1403.5556)
D. Russo, B. Van Roy, Learning to Optimize via Posterior Sampling, Mathematics of Operations Research 39(4):1221–1243, 2014. https://doi.org/10.1287/moor.2014.0650
W. R. Thompson, On the likelihood that one unknown probability exceeds another in view of the evidence of two samples, Biometrika 25(3/4):285–294, 1933. https://doi.org/10.2307/2332286

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Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

Distributionally Robust Chance-Constrained Programs with Right-Hand Side Uncertainty under Wasserstein Ambiguity: The Reduced Quantile-Strengthened MIP (20) Is an Exact Reformulation of DR-CCPResearch Paper

Motivation

A chance-constrained program chooses a decision $x$ so that a random vector $\xi$ falls outside a decision-dependent safety set $\mathcal S(x)$ with probability at most $\epsilon$ . Chance constraints are a standard model for reliability requirements in power systems, supply chains and transportation, where a plan must survive random demand or supply with high probability. In practice the distribution of $\xi$ is known only through samples $\xi_1, \dots, \xi_N$ , and the sample average approximation (SAA) that replaces it by the empirical distribution $\mathbb P_N$ is sensitive to the particular sample.

Distributionally robust chance constraints regularize SAA by requiring the probability bound for every distribution within Wasserstein distance $\theta$ of $\mathbb P_N$ . Chen, Kuhn and Wiesemann (arXiv:1809.00210) and Xie (doi:10.1007/s10107-019-01445-5) showed that, for safety sets defined by linear inequalities, the resulting program (DR-CCP) is a mixed-integer linear program with a big-M constant. These big-M formulations are hard for solvers to close when $\theta$ is small. Ho-Nguyen, Kılınç-Karzan, Küçükyavuz and Lee (doi:10.1007/s10107-020-01605-y) connect the formulation to the mixing sets of SAA and to robust 0–1 programming, and derive a smaller and tighter formulation that is still exact.

Setting

The random vector lives in $\mathbb R^K$ with an arbitrary norm $\|\cdot\|$ ; in the formalization it is a finite-dimensional real normed space $E$ . The dual norm of a linear functional $b$ is $\|b\|_* = \sup_{\|\xi\| \le 1} b^\top \xi$ . The decision $x$ ranges over a set $\mathcal X \subseteq \mathbb R^L$ .

The 1-Wasserstein distance between distributions $\mathbb P, \mathbb P'$ is the infimum of $\mathbb E\|\xi - \xi'\|$ over couplings of $\mathbb P$ and $\mathbb P'$ . The ambiguity set $\mathcal F_N(\theta)$ is the set of distributions within distance $\theta$ of $\mathbb P_N = \frac1N \sum_{i} \delta_{\xi_i}$ . The feasible region of (DR-CCP) is

\mathcal X_{\mathrm{DR}}(\mathcal S) = \Big\{x \in \mathcal X : \sup_{\mathbb P \in \mathcal F_N(\theta)} \mathbb P[\xi \notin \mathcal S(x)] \le \epsilon\Big\}.

Throughout, $\epsilon \in (0,1)$ and $\theta > 0$ are fixed.

The paper studies joint chance constraints with right-hand side uncertainty: for $p \in [P]$ there are $a_p \in \mathbb R^L$ , $b_p$ and $d_p \in \mathbb R$ , and

\mathcal S(x) = \{\xi : b_p^\top \xi + d_p - a_p^\top x > 0,\ p \in [P]\}.

Write $g_{i,p}(x) = (b_p^\top\xi_i + d_p - a_p^\top x)/\|b_p\|_*$ and $k = \lfloor \epsilon N \rfloor$ . Let $q_p$ be the $(k+1)$ -th largest value among $\{-b_p^\top\xi_i\}_{i\in[N]}$ , and let $[N]_p = \{i : -b_p^\top \xi_i > q_p\}$ .

The big-M formulation (5) of Chen et al. uses binaries $z \in \{0,1\}^N$ , slacks $r \ge 0$ and a scalar $t \ge 0$ with

\epsilon t \ge \theta + \tfrac1N \textstyle\sum_i r_i,\qquad M(1 - z_i) \ge t - r_i,\qquad g_{i,p}(x) + M z_i \ge t - r_i.

The paper's improved formulation (20) keeps the first two families, adds the knapsack constraint $\sum_i z_i \le k$ , replaces $M$ in the third family by $h_{i,p} = (-b_p^\top\xi_i - q_p)/\|b_p\|_*$ , imposes it only for $i \in [N]_p$ , and adds $(-q_p + d_p - a_p^\top x)/\|b_p\|_* \ge t$ for every $p$ .

Formalization targets

Goal: Theorem 3 (p. 655)

\mathcal X_{\mathrm{DR}}(\mathcal S) = \{x \in \mathcal X : \exists\,(z, r, t) \text{ satisfying (20b)–(20d)}\}.

The goal states equality of feasible sets of $x$ . Because the objective $c^\top x$ depends on $x$ alone, this is the paper's "exact reformulation".

Milestones, in attack order

(4b), p. 646: $\operatorname{dist}(\xi, \mathcal S(x)) = \max\{0, \min_p (b_p^\top\xi + d_p - a_p^\top x)/\|b_p\|_*\}$ , where $\operatorname{dist}(\xi, \mathcal S) = \inf\{\|\xi - \xi'\| : \xi' \notin \mathcal S\}$ is (1).
(3), p. 645: for open $\mathcal S(x)$ , $\mathcal X_{\mathrm{DR}}(\mathcal S)$ is the set of $x \in \mathcal X$ admitting $t \ge 0$ , $r \ge 0$ with $\operatorname{dist}(\xi_i, \mathcal S(x)) \ge t - r_i$ and $\epsilon t \ge \theta + \frac1N\sum_i r_i$ .
(6), p. 646: $\mathcal X_{\mathrm{DR}}(\mathcal S) = \{x \in \mathcal X : \text{(5b)–(5e)}\}$ .
Theorem 1, p. 648: adding $\sum_i z_i \le k$ and $g_{i,p}(x) + M z_i \ge 0$ to (5) keeps it exact.
Theorem 2, p. 651: replacing $M$ in (5e) by $h_{i,p}$ , with the knapsack constraint, keeps it exact.
Lemma 1, p. 653: for $x \in \mathcal X_{\mathrm{DR}}(\mathcal S)$ , $t$ may be taken equal to the $(k+1)$ -th smallest of the distances $\operatorname{dist}(\xi_i, \mathcal S(x))$ .
Proposition 1, p. 654: for $x \in \mathcal X_{\mathrm{DR}}(\mathcal S)$ , some solution of (17) also satisfies $(-q_p + d_p - a_p^\top x)/\|b_p\|_* - t \ge 0$ for every $p$ .

Milestones 2 and 3 are results of Chen, Kuhn and Wiesemann that this paper cites and uses, not results it proves.

Significance

Theorem 3 says that the scenario-wise big-M rows of the Wasserstein chance constraint can be replaced by at most $\lfloor\epsilon N\rfloor$ rows per $p$ , with coefficients computed from the data. Remark 2 of the paper counts at least $((1-\epsilon)N - 1)P$ fewer rows than (17). The reduced form exposes a robust 0–1 substructure, and the paper adapts Atamtürk's valid inequalities for it. Its computational study reports order-of-magnitude speed-ups on stochastic transportation problems.

The results are proved on paper; none is machine-checked. The formalization would give a verified chain from the Wasserstein worst-case probability to a finite mixed-integer system. Milestone (3), the dual reformulation, is reusable for any distributionally robust chance constraint over a 1-Wasserstein ball around an empirical distribution, and (4b) for any polyhedral safety set. A second formalization layer would verify the order-statistic arguments (Lemma 1, Proposition 1) used throughout the quantile-strengthening literature.

Difficulty

The deterministic parts — Theorems 1–3 given (3) and (4b) — are finite case analyses on $z_i \in \{0,1\}$ , but each step relies on order statistics of the sample with ties. Lemma 1 rests on the paper's "without loss of generality, sort the distances", and a formal proof must handle ties and the floor $\lfloor\epsilon N\rfloor$ exactly. The paper's proof of Theorem 2 argues "at optimality"; for a statement about feasible sets that argument has to be replaced by an explicit choice of $z$ .

The hard step is (3). It exchanges a supremum over all Borel probability measures in a Wasserstein ball for a scalar system. The naive attempt, which perturbs $\mathbb P_N$ by moving mass from the samples to the nearest unsafe points, proves only one inclusion. The other needs strong duality for worst-case probabilities over transport balls, together with attainment of the dual infimum, which uses $\theta > 0$ .

Formalization scope

The ξ-space is a type E with [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] and its Borel σ-algebra. The norm is arbitrary, not the sup norm of Fin K → ℝ.
Each $b_p$ is a continuous linear functional (StrongDual ℝ E), so $\|b_p\|_*$ is its operator norm.
Decisions are Fin L → ℝ, with $a_p^\top x$ written a p ⬝ᵥ x. Indices are 0-based.
The worst-case probability is the published ModelRiskOT.WorstProb.worstProb with cost $\|u - v\|$ , valued in $[0,\infty]$ , around the published WassersteinDRO.Duality.empiricalDistribution.
The distance (1) is Metric.infDist.
Binaries are reals with $z_i \in \{0,1\}$ .
Order statistics sort the multiset of values, so ties count with multiplicity.

Standing assumptions and pins in every statement:

$N \ge 1$ , $\epsilon \in (0,1)$ and $\theta > 0$ (p. 645);
$P \ge 1$ and $b_p \neq 0$ , which (4) needs;
for every statement involving $M$ , one bound $|b_p^\top\xi_i + d_p - a_p^\top x|/\|b_p\|_* \le M$ over $x \in \mathcal X$ and all $i, p$ . This is Remark 1's (10), taken uniform in $i$ . It replaces the compactness of $\mathcal X$ assumed on p. 642, which is not assumed;
for the general- $\mathcal S$ milestone (3), $\mathcal S(x)$ open with nonempty complement.

In (17b), the printed "(5b)–(5d) and (5c)" is read as "(5b)–(5d) and (8c)", following the proof of Theorem 2 and (20b).

Several trivializing formalizations are ruled out:

an empty or degenerate ball: $N \ge 1$ makes $\mathbb P_N$ a probability measure inside its own ball;
a junk distance: the unsafe set is nonempty under the pins, so infDist never sees the empty set;
a junk order statistic: the rank $k$ is below $N$ ;
a fixed norm;
a relaxation of $z$ to $[0,1]$ , which is a different theorem.

Welcome contributions:

the 1-Wasserstein duality for indicator losses on empirical distributions;
order-statistic lemmas for the multiset sort;
the distance from a point to the complement of an open polyhedron in a normed space.

Selected references

N. Ho-Nguyen, F. Kılınç-Karzan, S. Küçükyavuz, D. Lee, Distributionally robust chance-constrained programs with right-hand side uncertainty under Wasserstein ambiguity, Mathematical Programming Ser. B 196 (2022) 641–672. https://doi.org/10.1007/s10107-020-01605-y
Z. Chen, D. Kuhn, W. Wiesemann, Data-driven chance constrained programs over Wasserstein balls, Operations Research (2024); preprint 2018. https://arxiv.org/abs/1809.00210
W. Xie, On distributionally robust chance constrained programs with Wasserstein distance, Mathematical Programming 186 (2021) 115–155. https://doi.org/10.1007/s10107-019-01445-5
J. Blanchet, K. Murthy, Quantifying distributional model risk via optimal transport, Mathematics of Operations Research 44 (2019) 565–600. https://doi.org/10.1287/moor.2018.0936
A. Atamtürk, Strong formulations of robust mixed 0–1 programming, Mathematical Programming 108 (2006) 235–250. https://doi.org/10.1007/s10107-006-0709-5

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Operations Research Optimal Transport Optimization+1·Captain: mikedeng1

Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes 5: Worst-Case Displacements √δ G_δ Grow Monotonically in δ Along Straight LinesResearch Paper

Why radius sensitivity matters

A distributionally robust decision evaluates a rule against every data distribution within a specified transport budget of a baseline law. The budget is a modeling choice: increasing it expresses more uncertainty about the data. A robust objective value can increase with that budget, but the value alone does not describe how the adverse distribution changes. For a fixed linear decision, Blanchet, Murthy, and Zhang identify the worst-case distribution and ask what happens to each observation as the budget increases. Their Theorem 7 gives a sample-wise answer in a small-radius regime: adverse observations move along straight lines, and their distance from the original observations grows monotonically.

This matters when a robust solution is interpreted through its adverse scenarios. The result says that two nearby ambiguity budgets do not produce unrelated worst-case data sets. They produce a coherent family of transported observations whose directions are determined by the sign of the loss derivative. The theorem is a structural property of an optimal-transport ambiguity set, beyond a bound on its optimal value. It appears in a paper that also develops dual formulas and iterative optimization schemes for the same model Blanchet, Murthy, and Zhang, 2022.

Model and notation

Let $X$ take values in $\mathbb R^d$ with baseline law $P_0$ . A decision $\beta$ belongs to a convex set $B\subseteq\mathbb R^d$ and incurs loss $\ell(\beta^\top X)$ . At each source point $x$ , a positive-definite matrix $A(x)$ defines the transport cost

c(x,x')=(x-x')^\top A(x)(x-x').

The matrix may depend on $x$ , so movement can be cheaper in different directions and at different observations. The model requires lower semicontinuity of $c$ and almost-sure lower and upper spectral bounds for $A(x)$ . A distribution $P$ is within the ambiguity ball of radius $\delta>0$ when it can be coupled with $P_0$ at expected cost at most $\delta$ . The worst-case objective is the supremum of $E_P[\ell(\beta^\top X)]$ over that ball. In Lean, the equivalent coupling formulation records a joint law $\pi$ with first marginal $P_0$ , second marginal $P$ , and cost at most $\delta$ .

The paper reduces the worst-case calculation to a dual objective $f_\delta(\beta,\lambda)$ , where $\lambda\ge0$ prices the transport budget. Its pointwise term $\ell_{\rm rob}(\beta,\lambda;x)$ is the supremum over a scalar $\gamma$ of the function $F(\gamma,\beta,\lambda;x)$ in display (7). The set of maximizing scalars is $\Gamma^*(\beta,\lambda;x)$ . A scalar maximizer $G_\delta(x)$ produces a transported observation

X^*_\delta=x+\sqrt\delta\,G_\delta(x)A(x)^{-1}\beta.

Assumptions 2–4 require a convex loss with controlled growth, a fourth moment of $P_0$ , bounded second derivatives of the loss, a nondegenerate loss derivative, and compactness of $B$ . The statement here also uses Assumption 5, local strong convexity with a positive-probability nondegeneracy condition. The paper’s proof of Theorem 7 uses that condition through its small-radius strong-convexity result Theorems 4 and 7.

Formalization targets

For each fixed $\beta\in B\setminus\{0\}$ , the goal asserts that there is a common threshold $\delta_1>0$ and a family $G_\delta$ such that the law of $X^*_\delta$ is the unique worst-case distribution for every $0<\delta<\delta_1$ . For any pair $0<\delta<\delta'<\delta_1$ , it asserts, $P_0$ -almost surely,

\begin{aligned} 0&<\sqrt\delta\,G_\delta<\sqrt{\delta'}\,G_{\delta'} &&\text{if }\ell'(\beta^\top X)>0,\\ \sqrt{\delta'}\,G_{\delta'}&<\sqrt\delta\,G_\delta<0 &&\text{if }\ell'(\beta^\top X)<0,\\ G_\delta&=0 &&\text{if }\ell'(\beta^\top X)=0. \end{aligned}

Consequently,

\|X^*_\delta-X\|\le\|X^*_{\delta'}-X\|.

The goal keeps the uniqueness and primal attainment clauses together with these comparisons. The milestones follow the paper’s prerequisites: the optimizer and curvature bounds of Proposition 9, the unique worst-case law of Theorem 6(e), uniqueness of the dual minimizer at small radius, and the sign of the pointwise maximizer. Each has its own statement so solvers can use it elsewhere.

What the result establishes

The theorem gives a consistent interpretation of changes in the ambiguity radius. If the loss increases locally in the decision direction, an adverse observation moves farther in that direction; if the loss decreases, it moves farther in the opposite direction; at a stationary score it stays put. The corresponding displacement norm is monotone for each radius pair, almost surely. This is stronger than monotonicity of the worst-case objective value: it describes the geometry of a family of optimizers, not only their values.

The paper proves the result in §5.4. The work of this mission is to formalize its hypotheses, optimality claims, and sample-wise inequalities in Lean. The published ModelRiskOT.Duality coupling definitions already give a reusable primal interface; this mission adds the state-dependent quadratic cost, the scalar dual model, its small-radius regions, and the comparative-statics statements. The proof obligations remain open in this draft. Their formalization would also make the dual optimizer bounds and the deterministic transport representation reusable in later optimal-transport DRO developments.

Where the mathematical difficulty sits

The worst-case value being monotone in $\delta$ does not force a particular worst-case distribution, or each observation in one, to move monotonically. The theorem needs a unique worst-case law and a coherent choice of pointwise maximizers as $\delta$ varies. The scalar maximizer depends on both the radius and the dual price, while the optimal dual price itself changes with the radius. The paper’s small-radius conditions keep this joint variation controlled. Without uniqueness, two optimal laws could differ even at the same radius, and a sample-wise comparison would have no fixed object to compare.

The result also involves several layers of almost-sure reasoning. Spectral bounds on $A(x)$ hold only outside a null set; the relevant maximizer and its transported law must be measurable enough to define pushforwards and expectations. These conditions must be carried through the theorem without silently replacing an almost-sure claim by a pointwise one.

Formalization scope

Vectors use EuclideanSpace ℝ (Fin d), and the baseline is a Borel probability measure. The decision set is convex and compact and is explicitly nonempty. Assumption 1 gives positive definiteness of $A(x)$ everywhere, lower semicontinuity of the cost, and almost-sure uniform spectral bounds. The loss satisfies Assumptions 2–5. The fourth moment in Assumption 2 and the bounds on $\ell''$ control the integrals used by the dual theory. A positive ambiguity radius is required whenever its square root appears.

The dual terms $\ell_{\rm rob}$ and $f_\delta$ use extended-real values, since the pointwise supremum can be infinite. The worst-case value is a supremum over feasible couplings, with signed loss integrated using the published extended-real integral. A graph coupling witnesses attainment, and every optimal coupling must have the same second marginal. This ties the family $G_\delta$ to the actual worst-case distribution rather than an arbitrary monotone scalar field. Almost-everywhere measurability of $G_\delta$ and the transport map is part of the conclusion.

The radius $\delta$ varies inside the goal. The threshold $\delta_1$ is chosen before any radius pair, and the comparisons are stated almost surely for each pair. The paper prints Theorem 7 under Assumptions 1–4 while referring to $\delta_1$ and invoking Theorem 4, which uses Assumption 5; this mission includes Assumption 5 explicitly. It also reads the singleton clause of Theorem 6(e) almost surely, because its threshold is an essential supremum, and uses a non-strict lower curvature bound in Proposition 9(b), as its proof permits. Contributions on the small-radius dual theory, measurable maximizers, primal attainment, and the radius comparisons are all within scope.

Selected references

J. Blanchet, K. Murthy, and F. Zhang, Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes, Mathematics of Operations Research 47(2), 2022. arXiv:1810.02403v3.
J. Blanchet and K. Murthy, Quantifying Distributional Model Risk via Optimal Transport, Mathematics of Operations Research 44(2), 2019. arXiv:1604.01446v2.

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Combinatorics Complexity Theory Graph Theory+1·Captain: mikedeng1

Algebraic Approach to Promise Constraint Satisfaction 2: Pol(K_k, K_{2k−1}) Has No Olšák Function, So It Maps by a Minion Homomorphism to Pol(H₂, H_K)Research Paper

Approximate graph colouring

Given a graph that is promised to be $k$ -colourable, how many colours are needed to find a proper colouring in polynomial time? The decision version asks to distinguish $k$ -colourable graphs from graphs that are not even $c$ -colourable. It has been studied since Garey and Johnson (1976) and is conjectured to be NP-hard for all fixed $3 \le k \le c$ . This is the approximate graph colouring problem, and it is one of the motivating examples of promise constraint satisfaction problems (PCSPs).

Timeline of the NP-hardness results without extra assumptions, as surveyed in Example 2.9 and §6.2 of the paper:

2000–2004, Khanna–Linial–Safra and Guruswami–Khanna: 3 versus 4 colours.
2013, Huang (cited as [Hua13] in the paper): $k$ versus $2^{\Omega(k^{1/3})}$ colours for large $k$ .
2016, Brakensiek–Guruswami: $k$ versus $2k - 2$ colours for every $k \ge 3$ , via polymorphisms.
2018/2019, Barto–Bulín–Krokhin–Opršal (this paper): $k$ versus $2k - 1$ colours for every $k \ge 3$ , in particular 3 versus 5. The argument is an instance of a general algebraic theory of PCSPs.

The method also applies to the question of 5-cycle versus 3-colouring, which Brakensiek and Guruswami had raised as an open case.

Setting

A relational structure $\mathbf A$ consists of a finite domain $A$ and relations $R^{\mathbf A} \subseteq A^{\mathrm{ar}(R)}$ , one for each symbol $R$ of a signature. A PCSP template is a pair $(\mathbf A, \mathbf B)$ of structures with the same signature and a homomorphism $\mathbf A \to \mathbf B$ . Write $E_k = \{0, \dots, k-1\}$ .

$\mathbf K_k = (E_k; \neq_k)$ is the complete graph: one binary relation $\{(a, b) \mid a \ne b\}$ .
$\mathbf H_k = (E_k; \mathrm{NAE}_k)$ carries the ternary not-all-equal relation $E_k^3 \setminus \{(a, a, a)\}$ ; $\mathrm{PCSP}(\mathbf H_k, \mathbf H_c)$ is approximate colouring of 3-uniform hypergraphs.

An $n$ -ary polymorphism from $\mathbf A$ to $\mathbf B$ is a map $f : A^n \to B$ that, applied coordinatewise to any $n$ tuples of $R^{\mathbf A}$ , returns a tuple of $R^{\mathbf B}$ . For $\mathbf K_k$ and $\mathbf K_c$ these are exactly the $c$ -colourings of the power $\mathbf K_k^n$ . The set $\mathrm{Pol}(\mathbf A, \mathbf B)$ of polymorphisms of all arities $n \ge 1$ is closed under minors: $f(x_1, \dots, x_n) = g(x_{\pi(1)}, \dots, x_{\pi(m)})$ for $\pi : [m] \to [n]$ . A minion is any nonempty set of functions of arities $n \ge 1$ that is closed under minors. A minion homomorphism $\xi : \mathscr M \to \mathscr N$ preserves arities and commutes with minors. Write $\mathscr H_K = \mathrm{Pol}(\mathbf H_2, \mathbf H_K)$ .

An Olšák function is a 6-ary function $o$ with

o(x, x, y, y, y, x) = o(x, y, x, y, x, y) = o(y, x, x, x, y, y) \quad \text{for all } x, y .

The free structure $\mathbf F_{\mathscr M}(\mathbf A)$ has the $|A|$ -ary members of $\mathscr M$ as its domain and relations defined by minor identities that follow the tuples of $\mathbf A$ .

Formalization targets

Goal: the algebraic core of Theorem 6.5

For every $k \ge 3$ ,

\exists\, K \ge 2:\quad \text{there is a minion homomorphism } \mathrm{Pol}(\mathbf K_k, \mathbf K_{2k-1}) \to \mathrm{Pol}(\mathbf H_2, \mathbf H_K).

$K$ is left unspecified, as in the paper. This is the statement from which the paper deduces that $k$ versus $2k - 1$ colouring is NP-hard.

Milestones

Example 3.5: the Olšák minor condition is not satisfied in $\mathscr H_k$ for any $k \ge 2$ .
Lemma 4.4: homomorphisms $\mathbf F_{\mathscr M}(\mathbf A) \to \mathbf B$ correspond one-to-one to minion homomorphisms $\mathscr M \to \mathrm{Pol}(\mathbf A, \mathbf B)$ .
Theorem 6.2: for a minion $\mathscr M$ on finite sets,

\bigl(\exists K \ge 2,\ \exists\ \text{minion homomorphism } \mathscr M \to \mathscr H_K\bigr) \iff \mathscr M \text{ contains no Olšák function}.

Lemma 6.4: $\mathrm{Pol}(\mathbf K_k, \mathbf K_{2k-1})$ contains no Olšák function, $k \ge 3$ .

One further item, not a milestone, shows the bound is sharp. Proposition 10.1 says that $\mathrm{Pol}(\mathbf K_k, \mathbf K_{2k})$ contains an Olšák function.

Significance

Theorem 6.2 is a criterion stated entirely inside the minion. Together with the reduction theory of the paper (a minion homomorphism $\mathrm{Pol}(\mathbf A, \mathbf B) \to \mathrm{Pol}(\mathbf A', \mathbf B')$ gives a log-space reduction from $\mathrm{PCSP}(\mathbf A', \mathbf B')$ to $\mathrm{PCSP}(\mathbf A, \mathbf B)$ ) and the NP-hardness of approximate hypergraph colouring (Dinur, Regev and Smyth 2005), it shows that every finite template without an Olšák polymorphism has an NP-hard PCSP (Corollary 6.3). Lemma 6.4 then settles the 3 versus 5 colouring case and improves the $2k - 2$ bound to $2k - 1$ for every $k \ge 3$ . Proposition 10.1 shows that this criterion cannot go beyond $2k - 1$ .

The results are proved in the paper. None of them has a machine-checked proof that this mission knows of. This mission produces a Lean statement of the minion side of the argument. That includes the free-structure correspondence (Lemma 4.4), which every later application of the theory uses, and Theorem 6.2 for arbitrary finite minions. A finished formalization would include the clique argument behind Lemma 6.4 and would check the key hardness inequality $c \ge 2k - 1$ of the paper by machine.

Difficulty

Lemma 6.4 is a statement about every $(2k-1)$ -colouring of $\mathbf K_k^6$ that is constant on certain triples of vertices. The power graph has $k^6$ vertices, so exhaustive search is out of reach for general $k$ , and case checks for $k = 3$ do not give the general claim. Showing that $2k - 1$ colours do not suffice requires an explicit obstruction, uniform in $k$ , in the graph obtained by identifying those triples; the identification creates edges that are not present in $\mathbf K_k^6$ , and keeping track of them is the combinatorial core. The bound is tight (Proposition 10.1), so no argument that would also apply to $2k$ colours can work.

In Theorem 6.2 the direction from "no Olšák function" to a minion homomorphism must build a minion homomorphism into $\mathscr H_K$ out of nothing but the absence of a single identity. That direction depends on Lemma 4.4, which relates maps defined on functions of a single arity to minion homomorphisms defined on all arities; in Lean the bookkeeping of enumerations, minors and dependent arities is substantial.

Formalization scope

The development reuses the published relational-structure layer PCSPBLPAff_Symmetric_Setting (RelStruct, IsHom, IsPolymorphism, IsPromiseTemplate). Domains $E_k$ are Fin k. $\mathbf K_k$ and $\mathbf H_k$ have one-symbol signatures of arity 2 and 3. Polymorphisms apply $f$ to the columns of an $n \times \mathrm{ar}(R)$ matrix whose rows lie in $R^{\mathbf A}$ , which is the paper's matrix transposed. A minion is a family mem n of $n$ -ary functions (Fin n → A) → B with no nullary members, nonempty and closed under the minors fun x => g (x ∘ π). A minion homomorphism is a family of maps that preserves membership and minors on members, with unconstrained values outside the minion, so Lemma 4.4's "one-to-one" is stated up to agreement on the minion. Theorem 6.2 assumes the minion's sets are finite, which is the paper's standing convention (Definition 2.20). The free structure takes an enumeration $e : \mathrm{Fin}\ n \simeq A$ of the generating domain and quantifies the enumeration of each relation existentially. In the goal, $k \ge 3$ makes $2k - 1$ free of natural-number truncation.

The codomain of the goal is exactly $\mathrm{Pol}(\mathbf H_2, \mathbf H_K)$ with $K \ge 2$ . A minion homomorphism into an arbitrary or existentially chosen minion, or into $\mathscr H_K$ with $K \le 1$ , would be trivial and is ruled out by the statement.

Not formalized. The NP-hardness conclusions of Theorem 6.5 and Corollary 6.3 are left out, and so are the log-space reduction of Theorem 3.1 and the cited hardness of approximate hypergraph colouring (Theorem 5.23, [DRS05]). There is no complexity-theoretic substrate in Mathlib for them. Lemma 6.6 ( $\mathrm{Pol}(\mathbf C_5, \mathbf K_3)$ ) is a natural follow-up item. Contributions are welcome on every milestone. Lemma 4.4 and the minion layer can be reused beyond this mission, for any minion-homomorphism argument in the PCSP literature.

Selected references

L. Barto, J. Bulín, A. Krokhin, J. Opršal, Algebraic approach to promise constraint satisfaction, arXiv:1811.00970v3, 2019; J. ACM 68(4), 2021. https://arxiv.org/abs/1811.00970v3
J. Brakensiek, V. Guruswami, New hardness results for graph and hypergraph colorings, CCC 2016. https://doi.org/10.4230/LIPIcs.CCC.2016.14
I. Dinur, O. Regev, C. Smyth, The hardness of 3-uniform hypergraph coloring, Combinatorica 25(5), 2005. https://doi.org/10.1007/s00493-005-0032-4
M. Olšák, The weakest nontrivial idempotent equations, Bull. London Math. Soc. 49(6), 2017. https://doi.org/10.1112/blms.12097
S. Khanna, N. Linial, S. Safra, On the hardness of approximating the chromatic number, Combinatorica 20(3), 2000. https://doi.org/10.1007/s004930070013

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Optimization·Captain: mikedeng1

A Unified Convergence Analysis of Block Successive Minimization Methods for Nonsmooth Optimization I: Every Limit Point of the SUM Algorithm Is a Stationary PointResearch Paper

Motivation

Many algorithms in signal processing, statistics and machine learning share one template: replace a hard objective $f$ by a simpler function $u(\cdot,y)$ that lies above $f$ and touches it at the current point $y$ , minimise that surrogate, and repeat. The expectation–maximisation (EM) algorithm, difference-of-convex (DC) programming, proximal minimisation and many majorisation–minimisation schemes are of this form. Razaviyayn, Hong and Luo (arXiv:1209.2385; SIAM J. Optim. 23 (2013) 1126–1153) call the single-block version Successive Upper-bound Minimization (SUM) and give a short set of conditions on the surrogate under which every limit point of the iterates is a stationary point, without convexity or smoothness of $f$ .

Their single-block result, Theorem 1, is the base case of the paper. The block versions (BSUM, and block successive convex approximation with Armijo steps) are the subjects of the next two missions in this series. Earlier convergence results for related schemes were weaker: the inner approximation algorithm of Marks and Wright (1978), cited on p. 5–6 of the paper, guarantees stationarity only if the whole sequence converges, and only for smooth objectives.

Setting

Let $\mathcal X\subseteq\mathbb R^m$ be a closed convex set and $f:\mathbb R^m\to\mathbb R$ an objective whose relevant values are those on $\mathcal X$ (the paper assumes $\operatorname{dom} f=\mathcal X$ ). The problem is

\min_x\ f(x)\quad\text{s.t.}\quad x\in\mathcal X. \tag{2}

The directional derivative of a function $h$ at $x$ in direction $d$ is the lower limit

h'(x;d)=\liminf_{\lambda\downarrow 0}\frac{h(x+\lambda d)-h(x)}{\lambda},

an extended real number. A point $z$ is a stationary point of (2) if $z\in\mathcal X$ and $f'(z;d)\ge 0$ for every $d$ with $z+d\in\mathcal X$ ; the set of stationary points is $\mathcal X^*$ .

An approximation function $u:\mathbb R^m\times\mathbb R^m\to\mathbb R$ satisfies Assumption 1 if

(A1) $u(y,y)=f(y)$ for all $y\in\mathcal X$ ;
(A2) $u(x,y)\ge f(x)$ for all $x,y\in\mathcal X$ ;
(A3) $u'(x,y;d)\big|_{x=y}=f'(y;d)$ for all $y\in\mathcal X$ and all $d$ with $y+d\in\mathcal X$ , the derivative being taken in the first argument only;
(A4) $u$ is continuous in $(x,y)$ .

The SUM algorithm starts from $x^0\in\mathcal X$ and, for $r=0,1,2,\dots$ , sets $x^{r+1}$ to an arbitrary element of $\arg\min_{x\in\mathcal X}u(x,x^r)$ . A limit point of the iterates is the limit of some subsequence $x^{r_j}$ .

Formalization targets

Goal: Theorem 1 (p. 7)

Under Assumption 1, every limit point $z$ of the iterates generated by the SUM algorithm is a stationary point of (2):

z\in\mathcal X\quad\text{and}\quad f'(z;d)\ge 0\qquad\forall\, d\in\mathbb R^m\text{ with } z+d\in\mathcal X .

The theorem asserts nothing about existence of limit points or of minimisers of the subproblems; it holds for every run the algorithm can produce.

Milestones (proof of Theorem 1, pp. 7–8)

Descent, (10)–(11): $f(x^{r+1})\le u(x^{r+1},x^r)\le u(x^r,x^r)=f(x^r)$ for all $r$ , hence $f(x^0)\ge f(x^1)\ge\cdots$ .
Limit minimality: a limit point $z$ lies in $\mathcal X$ and $u(z,z)\le u(x,z)$ for all $x\in\mathcal X$ .
First-order condition: if $z\in\mathcal X$ minimises $u(\cdot,z)$ over $\mathcal X$ then $u'(x,z;d)\big|_{x=z}\ge 0$ for all $d$ with $z+d\in\mathcal X$ .

Companion statements

Corollary 1 (p. 8): if the level set $\mathcal X^0=\{x\mid f(x)\le f(x^0)\}$ is compact, then $\lim_{r\to\infty} d(x^r,\mathcal X^*)=0$ .
Proposition 1 (p. 6): if $f=f_0+f_1$ with $f_0$ continuously differentiable and $f_1$ directionally differentiable on $\mathcal X$ , and $u=u_0+f_1$ with $u_0$ continuously differentiable, $u_0(y,y)=f_0(y)$ and $u_0(x,y)\ge f_0(x)$ on $\mathcal X$ , then (A1)–(A3) hold.

Significance

Theorem 1 reduces the convergence analysis of an entire family of algorithms to checking four properties of a surrogate. Proposition 1 makes (A3) checkable in the common composite case "smooth plus nonsmooth": majorise only the smooth part and keep the nonsmooth part unchanged. Corollary 1 turns the limit-point statement into convergence of the whole sequence to the stationary set when the level set is compact. The block-coordinate results of the same paper (Theorems 2–4) reuse the structure of this argument block by block.

The results are proved on paper; to our knowledge none of them has a machine-checked proof. A formal development would supply a reusable account of extended-real lower directional derivatives on convex sets, the first-order necessary condition for constrained minimisers in that generality, and the limit-point argument for descent methods with a continuous surrogate, all of which recur in the analysis of block coordinate and majorisation–minimisation methods.

Difficulty

The argument is short, and the difficulty lies in its precise use of the hypotheses. The surrogate inequality must be passed to the limit along a subsequence, and only joint continuity of $u$ (A4) on $\mathcal X\times\mathcal X$ allows this: $f$ itself is not assumed continuous, so the natural first idea of passing $f(x^{r_j})$ to the limit is not available. The derivative is a liminf in the extended reals, so it may be $\pm\infty$ , and the comparison between $u$ and $f$ is made only through (A3) at the single point $x=y$ and only in feasible directions. For Proposition 1, (4)–(5) give only a one-sided inequality between the derivatives of $u_0(\cdot,y)$ and $f_0$ at $y$ , because $y$ may lie on the boundary of $\mathcal X$ ; equality needs an argument at interior points of a feasible segment and the continuity of the derivatives.

Formalization scope

The space is EuclideanSpace ℝ (Fin m) (the 2-norm of p. 4). The directional derivative is the published TsengBCD.Stationary.dirDeriv (Tseng 2001), an EReal-valued liminf over $\lambda\downarrow 0$ , used as a reference item. The definitions module BSUMConv.SUM.Setting provides IsStationaryOn, stationarySet, AssumptionA1–AssumptionA4, Assumption1, the run predicate IsSUMRun and the level set levelSet0.

Conventions committed to:

$f$ and $u$ are real functions on the whole space; the page's $\operatorname{dom} f=\mathcal X$ is reflected by testing only feasible directions and by intersecting the level set $\mathcal X^0$ with $\mathcal X$ .
(A4) is continuity of $u$ on $\mathcal X\times\mathcal X$ , the only domain on which the page uses $u$ .
A SUM run is a predicate on sequences: $x^0\in\mathcal X$ and each $x^{r+1}$ is some minimiser of $u(\cdot,x^r)$ over $\mathcal X$ . Existence of minimisers is not assumed.
"Limit point" is a cluster point of the sequence (MapClusterPt), not the limit of a convergent sequence.
$\lim_r d(x^r,\mathcal X^*)=0$ is stated as "for every $\varepsilon>0$ , eventually some stationary point is within $\varepsilon$ of $x^r$ ", which avoids a distance to a possibly empty set.
In Proposition 1, "continuously differentiable" means $C^1$ on the whole space (jointly in $(x,y)$ for $u_0$ ), and "the directional derivative of $f_1$ exists at every point $x\in\mathcal X$ " means a finite one-sided limit in every feasible direction.
Closedness and convexity of $\mathcal X$ , standing assumptions of (2), are hypotheses of Theorem 1, Corollary 1 and Proposition 1; the milestones carry only the ones their step uses.
The text of (10) attributes step (i) to (A1) and the final equality to (A2) and writes $x^{t+1}$ ; the formal statements use the hypotheses where they are actually needed.

The statements cannot be trivialised: Assumption 1 is satisfiable (for instance by the proximal surrogate $u(x,y)=f(x)+\tfrac1{2c}\|x-y\|^2$ of §VIII-B for suitable $f$ ), stationarity is a genuine condition (it fails for $\|x\|$ at any $x\neq 0$ on $\mathbb R^m$ ), and Theorem 1 is about every cluster point, not about runs assumed convergent. Contributions welcome: proofs of the three milestones, of Theorem 1 from them, and of the two companion statements; general lemmas about dirDeriv (constants, sums with a function having a finite one-sided derivative, first-order conditions on convex sets) are reusable across this series.

Selected references

M. Razaviyayn, M. Hong, Z.-Q. Luo, A unified convergence analysis of block successive minimization methods for nonsmooth optimization, arXiv:1209.2385v1, 2012; SIAM J. Optim. 23(2) (2013) 1126–1153. https://arxiv.org/abs/1209.2385 , https://doi.org/10.1137/120891009
P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization, J. Optim. Theory Appl. 109 (2001) 475–494. https://doi.org/10.1023/A:1017501703105
B. R. Marks, G. P. Wright, A general inner approximation algorithm for nonconvex mathematical programs, Oper. Res. 26(4) (1978) 681–683. https://doi.org/10.1287/opre.26.4.681
A. P. Dempster, N. M. Laird, D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc. B 39 (1977) 1–38. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x

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Combinatorics Operations Research Optimal Transport+1·Captain: mikedeng1

Computational Optimal Transport VI: On Termination the Auction Algorithm Returns an Assignment Whose Cost Is Within nε of the OptimumTextbook

Motivation

The optimal assignment problem asks for a one-to-one matching of $n$ points to $n$ objects of least total cost. It is the discrete optimal transport problem between two uniform measures with the same number of atoms, and by the Birkhoff–von Neumann theorem it coincides with the Kantorovich linear program on that instance (Peyré and Cuturi, Computational Optimal Transport, Proposition 2.1). Assignment solvers are therefore the inner routine of exact discrete optimal transport, of matching-based estimators in statistics and of many combinatorial optimization pipelines.

The auction algorithm was introduced by Bertsekas in 1981 (Bertsekas 1981) and refined with ε-scaling by Bertsekas and Eckstein in 1988 (Bertsekas & Eckstein 1988). It is a dual coordinate method with an economic reading: unassigned points bid for objects, prices of contested objects move, and an object changes hands when outbid. Its practical appeal is that each iteration is local and cheap, and that it parallelizes well. Peyré and Cuturi present it in §3.7 of their monograph (pp. 417–422) as an alternative use of the machinery of C-transforms, and as a precursor to the Sinkhorn algorithm of Chapter 4.

Setting

Fix $n\ge 2$ , write $[\![n]\!]=\{1,\dots,n\}$ , and let $\mathbf C\in\mathbb R^{n\times n}$ be a cost matrix: $\mathbf C_{i,j}$ is the cost of assigning point $i$ to object $j$ . A permutation $\sigma$ of $[\![n]\!]$ is an assignment, of cost $\sum_i \mathbf C_{i,\sigma_i}$ .

A dual vector (price vector) $\mathbf g\in\mathbb R^n$ is indexed by the objects. Its $\bar{\mathbf C}$ -transform is $(\mathbf g^{\bar{\mathbf C}})_i=\min_j \mathbf C_{i,j}-\mathbf g_j$ , the best adjusted cost available to point $i$ ; a pair $(\mathbf f,\mathbf g)$ is dual feasible if $\mathbf f_i+\mathbf g_j\le \mathbf C_{i,j}$ for all $i,j$ .

The algorithm maintains a triplet $(S,\xi,\mathbf g)$ : a set $S\subseteq[\![n]\!]$ of assigned points, a partial assignment vector $\xi$ (an injective map from $S$ to $[\![n]\!]$ ) and a dual vector $\mathbf g$ . The state satisfies ε-complementary slackness (ε-CS) if

\forall i\in S,\qquad \mathbf C_{i,\xi_i}-\mathbf g_{\xi_i}\le \varepsilon+\min_j\big(\mathbf C_{i,j}-\mathbf g_j\big).

One iteration picks a point $i\notin S$ , a lowest adjusted cost index $j^1_i\in\operatorname{argmin}_j \mathbf C_{i,j}-\mathbf g_j$ and a second lowest $j^2_i\in\operatorname{argmin}_{j\neq j^1_i}\mathbf C_{i,j}-\mathbf g_j$ , and performs

\mathbf g_{j^1_i}\leftarrow \mathbf C_{i,j^1_i}-\big(\mathbf C_{i,j^2_i}-\mathbf g_{j^2_i}\big)-\varepsilon, \tag{3.9}

removes from $S$ the point $i'$ with $\xi_{i'}=j^1_i$ (if there is one), sets $\xi_i=j^1_i$ and adds $i$ to $S$ . The algorithm starts from $S=\emptyset$ , $\mathbf g=0_n$ , and terminates when $S=[\![n]\!]$ .

Formalization targets

Goal: Proposition 3.9 (p. 422)

For every $\varepsilon>0$ and every run of the algorithm that terminates, the final $\xi$ is a permutation and

\sum_i \mathbf C_{i,\xi_i}\;\le\;\sum_i \mathbf C_{i,\sigma_i}+n\varepsilon\qquad\text{for every permutation }\sigma.

The goal fixes no particular tie-breaking rule and no order in which unassigned points bid: it holds for all of them.

Milestones

(3.8), p. 419. If $\mathbf C_{i,\sigma_i}-\mathbf g_{\sigma_i}=\min_j \mathbf C_{i,j}-\mathbf g_j$ for every $i$ , then $\sigma$ is an optimal assignment and $(\mathbf g^{\bar{\mathbf C}},\mathbf g)$ is an optimal dual pair.
Properties (b)–(c), pp. 419–420. At each iteration the size of $S$ does not decrease, some price decreases by at least $\varepsilon$ , and no price increases.
Proposition 3.7, p. 420. Every state of every run satisfies ε-CS.

Companion: Proposition 3.8 (p. 421), corrected

For $\mathbf C\ge 0$ , every run has at most $n\big(\lfloor\|\mathbf C\|_\infty/\varepsilon\rfloor+1\big)$ iterations. This is the termination statement; the goal does not depend on it.

Significance

Proposition 3.9 is the accuracy guarantee of the auction algorithm. With integer costs and $\varepsilon<1/n$ it yields an exactly optimal assignment, which is how the algorithm is used as an exact solver; combined with ε-scaling it gives the complexity bounds quoted in Remark 3.3. The argument is also the template for every ε-relaxed primal–dual method: an approximate optimality certificate maintained by local moves, turned into a global suboptimality bound by weak duality.

The results are classical and proved in the book and in Bertsekas's monographs; they are not open. What this mission adds is a machine-checked account of the algorithm with all of its nondeterminism (ties in the argmins, the order of bidders) quantified explicitly, together with the corrected iteration bound. To the best of the mission's prior-art search, no Lean formalization of the auction algorithm exists on the platform or in Mathlib, which has the assignment problem only implicitly (through permutations and doubly stochastic matrices).

Difficulty

The obvious argument for the goal sums the ε-CS inequality over $i$ and compares with an arbitrary permutation $\sigma$ . The sum telescopes only if the final $\xi$ is a bijection, so the proof must carry, through every iteration, the invariant that $\xi$ is injective on $S$ , and this invariant interacts with the removal of the displaced point in step 2: the set update has to remove exactly the point holding $j^1_i$ , which needs injectivity at the previous step.

Preserving ε-CS for the points that are not bidding relies on prices only ever decreasing, which in turn needs $\varepsilon\ge 0$ and the specific form of (3.9); preserving it for the bidder needs the second-best index $j^2_i$ to realize the minimum over $j\neq j^1_i$ , including in the presence of ties. The printed termination bound is false (see below), so the companion requires reconstructing the counting argument: an object that is still unassigned has never been bid on and keeps price $0$ , which caps the number of bids on any other object.

Formalization scope

$[\![n]\!]$ is Fin n; $\mathbf C$ is a Matrix (Fin n) (Fin n) ℝ; prices are Fin n → ℝ; $\min_j$ is the infimum over the finite nonempty index set.
A state is a structure with fields S : Finset (Fin n), ξ : Fin n → Fin n and g : Fin n → ℝ. Only the values of ξ on S are meaningful; injectivity on S is a property to be proved, not a field.
An iteration is a relation between consecutive states (AuctionStep), with the choice of bidder, of $j^1_i$ and of $j^2_i$ existential. A run of $T$ iterations is a sequence of states starting from $S=\emptyset$ , $\mathbf g=0$ ; it is terminated when $S=[\![n]\!]$ at step $T$ . The results hold for every run, whatever the tie-breaking.
Costs are the unnormalized sums $\sum_i \mathbf C_{i,\sigma_i}$ , as in the book's proof of Proposition 3.9. Under the $\frac1n$ normalization of (2.2) the gap $n\varepsilon$ becomes $\varepsilon$ .
Added hypotheses: $n\ge2$ in the goal (the second-best index requires it); $\varepsilon>0$ throughout; $\mathbf C\ge0$ in the corrected Proposition 3.8, which its proof uses.
Printed slips. Proposition 3.8 as printed ( $N=n\|\mathbf C\|_\infty/\varepsilon$ ) fails for $\mathbf C=0$ ; the companion states the bound the proof gives. Property (b) "can only increase" is stated as non-decrease; property (c) refers to an object index $j$ .
Trivialization ruled out. The run predicate forces the initial state, the exact update (3.9) and the set update of step 2 at every iteration, and the goal requires $S=[\![n]\!]$ at the end; a terminated run exists already for $n=2$ , so the goal is not vacuous.

Contributions welcome: the injectivity invariant of ξ on S as a reusable lemma; the milestones in any order; a proof of termination (the companion) and of the existence of a terminated run for every $n\ge2$ .

Selected references

G. Peyré and M. Cuturi, Computational Optimal Transport, Foundations and Trends in Machine Learning 11(5–6):355–607, 2019, §3.7. https://doi.org/10.1561/2200000073
D. P. Bertsekas, A new algorithm for the assignment problem, Mathematical Programming 21:152–171, 1981. https://doi.org/10.1007/BF01584237
D. P. Bertsekas and J. Eckstein, Dual coordinate step methods for linear network flow problems, Mathematical Programming 42:203–243, 1988. https://doi.org/10.1007/BF01589417
D. P. Bertsekas, Auction algorithms for network flow problems: A tutorial introduction, Computational Optimization and Applications 1:7–66, 1992. https://doi.org/10.1007/BF00247653

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Machine Learning Numerical Analysis Optimal Transport+1·Captain: mikedeng1

Computational Optimal Transport VIII: Sinkhorn's Scalings Contract by λ(K)² per Iteration in Hilbert's Projective MetricTextbook

Why Sinkhorn's algorithm needs a convergence rate

Sinkhorn's algorithm is the standard method for computing entropically regularized optimal transport between two histograms. It is used throughout machine learning, statistics, imaging and graphics because each iteration is two matrix–vector products followed by entrywise divisions, which parallelize on GPUs and differentiate easily. The same iteration, under the names matrix scaling, RAS or iterative proportional fitting, has been used since the 1930s in transportation planning, economics and contingency-table analysis. Whether the iteration converges, and how fast, determines how many iterations a practitioner must run and which stopping criteria are reliable.

This mission formalizes the global convergence analysis of Sinkhorn's algorithm as presented in Chapter 4 (§4.2, Remarks 4.12–4.14) of G. Peyré and M. Cuturi, Computational Optimal Transport (2019).

Timeline. Sinkhorn (1964) proved that a positive matrix can be scaled to be doubly stochastic, and Sinkhorn and Knopp (1967) extended this to prescribed marginals. Birkhoff (1957) and Samelson (1957) independently showed that a positive linear map is a strict contraction of the positive cone for Hilbert's projective metric, with an explicit ratio. Franklin and Lorenz (1989) combined the two to prove that Sinkhorn's scalings converge linearly in Hilbert's metric at rate $\lambda(K)^2$ . Linial, Samorodnitsky and Wigderson (1998) related the iteration to the permanent.

Setting

Fix $n, m \ge 1$ , a matrix $K \in \mathbb{R}^{n\times m}$ with positive entries (in the book, the Gibbs kernel $K_{i,j} = e^{-C_{i,j}/\varepsilon}$ ), and vectors $a \in \mathbb{R}^n$ , $b \in \mathbb{R}^m$ with positive entries. Products $\odot$ and quotients of vectors are entrywise. The regularized transport plan has the form $P = \mathrm{diag}(u)K\mathrm{diag}(v)$ , where the scalings $u, v$ solve

u \odot (Kv) = a, \qquad v \odot (K^\top u) = b. \tag{4.14}

Sinkhorn's algorithm alternately enforces each equation:

v^{(0)} = \mathbb{1}_m,\qquad u^{(\ell+1)} = \frac{a}{Kv^{(\ell)}},\qquad v^{(\ell+1)} = \frac{b}{K^\top u^{(\ell+1)}}. \tag{4.15}

For vectors with positive entries, Hilbert's projective metric is

d_{\mathcal H}(u,u') = \log\max_{i,j}\frac{u_i u'_j}{u_j u'_i},

which vanishes exactly when $u$ and $u'$ are proportional, and equals the variation seminorm $\|\log u - \log u'\|_{\mathrm{var}}$ with $\|f\|_{\mathrm{var}} = \max_i f_i - \min_i f_i$ . Birkhoff's constants of $K$ are

\eta(K) = \max_{i,j,k,\ell}\frac{K_{i,k}K_{j,\ell}}{K_{j,k}K_{i,\ell}},\qquad \lambda(K) = \frac{\sqrt{\eta(K)}-1}{\sqrt{\eta(K)}+1}.

Write $P^{(\ell)} = \mathrm{diag}(u^{(\ell)})K\mathrm{diag}(v^{(\ell)})$ and $P^\star = \mathrm{diag}(u^\star)K\mathrm{diag}(v^\star)$ for a positive solution $(u^\star, v^\star)$ of (4.14).

Formalization targets

Goal: Theorem 4.2 (Franklin–Lorenz)

For every $\ell$ ,

d_{\mathcal H}(v^{(\ell)}, v^\star) \le \lambda(K)^{2\ell}\, d_{\mathcal H}(v^{(0)}, v^\star),\qquad d_{\mathcal H}(u^{(\ell+1)}, u^\star) \le \lambda(K)^{2\ell+1}\, d_{\mathcal H}(v^{(0)}, v^\star),

so both distances tend to $0$ ; for $\ell \ge 1$ ,

d_{\mathcal H}(u^{(\ell)}, u^\star) \le \frac{d_{\mathcal H}(P^{(\ell)}\mathbb{1}_m, a)}{1-\lambda(K)^2},\qquad \max_{i,j}\big|\log P^{(\ell)}_{i,j} - \log P^\star_{i,j}\big| \le d_{\mathcal H}(u^{(\ell)},u^\star) + d_{\mathcal H}(v^{(\ell)},v^\star),

and for $\ell \ge 0$ the mirrored bound for $v^{(\ell)}$ with the half-step coupling $\mathrm{diag}(u^{(\ell+1)})K\mathrm{diag}(v^{(\ell)})$ . The book writes the rate as $O(\lambda(K)^{2\ell})$ ; the constants stated are those its proof yields.

Milestones, in attack order

(4.21): $d_{\mathcal H}(u,u') = \|\log u - \log u'\|_{\mathrm{var}}$ .
Remark 4.12: $d_{\mathcal H}$ is a distance on rays (triangle inequality; zero exactly on proportional vectors).
Invariances: $d_{\mathcal H}(v,v') = d_{\mathcal H}(v/v',\mathbb{1}) = d_{\mathcal H}(\mathbb{1}/v,\mathbb{1}/v')$ and invariance under positive diagonal scaling.
Theorem 4.1 (Birkhoff–Hopf): $d_{\mathcal H}(Kv,Kv') \le \lambda(K)\,d_{\mathcal H}(v,v')$ and $\lambda(K) < 1$ .
One half-step contracts by $\lambda(K)$ : $d_{\mathcal H}(u^{(\ell+1)},u^\star) \le \lambda(K)\,d_{\mathcal H}(v^{(\ell)},v^\star)$ , and the $v$ -analogue.
The display behind (4.23): $d_{\mathcal H}(u^{(\ell)},u^\star) \le d_{\mathcal H}(a, u^{(\ell)}\odot Kv^{(\ell)}) + \lambda(K)^2 d_{\mathcal H}(u^{(\ell)},u^\star)$ , and its mirror.

Significance

The result. Theorem 4.2 gives a global linear rate that holds from the first iteration, independent of the starting point, with a ratio computable from $K$ alone. Through (4.21) the rate transfers to the dual potentials $\varepsilon\log u^{(\ell)}$ , and the bounds (4.23) turn the marginal violations $P^{(\ell)}\mathbb{1}_m$ versus $a$ — which are computed anyway — into certified stopping criteria. Theorem 4.1 alone also yields a quantitative Perron–Frobenius theorem (Remark 4.13).

Formalizing it. All results here are classical and proved in the literature (Birkhoff 1957; Franklin–Lorenz 1989); Theorem 4.1 is quoted by the book without proof. To our knowledge neither Hilbert's projective metric on the positive cone, the Birkhoff–Hopf theorem, nor the convergence rate of Sinkhorn's algorithm has a machine-checked proof in Mathlib. The mission produces a reusable development of $d_{\mathcal H}$ and Birkhoff's contraction theorem, and a certified rate and stopping rule for the most widely used optimal-transport solver.

Difficulty

Milestones 1–3 and 5–6 are finite manipulations of maxima of ratios. The central difficulty is Theorem 4.1. The obvious argument — bound $(Kv)_i/(Kv')_i$ entrywise by $\max_k v_k/v'_k$ — only shows that $K$ is non-expansive ( $\lambda \le 1$ ); it loses the strict contraction entirely. The sharp ratio $(\sqrt{\eta}-1)/(\sqrt{\eta}+1)$ depends on how $K$ acts on the two vectors jointly, which the entrywise argument does not see. The book quotes the theorem without proof, so the argument has to be supplied from Birkhoff's original paper or later expositions.

Formalization scope

Lean namespace CompOT.Sinkhorn. Indices are $0$ -based (Fin n, Fin m); vectors are functions Fin n → ℝ, matrices Matrix (Fin n) (Fin m) ℝ, and $\odot$ and $/$ are the pointwise operations on functions. $d_{\mathcal H}$ , $\eta$ and $\|\cdot\|_{\mathrm{var}}$ use real iSup/iInf over the finite index sets. Every theorem assumes $0 < n$ , $0 < m$ and positive entries for $K$ , $a$ , $b$ and every vector passed to $d_{\mathcal H}$ , which excludes Lean's junk values for division by zero and logarithms of non-positive numbers. Sinkhorn's algorithm is a run predicate (IsSinkhornRun) stating the recursion (4.15) with $v^{(0)} = \mathbb{1}_m$ ; $u^{(0)}$ is not an iterate and is unconstrained. The solution $(u^\star, v^\star)$ is any positive solution of (4.14), taken as a hypothesis (it exists iff $\sum_i a_i = \sum_j b_j$ ); the book's histograms are probability vectors, but only positivity is used. $P^\star$ in (4.24) is $\mathrm{diag}(u^\star)K\mathrm{diag}(v^\star)$ , the solution of the regularized problem (4.2) by Proposition 4.3.

Deviations from the print, all disclosed in the items: the $O(\cdot)$ of (4.22) is replaced by the proof's explicit constants; the limit $(u^{(\ell)}, v^{(\ell)}) \to (u^\star, v^\star)$ is stated as $d_{\mathcal H} \to 0$ , since scalings are determined only up to $(ru, v/r)$ ; the second inequality of (4.23) is false as printed (its right side is $0$ for $\ell \ge 1$ ) and is stated with the half-step coupling the mirrored proof gives.

A definition of $d_{\mathcal H}$ without the logarithm, or as a maximum of single-entry ratios, would make the statements different theorems; the definition here is the book's, and (4.21) is a milestone checking it against the variation seminorm. A run predicate not tied to $K$ , $a$ , $b$ , or an unsatisfiable one, is ruled out: the predicate is the recursion itself and is inhabited for every $K$ , $a$ , $b$ .

Contributions welcome: the metric properties of $d_{\mathcal H}$ (reusable for Perron–Frobenius and nonlinear consensus), a proof of the Birkhoff–Hopf theorem, and positivity lemmas for Sinkhorn iterates.

Selected references

G. Peyré, M. Cuturi, Computational Optimal Transport, Foundations and Trends in Machine Learning 11(5–6):355–607, 2019. https://doi.org/10.1561/2200000073 (§4.2, pp. 431–443)
J. Franklin, J. Lorenz, On the scaling of multidimensional matrices, Linear Algebra and its Applications 114/115:717–735, 1989. https://doi.org/10.1016/0024-3795(89)90490-4
G. Birkhoff, Extensions of Jentzsch's theorem, Transactions of the AMS 85(1):219–227, 1957. https://doi.org/10.1090/S0002-9947-1957-0087058-6
H. Samelson, On the Perron–Frobenius theorem, Michigan Mathematical Journal 4(1):57–59, 1957. https://doi.org/10.1307/mmj/1028990177
R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Annals of Mathematical Statistics 35(2):876–879, 1964. https://doi.org/10.1214/aoms/1177703591
N. Linial, A. Samorodnitsky, A. Wigderson, A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents, STOC 1998. https://doi.org/10.1145/276698.276880

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Convex Optimization Optimization·Captain: mikedeng1

The Direct Extension of ADMM for Multi-block Convex Minimization Problems is Not Necessarily Convergent III: With a Strongly Convex Objective and β = 1 the Extended ADMM Can Still DivergeResearch Paper

Motivation

The alternating direction method of multipliers (ADMM) is a standard method for convex problems whose objective splits into two blocks coupled by a linear constraint (Gabay & Mercier 1976; Boyd et al. 2011). Many models in imaging, statistics and sparse or low-rank recovery have three or more blocks, and the scheme practitioners reach for first is the direct extension of ADMM: minimize the augmented Lagrangian over each block in turn, then update the multiplier. For years it was unknown whether this scheme converges in general.

Chen, He, Ye and Yuan (Math. Program., 2016) settled the question negatively: on a three-variable linear system with zero objective, the direct extension diverges for every penalty parameter (their Theorem 3.1). Their Section 4.1 then asks whether strong convexity of the objective rescues convergence. Han and Yuan (J. Optim. Theory Appl., 2012) had proved convergence for strongly convex objectives under a restriction on the penalty parameter determined by the strong convexity moduli. Theorem 4.1 shows that this restriction cannot simply be removed. This mission formalizes Theorem 4.1 and the three facts on p. 14 that support it.

Timeline:

1975–1976: Glowinski–Marrocco and Gabay–Mercier introduce two-block ADMM and prove its convergence.
2012: Han and Yuan prove convergence of the multi-block direct extension when every $\theta_i$ is strongly convex and $\beta$ is small enough.
2014 (authors' version), published 2016: Chen, He, Ye and Yuan give the first divergent example (Theorem 3.1) and its strongly convex variant (Theorem 4.1).

Setting

The three-block problem (1.1) is

\min\ \theta_1(x_1)+\theta_2(x_2)+\theta_3(x_3)\quad\text{s.t.}\quad A_1x_1+A_2x_2+A_3x_3=b,\quad x_i\in\mathcal X_i,

with $A_i\in\mathbb R^{p\times n_i}$ , $b\in\mathbb R^p$ , closed convex sets $\mathcal X_i\subseteq\mathbb R^{n_i}$ and convex functions $\theta_i:\mathbb R^{n_i}\to\mathbb R$ ; the solution set is assumed nonempty.

For a penalty parameter $\beta>0$ and a multiplier $\lambda\in\mathbb R^p$ , the augmented Lagrangian (1.6) is

\mathcal L_{\mathcal A}(x_1,x_2,x_3,\lambda)=\sum_{i=1}^3\theta_i(x_i)-\lambda^T(A_1x_1+A_2x_2+A_3x_3-b)+\frac\beta2\|A_1x_1+A_2x_2+A_3x_3-b\|^2 .

The direct extension of ADMM (1.5) starts from $(x_2^0,x_3^0,\lambda^0)$ and, for $k\ge0$ , sets $x_1^{k+1}$ , $x_2^{k+1}$ , $x_3^{k+1}$ to minimizers of $\mathcal L_{\mathcal A}$ over $\mathcal X_1,\mathcal X_2,\mathcal X_3$ in that order, each with the newest values of the other blocks, and then sets $\lambda^{k+1}=\lambda^k-\beta(A_1x_1^{k+1}+A_2x_2^{k+1}+A_3x_3^{k+1}-b)$ .

A function $f$ is $m$ -strongly convex if $f-\frac m2\|\cdot\|^2$ is convex.

The instance (4.1) is

\min\ 0.05x_1^2+0.05x_2^2+0.05x_3^2\quad\text{s.t.}\quad\begin{pmatrix}1&1&1\\1&1&2\\1&2&2\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=0,

with scalar $x_i$ , $\mathcal X_i=\mathbb R$ and $b=0$ . Its unique solution is $x=0$ . For $\beta=1$ the state after an iteration is $z=(x_2,x_3,\lambda_1,\lambda_2,\lambda_3)\in\mathbb R^5$ , and $M_{41}$ denotes the $5\times5$ iteration matrix that maps $z^k$ to $z^{k+1}$ .

Formalization targets

Goal: Theorem 4.1

For the model (1.1) with strongly convex objective, (1.5) is not necessarily convergent for all $\beta>0$ . Formally, (4.1) satisfies the standing assumptions, each $\theta_i$ is $\tfrac1{10}$ -strongly convex, and at $\beta=1$ there is a starting point $(x_2^0,x_3^0,\lambda^0)$ from which a run of (1.5) exists and

\text{no run }(x_1^k,x_2^k,x_3^k,\lambda^k)_{k\ge0}\text{ of (1.5) from it converges.}

The goal fixes the witness instance and $\beta=1$ , because those are what the paper computes. It does not fix the starting point.

Milestones

Fixed matrix mapping. Every run of (1.5) on (4.1) with $\beta=1$ satisfies $z^{k+1}=M_{41}z^k$ for all $k$ , with the explicit rational matrix $M_{41}$ .
Spectral radius. $\rho(M_{41})$ , the largest modulus of a complex eigenvalue, lies in $[1.00865,1.00875)$ , i.e. equals $1.0087$ to the four decimals of the page.
Diverging powers. Some real $z\in\mathbb R^5$ has $\|M_{41}^kz\|\to\infty$ .

Significance

Theorem 4.1 says that strong convexity of each $\theta_i$ , on its own, does not make the three-block direct extension of ADMM converge. Convergence results for strongly convex models therefore have to restrict $\beta$ , as Han and Yuan do, or modify the scheme. Examples of modified schemes are a shrunken multiplier step (discussed in §4.2 of the paper) and Gaussian back substitution (He, Tao & Yuan 2012). The example is also a concrete test case: any proposed sufficient condition for convergence has to exclude (4.1) at $\beta=1$ .

The theorem is proved in the paper, but only in outline: the iteration matrix is not printed, the spectral radius is reported as the outcome of "a simple calculation", and the construction of a divergent starting point is "omitted for succinctness". A formalization makes each step explicit and machine-checked. That means the exact iteration matrix, a certified enclosure of its spectral radius, and a real starting point whose iterates are unbounded. As far as is known, none of these results has been machine-checked.

Difficulty

There are three separate difficulties.

Reduction to a matrix. The method is defined by three minimizations, not by a formula, so the matrix recursion has to be derived from those minimizations.
Certifying the spectral radius. The spectral radius is a property of a degree-5 characteristic polynomial with large rational coefficients. Its dominant roots are a non-real conjugate pair of modulus about $1.00874$ , so the margin above $1$ is under one percent, and the rounding interval requested is narrow.
From eigenvalue to divergence. A spectral radius above $1$ does not by itself give a divergent real trajectory. The dominant eigenvectors are complex, and the starting vector must be real yet have a nonzero component along the expanding pair.

A natural first idea is to argue that strong convexity of the objective forces the iterates toward the unique solution. This fails because the multiplier and the coupling between blocks are not controlled by the objective's curvature once $\beta$ is large relative to the moduli.

Formalization scope

Vectors in $\mathbb R^n$ are Fin n → ℝ and matrices are Matrix (Fin p) (Fin n) ℝ. The Euclidean square norm in (1.6) is written $r\cdot r$ (⬝ᵥ), never Mathlib's sup norm. Norms appear otherwise only in " $\|\cdot\|\to\infty$ ", where every norm on $\mathbb R^5$ gives the same meaning. Blocks are separate fields, and multiplier components are 0-based. A run is a predicate on sequences: " $x^{k+1}=\operatorname{Argmin}$ " means " $x^{k+1}$ is a minimizer", and $x_1^0$ is never read. "Converges" means convergence of $(x_1^k,x_2^k,x_3^k,\lambda^k)$ in the product topology.

Strong convexity is Mathlib's StrongConvexOn Set.univ (1/10). The paper's "closed convex" $\theta_i$ are real-valued on all of $\mathbb R^{n_i}$ , hence continuous, so only convexity is stated.

The quantifier "for all $\beta>0$ " is read as "not for every $\beta>0$ ", witnessed at $\beta=1$ . The stronger reading, divergence for every $\beta$ , is false for (4.1), where the spectral radius is below $1$ for small $\beta$ .

Trivializations ruled out. The goal asserts the existence of a run, so non-convergence cannot hold vacuously. Runs are characterized by minimizer conditions rather than an argmin choice function with junk values. Spectral statements are over $\mathbb C$ rather than over real eigenvalues only; the only real eigenvalue of $M_{41}$ is $0$ .

What a complete development needs: first-order optimality for strictly convex scalar quadratics; existence and uniqueness of the subproblem minimizers; a certified root enclosure for a quartic with rational coefficients (via Rouché-type bounds, interval arithmetic or an explicit factorization over $\mathbb R$ ); and the passage from a dominant complex eigenpair to a real unbounded orbit. The last two are reusable for any linear-iteration divergence argument, including Theorem 3.1 of the same paper. Contributions of any of these lemmas, as separate theorems, are welcome.

Selected references

C. Chen, B. He, Y. Ye, X. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program. 155 (2016) 57–79. https://doi.org/10.1007/s10107-014-0826-5
D. Han, X. Yuan, A note on the alternating direction method of multipliers, J. Optim. Theory Appl. 155 (2012) 227–238. https://doi.org/10.1007/s10957-012-0003-z
D. Gabay, B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl. 2 (1976) 17–40. https://doi.org/10.1016/0898-1221(76)90003-1
S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn. 3 (2011) 1–122. https://doi.org/10.1561/2200000016

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Algebra Combinatorics Complexity Theory+1·Captain: mikedeng1

Algebraic Approach to Promise Constraint Satisfaction 4: No Finite D with 1-in-3 → D → NAE Has a Cyclic Polymorphism of Prime Arity p > 60|D|Research Paper

Motivation

A promise constraint satisfaction problem (PCSP) is given by a pair of relational structures $(\mathbf A,\mathbf B)$ with a homomorphism $\mathbf A\to\mathbf B$ : an instance is promised to be satisfiable in $\mathbf A$ , and it suffices to satisfy it in $\mathbf B$ . The paper of Barto, Bulín, Krokhin and Opršal (arXiv:1811.00970v3) develops the algebraic theory of PCSPs, in which the complexity of $\mathrm{PCSP}(\mathbf A,\mathbf B)$ is governed by its polymorphisms.

The most prominent example is 1-in-3 versus Not-All-Equal-SAT: given a system of ternary constraints that has an assignment making exactly one variable in each constraint true, find an assignment making the variables in each constraint not all equal. Brakensiek and Guruswami (arXiv:1704.01937) showed that this problem is solvable in polynomial time, although both 1-in-3-SAT and NAE-SAT are NP-complete on their own. Their algorithm, like every tractability result for PCSPs known when the paper was written, works by finding a tractable CSP template $\mathbf C$ with $\mathbf T\to\mathbf C\to\mathbf H_2$ (a "sandwich") and solving the CSP over $\mathbf C$ . For 1-in-3 vs NAE the template used is infinite: the integers with the relation $x+y+z=1$ .

Section 8 of the paper shows that this is unavoidable: no finite tractable template can be sandwiched, or more generally pp-constructed, in this way. This separates PCSPs from CSPs: infinite domains are necessary in the sandwich method.

Timeline: Feder and Vardi conjectured the CSP dichotomy (1998); Barto and Kozik characterised the borderline by cyclic polymorphisms (BK12, 2012); Bulatov and Zhuk proved the dichotomy (arXiv:1703.03021, arXiv:1704.01914, 2017); Brakensiek and Guruswami proved 1-in-3 vs NAE tractable (2018); Barto, Bulín, Krokhin and Opršal proved Theorem 8.1 (2019).

Setting

All structures here have one ternary relation. For a set $D$ and $R\subseteq D^3$ , $(D;R)$ is the structure with domain $D$ and relation $R$ . The 1-in-3 structure is $\mathbf T=(\{0,1\};\{(1,0,0),(0,1,0),(0,0,1)\})$ and the not-all-equal structure is $\mathbf H_2=(\{0,1\};\{0,1\}^3\setminus\{(0,0,0),(1,1,1)\})$ .

A homomorphism $h:(A;R_A)\to(B;R_B)$ is a map with $(h(a),h(b),h(c))\in R_B$ whenever $(a,b,c)\in R_A$ . A polymorphism of $(D;R)$ of arity $p$ is a map $s:D^p\to D$ such that, applied coordinatewise to any $p$ triples from $R$ , it yields a triple in $R$ . An operation $s:D^p\to D$ is cyclic if $s(a_1,a_2,\dots,a_p)=s(a_2,\dots,a_p,a_1)$ for all $a\in D^p$ (Definition 8.2).

The proof of Theorem 8.1 works in the following situation: $D$ is finite, $f:\mathbf T\to(D;R)$ and $g:(D;R)\to\mathbf H_2$ are homomorphisms, $p>60|D|$ is a prime and $s$ is a cyclic polymorphism of $(D;R)$ of arity $p$ . From $s$ it builds the $p^2$ -ary operation

t(X)=s\big(s(x_{11},\dots,x_{p1}),\dots,s(x_{1p},\dots,x_{pp})\big)

on $p\times p$ matrices $X=(x_{ij})$ , and studies zero-one matrices, evaluated in $D$ through $f$ : their area $\lambda(X)$ (the fraction of ones), $g$ -equivalence $X\sim Y\iff g(t(X))=g(t(Y))$ , tameness ( $X\sim 0_{p\times p}$ with $\lambda(X)<1/3$ , or $X\sim 1_{p\times p}$ with $\lambda(X)>1/3$ ), covers (three matrices with exactly one $1$ in each position), the line segments $\langle i\rangle$ (ones in the first $i$ positions in row-major order), and almost rectangles $[k,\dots,k,l,\dots,l]$ (columns of heights $k$ then $l$ , with step $0\le k-l\le 5|D|$ ).

Formalization targets

Goal: the algebraic core of Theorem 8.1

\mathbf T\xrightarrow{f}(D;R)\xrightarrow{g}\mathbf H_2,\ \ D\text{ finite},\ \ p\text{ prime},\ p>60|D|\ \Longrightarrow\ (D;R)\text{ has no cyclic polymorphism of arity }p.

Milestones, in proof order

Lemma 8.7: the three matrices of a cover are not all $g$ -equivalent.
Lemma 8.8: $t$ is cyclic in its $p^2$ row-major arguments.
Lemma 8.9: with $p^2=3q+1$ , $\langle 0\rangle\sim\cdots\sim\langle q\rangle\not\sim\langle q+1\rangle\sim\cdots\sim\langle 2q+1\rangle$ .
Lemma 8.10: every $\langle i\rangle$ , $0\le i\le p^2$ , is tame, and $0_{p\times p}\not\sim 1_{p\times p}$ .
Proposition 8.12: every almost rectangle is tame.
§8.4: there are $l_1<l_2$ in $(p/3-2|D|,\,p/3)$ with $s(1^{l_1}0^{p-l_1})=s(1^{l_2}0^{p-l_2})$ .

Significance

Combined with the theorem of Barto and Kozik (a finite CSP template whose CSP is not NP-complete has cyclic polymorphisms of every prime arity $p>|D|$ ) and with the paper's Theorem 4.12, the core yields Theorem 8.1: if $(\mathbf T,\mathbf H_2)$ is pp-constructible from a finite $\mathbf D$ , then $\mathrm{CSP}(\mathbf D)$ is NP-complete. Hence the polynomial-time algorithm for 1-in-3 vs NAE cannot come from any finite tractable CSP, and the infinite template of Brakensiek and Guruswami is necessary. It is a concrete separation between PCSPs and CSPs: a tractable PCSP that is not explained by any finite tractable CSP.

The theorem is proved in the paper; nothing in this section has a machine-checked proof. The mission produces a formal proof of the combinatorial core, a reusable definition of cyclic operations on top of the platform's polymorphism library, and the matrix calculus of §8 (areas, covers, $t$ ), which is the part of the argument where index conventions are easiest to get wrong.

Difficulty

Polymorphisms of $(D;R)$ are not determined by their values on zero-one inputs, and nothing beyond cyclicity is known about $s$ . The obvious attempt, to show directly that $s$ restricted to $\{f(0),f(1)\}$ depends only on the number of ones, fails: this property does not follow from cyclicity (Remark 8.15 asks whether polymorphisms of that kind always exist, and it is open). The only information available about $s$ is the coarse two-valued invariant $g(t(\cdot))$ on zero-one matrices, and the contradiction has to be produced from that invariant alone, uniformly over all finite $D$ ; the margin $p>60|D|$ is what keeps the areas involved away from $1/3$ .

Formalization scope

Structures are instances of the published PCSPBLPAff.Symmetric.RelStruct with the single-symbol signature Unit of arity $3$ ; homomorphisms and polymorphisms are the published IsHom and IsPolymorphism. The domain $\{0,1\}$ is Fin 2; $D$ is a Fintype with decidable equality. Matrices are functions Fin p → Fin p → Fin 2, row index first; the $p^2$ arguments of $t$ are read row-major via finProdFinEquiv; areas are rationals. Cyclicity is s a = s (a ∘ finRotate p), the paper's $(a_2,\dots,a_p,a_1)$ . The paper renames $D$ so that $f(0)=0$ , $f(1)=1$ ; the formalization keeps $f$ and evaluates zero-one matrices through it. Definition 8.11 prints column heights $1\le k_i\le p$ , but its proofs use height $0$ , so heights range over $0\le k_i\le p$ .

Lemmas 8.7–8.10 are stated with only the hypotheses their proofs use (Lemma 8.8 needs only cyclicity; Lemmas 8.9–8.10 need $p>3$ prime rather than $p>60|D|$ ). Lemmas 8.13 and 8.14 concern a minimal counterexample inside the proof of Proposition 8.12 and are not separate items.

Left out: the NP-completeness conclusion of Theorem 8.1; the reduction from pp-constructibility to a homomorphic relaxation (Theorem 4.12, mission 1 of this series); the fact that a pp-power of a finite tractable CSP template is a finite tractable CSP template; and the cited Theorem 8.3 [BK12], which is never assumed or stated.

The goal ends in False, so it would be trivial if its hypotheses were contradictory for a reason unrelated to $s$ . They are not: $D=\{0,1\}$ , $R$ the 1-in-3 relation, $f=g=\mathrm{id}$ satisfies every hypothesis except the existence of $s$ . No restriction is placed on $s$ beyond being a cyclic polymorphism; assuming idempotence or dependence on counts only would assume away the content (Remark 8.15).

Contributions welcome: proofs of the milestones, a general library of cyclic operations and their compositions, and a formalization of Theorem 8.3 or of the pp-power step, which would turn the core into Theorem 8.1 itself.

Selected references

L. Barto, J. Bulín, A. Krokhin, J. Opršal, Algebraic approach to promise constraint satisfaction, arXiv:1811.00970v3, 2019; J. ACM 68(4), 2021. https://arxiv.org/abs/1811.00970
L. Barto, M. Kozik, Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem, Logical Methods in Computer Science 8(1:07), 2012. https://doi.org/10.2168/LMCS-8(1:7)2012
J. Brakensiek, V. Guruswami, Promise constraint satisfaction: structure theory and a symmetric Boolean dichotomy, SODA 2018. https://arxiv.org/abs/1704.01937
A. Bulatov, A dichotomy theorem for nonuniform CSPs, FOCS 2017. https://arxiv.org/abs/1703.03021
D. Zhuk, A proof of the CSP dichotomy conjecture, FOCS 2017; J. ACM 67(5), 2020. https://arxiv.org/abs/1704.01914

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Operations Research Optimization·Captain: mikedeng1

Responsible Sourcing in Supply Chains: When Each of the Four Sourcing Strategies Is OptimalResearch Paper

Motivation

A buyer can reduce production cost by using a supplier whose operations carry a risk of a responsibility violation. That choice can also bring a direct penalty and cause customers to leave. Guo, Lee, and Swinney study how the cost saving, consumer demand, and violation risk jointly determine the buyer’s sourcing decision in a transparent supply chain. Their model distinguishes customers who value responsible production from those who do not, making it possible to ask when the buyer uses a responsible supplier, a risky supplier, or both. The question concerns supplier selection when the buyer knows each supplier’s responsibility type and customers can react to a violation. Guo, Lee, and Swinney (2016) give the model and its four strategy regions.

Setting

A single buyer sells to a market whose total size is normalized to one. A responsible supplier has marginal cost $c_R$ and no responsibility violation in the model. A risky supplier has lower marginal cost $c_{NR}<c_R$ and a violation occurs with probability $\phi\in[0,1]$ . Write $\Delta=c_R-c_{NR}$ for the responsible supplier’s cost premium. A violation costs the buyer a fixed amount $c_{VP}$ .

Every consumer has baseline willingness to pay $v$ . A fraction $\theta\in[0,1]$ is socially conscious and has additional willingness to pay $r$ for a responsibly sourced product. Upon a violation, a fraction $\alpha\in[0,1]$ of this group exits the market. The remaining fraction $1-\theta$ has no responsible-sourcing premium and does not exit for that reason. These symbols and ranges follow the paper’s Section 3; the theorem adds no sign restriction to $v$ , $r$ , or $c_{VP}$ .

The buyer chooses among four sourcing strategies. Low-cost sourcing (LC) uses the risky supplier for the whole market. Dual sourcing (DS) serves socially conscious consumers with responsible units and other consumers with risky units. Responsible niche sourcing (RN) serves only socially conscious consumers with responsible units. Responsible mass market sourcing (RM) serves everyone with responsible units. With $\Pi^s$ denoting strategy $s$ ’s expected profit, Table 2 gives

\begin{aligned} \Pi^{LC}&=(1-\alpha\phi)\theta(v-c_{NR})+(1-\theta)(v-c_{NR})-\phi c_{VP},\\ \Pi^{DS}&=(1-\alpha\phi)\theta(v+r-c_R)+(1-\theta)(v-c_{NR})-\phi c_{VP},\\ \Pi^{RN}&=\theta(v+r-c_R),\\ \Pi^{RM}&=v-c_R. \end{aligned}

A strategy is optimal when its profit is at least the profit of each of the four strategies. This definition allows ties. In the statements below, $[x]^+=\max(x,0)$ , as defined immediately before Proposition 1. Section 4 and Table 2 are the sources for these formulas.

Formalization targets

Proposition 1: sufficient conditions for each strategy

The goal is all four clauses of Proposition 1, p. 2729, as printed. For LC, the sufficient conditions are

r<\Delta,\qquad \phi[\alpha\theta(v-c_{NR})+c_{VP}]<\Delta-[\theta r-(1-\theta)(v-c_R)]^+.

For DS, they are

r>\Delta,\qquad \phi[\alpha\theta(v+r-c_{NR})+c_{VP}]<\Delta(1-\theta)+\theta r-[\theta r-(1-\theta)(v-c_R)]^+.

For RN, the threshold is $r>(v-c_R)(1-\theta)/\theta$ ; for RM it is $r<(v-c_R)(1-\theta)/\theta$ . Each of these last two clauses also requires that neither LC nor DS be optimal. They are stated for $\theta>0$ , since their displayed quotient is undefined at zero. The six milestones record the pairwise strategy comparisons stated in the appendix proof, in its order: LC with DS, RM, and RN; RN with RM; and DS with RN and RM. Proposition 1 and its appendix proof supply these targets.

Significance

The proposition partitions the buyer’s decision into explicit sufficient regions. It identifies when the low cost of risky sourcing outweighs expected violation costs, when serving both segments separately pays, and when responsible sourcing should serve only the conscious segment or the full market. The four profits also provide the base for the paper’s later comparative statics on sourcing quantities, penalties, and transparency. Those later propositions are outside this mission; their exact treatment of strategy ties and sourcing quantities warrants a separate development. Guo, Lee, and Swinney (2016) establish these results in the published article.

The mathematical result is proved in that article. The formalization task is to give its parameter ranges, four profits, and optimality claims a machine-checkable statement, then prove the pairwise comparisons and the four-clause goal. The draft statements compile locally, but their proofs are open. The resulting model and comparisons can support later formalizations of the article’s other propositions.

Difficulty

The four choices must be compared under a single, consistent profit table. A comparison with one competitor does not establish optimality among all four, and the positive-part term in Proposition 1 compresses two distinct comparisons into one threshold. Boundary cases also matter: a strict “preferred to” comparison can disappear when its profit difference is zero, even though weak optimality remains meaningful. The printed sufficient condition in part (ii) uses $c_{NR}$ inside the violation term, while the appendix’s DS comparisons use $c_R$ . Both statements must retain their respective printed forms; treating them as the same expression would change the source claim.

Formalization scope

Lean represents the eight parameters by real fields of one record and the four strategies by a finite inductive type. The range assumptions are $0\leq\theta,\alpha,\phi\leq1$ and $c_{NR}<c_R$ . The cost premium is defined from the two costs. The theorem compares exactly the four Table 2 profits; the Section 3–4 derivation of prices and quantities from the underlying selling game is outside this mission. An optimal strategy is weakly best against all four strategies, including itself. A model with fewer competitors or profits that differ from Table 2 would not capture the proposition.

The clauses containing division by $\theta$ require $\theta>0$ . The appendix’s strict LC–DS equivalence additionally requires $\theta>0$ and $\alpha\phi<1$ ; otherwise equal profits can defeat its “if and only if” wording. Proposition 1 itself retains the full standing ranges. No positivity of $v$ , $r$ , or $c_{VP}$ is assumed. Part (ii) retains the printed $c_{NR}$ , which makes its hypothesis stronger than the corresponding appendix conditions with $c_R$ . Its conclusion remains valid under the standing ranges. These conventions are visible in the statements and can be checked independently of the paper’s informal derivation.

The development needs only real arithmetic, finite case analysis, the four profit definitions, and the maximum of a real number with zero. The parameter record and strategy comparison interface are reusable for later propositions in the same paper. Contributions that prove the exact milestone statements, establish the goal, or clarify behavior at ties are within scope.

Selected references

Guo, R., H. L. Lee, and R. Swinney, Responsible Sourcing in Supply Chains, Management Science 62(9):2722–2744, 2016. DOI: 10.1287/mnsc.2015.2256.

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Convex Optimization Optimization·Captain: mikedeng1

Error Bounds, Quadratic Growth, and Linear Convergence of Proximal Methods I: For the Proximal Gradient Method, the Error Bound Condition Is Equivalent to Quadratic GrowthResearch Paper

Motivation

A large share of the optimization problems solved in statistics, signal processing and machine learning have the composite form

\min_x\ \varphi(x) := f(x) + g(x),

where $f$ is smooth and $g$ is convex but possibly nonsmooth or extended-valued (a norm penalty, the indicator of a constraint set). The workhorse algorithm for such problems is the proximal gradient method, which takes a gradient step on $f$ followed by a proximal step on $g$ . Its worst-case rate on convex problems is sublinear, yet in practice it often converges linearly. The classical explanation, going back to Luo and Tseng (1993) Ann. Oper. Res. 46, is an error bound condition: near the solution set, the distance to the minimizers is controlled by the length of the algorithm's own step. Under such a condition the method converges linearly.

The difficulty is that the error bound is stated in terms of the algorithm, not the problem. Drusvyatskiy and Lewis (arXiv:1602.06661, Math. Oper. Res. 43(3), 2018) showed that, for convex problems, it is equivalent to a purely geometric property of $\varphi$ : quadratic growth away from the set of minimizers. This mission formalizes that equivalence (§3 of the paper) together with the three comparison results that build it.

Setting

Write $\mathbb R^n$ for Euclidean space with norm $\|\cdot\|$ , $\overline{\mathbb R} = \mathbb R\cup\{\pm\infty\}$ , and $\operatorname{dist}(x;Q) = \inf_{z\in Q}\|z-x\|$ . A function $g:\mathbb R^n\to\overline{\mathbb R}$ is closed if it is lower semicontinuous. For real $r$ , the sublevel set is $[\varphi\le r] = \{x:\varphi(x)\le r\}$ .

The problem is (3.1): $g$ is a closed convex function, and $f:\mathbb R^n\to\mathbb R$ is convex, $C^1$ -smooth, with $\beta$ -Lipschitz gradient, $\|\nabla f(x)-\nabla f(y)\|\le\beta\|x-y\|$ . Let $S$ be the (nonempty) set of minimizers of $\varphi = f+g$ and $\varphi^*$ its minimal value. For $t>0$ the proximal map is

\operatorname{prox}_{tg}(z) = \operatorname*{argmin}_y\Big\{g(y)+\tfrac1{2t}\|y-z\|^2\Big\},

the proximal gradient method iterates $x_{k+1} = \operatorname{prox}_{tg}(x_k - t\nabla f(x_k))$ , and the prox-gradient mapping is

\mathcal G_t(x) = t^{-1}\big(x - \operatorname{prox}_{tg}(x - t\nabla f(x))\big),

so that $x_{k+1} = x_k - t\,\mathcal G_t(x_k)$ and $\mathcal G_t(x) = 0$ exactly at minimizers.

For $\nu>0$ , $\varphi$ has quadratic growth with constant $\alpha>0$ if $\varphi(x)\ge\varphi^*+\frac\alpha2\operatorname{dist}^2(x;S)$ on $[\varphi\le\varphi^*+\nu]$ (3.12), and the error bound condition with parameters $(\gamma,\nu)$ holds if $\operatorname{dist}(x;S)\le\gamma\|\mathcal G_t(x)\|$ on the same sublevel set (Definition 3.1, (3.13)).

Formalization targets

Goal: Corollary 3.6 (p. 9)

\text{(3.12) with }\alpha\ \Longrightarrow\ \text{(3.13) with }\gamma = (2\alpha^{-1}+t)(1+\beta t),\qquad \text{(3.13) with }\gamma\ \Longrightarrow\ \text{(3.12) for every }\alpha\in(0,\gamma^{-1}),

for every step size $t>0$ and every $\nu>0$ .

Milestones

For a closed convex $h$ with minimizers $S\neq\emptyset$ and minimal value $h^*$ :

Theorem 3.3 (pp. 6–7): quadratic growth on $[h\le h^*+\nu]$ with constant $\alpha$ implies $\operatorname{dist}(x;S)\le L\operatorname{dist}(0;\partial h(x))$ there with $L = 2\alpha^{-1}$ ; conversely the latter implies quadratic growth with any $\alpha\in(0,1/L]$ .
Theorem 3.4 (p. 7): the subdifferential bound with $L$ implies $\operatorname{dist}(x;S)\le\widehat L\,t^{-1}\|x-\operatorname{prox}_{th}(x)\|$ with $\widehat L = L+t$ ; conversely the latter implies the former with $L = \widehat L$ .
Theorem 3.5 (p. 8): for maximal monotone $G,\Phi$ with $\Phi = F + G$ and $F$ single-valued, the forward-backward step satisfies $\|\mathcal G_t(x)\|\le\operatorname{dist}(0;\Phi(x))$ (3.10), and if $F$ is $\beta$ -Lipschitz,

(1-\beta t)\|\mathcal G_t(x)\|\ \le\ \|t^{-1}(x-\operatorname{prox}_{t\Phi}(x))\|\ \le\ (1+\beta t)\|\mathcal G_t(x)\|.\qquad(3.11)

Theorem 3.2, second claim (p. 5, at $t=\beta^{-1}$ ): under the error bound, the iterates converge R-linearly to any limit point $x^*$ , $\|x_{r+k}-x^*\|^2\le(1-\frac1{2\beta\gamma})^kC(\varphi(x_r)-\varphi^*)$ .

Significance

The result. Corollary 3.6 converts a condition on an algorithm into a condition on the objective. Quadratic growth can be checked by hand in many models (strongly convex losses, polyhedral problems, problems with a sharp minimum), and once it holds, Theorem 3.2 gives linear convergence of the proximal gradient method with an iteration count of order $\frac\beta\alpha\ln\frac1\epsilon$ , matching the rate familiar from strongly convex problems. The rest of the paper uses this equivalence as its main tool: §4 proves quadratic growth for structured problems $f(Ax)+g(x)$ under dual strict complementarity, and §§5–8 extend the comparison to prox-linear methods for nonconvex compositions. Theorems 3.3 and 3.4 are statements about a single closed convex function and are of independent use in the analysis of proximal point methods.

Formalizing it. All results here are proved on paper; none of them is machine-checked. The mission produces a Lean statement of the equivalence with the paper's explicit constants, and of the step-length comparison between the forward-backward and the proximal point methods for maximal monotone operators. Formal proofs require the existence, uniqueness and nonexpansiveness of resolvents and proximal maps, the subdifferential sum rule for a smooth plus a convex function, and a quadratic growth argument for convex functions; these are reusable well beyond this paper.

Difficulty

The forward direction is a chain of three inequalities, but each link compares objects that live in different worlds: the subdifferential of $\varphi$ , the proximal map of $\varphi$ , and the proximal map of $g$ evaluated at a gradient-shifted point. The proximal gradient step is not the proximal point step on $\varphi$ , and the obvious attempt to compare $\mathcal G_t(x)$ with $\partial\varphi(x)$ directly gives only one inequality (3.10); the reverse comparison (3.11) needs the resolvent of the sum $\nabla f + \partial g$ and the Lipschitz constant of $\nabla f$ . The converse half of Theorem 3.3 (error bound implies growth) is the deepest step: the paper omits its proof and cites Drusvyatskiy–Ioffe and Drusvyatskiy–Mordukhovich–Nghia. It has to pass from a pointwise bound on subgradients to a bound on function values over the whole sublevel set, which no single subgradient inequality provides.

Formalization scope

$\mathbb R^n$ is EuclideanSpace ℝ (Fin n); $\overline{\mathbb R}$ -valued functions are EReal-valued; $f$ is real-valued and $\varphi = f + g$ is computed in EReal. "Closed convex" is the published RockafellarMaxMono.Shared.ProperConvex (never $-\infty$ , somewhere finite, convex) together with LowerSemicontinuous.
$S$ is the set of minimizers, assumed nonempty as in the paper; $\varphi^*$ is a real number equal to $\varphi$ on $S$ ; $\operatorname{dist}$ is Metric.infDist; $\nu>0$ is real.
Proximal maps and resolvents are predicates, never chosen functions: $p=\operatorname{prox}_{tg}(z)$ is the published GoldenRatioVI.Shared.IsProxPoint (fun y => t * g y) z p (same argmin for $t>0$ ), and resolvents are maps satisfying the published ThreeOpSplitting.Convergence.IsResolvent. Conditions are stated for every such point; existence and uniqueness are facts a solver proves.
$\operatorname{dist}(0;\partial h(x))$ follows the convention $\operatorname{dist}(0;\emptyset) = +\infty$ : an inequality " $a\le L\operatorname{dist}(0;\partial h(x))$ " is stated as " $a\le L\|v\|$ for every $v\in\partial h(x)$ ". Using Metric.infDist 0 (∂h x), which is $0$ on the empty set, would turn (3.7) into the false claim that every point of the sublevel set where $\partial h$ is empty is a minimizer; this trivializing or falsifying encoding is ruled out.
The subdifferential is the published ProxAlg.FixedPoint.subdifferential, the gradient is Mathlib's gradient.
Theorem 3.2 is stated at $t = \beta^{-1}$ with $\beta>0$ : the printed range $t\le\beta^{-1}$ relies on the descent inequality (3.2), which fails for smaller $t$ . Its first claim (an iteration count) and Corollary 3.7 are not stated.

Contributions welcome: proofs of the milestones in any order, general lemmas on resolvents of maximal monotone operators and on proximal maps of EReal-valued convex functions, and a proof of the converse half of Theorem 3.3.

Selected references

D. Drusvyatskiy, A. S. Lewis, Error bounds, quadratic growth, and linear convergence of proximal methods, Math. Oper. Res. 43(3), 2018. arXiv:1602.06661v2 — https://arxiv.org/abs/1602.06661
Z.-Q. Luo, P. Tseng, Error bounds and convergence analysis of feasible descent methods: a general approach, Ann. Oper. Res. 46/47:157–178, 1993. https://doi.org/10.1007/BF02096261
D. Drusvyatskiy, A. D. Ioffe, Quadratic growth and critical point stability of semi-algebraic functions, Math. Program. 153(2):635–653, 2015 (reference [15] of the paper).
D. Drusvyatskiy, B. S. Mordukhovich, T. T. A. Nghia, Second-order growth, tilt stability, and metric regularity of the subdifferential, J. Convex Anal. 21(4):1165–1192, 2014 (reference [19] of the paper).
R. T. Rockafellar, R. J.-B. Wets, Variational Analysis, Springer, 1998. https://doi.org/10.1007/978-3-642-02431-3

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