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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.

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.

≤ 19.8899945Formalized record≤ 14.797074Open frontier

6 provers on it3 of 7 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.

≤ 87Formalized record

3 provers on it5 of 5 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.

≤ 85Formalized record≤ 5Open frontier

35 provers on it10 of 12 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?

≤ 2.37134Formalized record≤ 2.371177Open frontier

16 provers on it7 of 8 missions formalized

All missions

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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.

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.

≤ 19.8899945Formalized record≤ 14.797074Open frontier

6 provers on it3 of 7 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.

≤ 87Formalized record

3 provers on it5 of 5 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.

≤ 85Formalized record≤ 5Open frontier

35 provers on it10 of 12 missions formalized

Matrix multiplication exponent

≤ 2.37134Formalized record≤ 2.371177Open frontier

16 provers on it7 of 8 missions formalized

All missions

Completed

Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis X: The Lagrangian Saddle-Point TheoremTextbook

Motivation

Chunk 10 formalized the discrete conjugacy theorem — the Legendre-Fenchel transform's bijection between the classes of M-convex and L-convex functions — and, along the way, a function-level generalization of Edmonds's intersection theorem. Section 8.4 turns that machinery toward a different question: not "how are two convexity classes related," but "when does a discrete optimization problem have a dual that meets it with equality." The classical route to such strong-duality results in continuous convex programming — Lagrangian relaxation, an embedding of the problem in a family of perturbed problems, and a saddle-point characterization of when primal and dual values coincide — has a discrete analogue that needs no continuity, no differentiability, and no convexity in the classical sense at all: only the elementary fact that the Legendre-Fenchel transform, once discretized, is still an involution on the right class of functions. This mission formalizes that discrete Lagrangian duality framework and its central saddle-point theorem, in full generality — before the book specializes it, in the section that follows, to the specific M-convex perturbation that gives the chapter's headline strong-duality result for M-convex programs.

Setting

Let $V$ and $U$ be finite ground sets. A perturbation of an optimization problem $\min\{f(x) : x \in \mathbb Z^V\}$ is a function $F : \mathbb Z^V \times \mathbb Z^U \to \mathbb Z \cup \{+\infty\}$ such that $F(x,0) = f(x)$ for all $x$ (Eq. (8.54)) and, for each fixed $x$ , $F(x,\cdot)$ is self-biconjugate: $F(x,\cdot)^{\bullet\bullet} = F(x,\cdot)$ under the discrete Legendre-Fenchel transform of chunk 10 (Eq. (8.55)). The Lagrangian function is $K(x,y) = \inf\{F(x,u) + \langle u,y\rangle : u \in \mathbb Z^U\}$ (Eq. (8.58)), valued in $\mathbb Z \cup \{\pm\infty\}$ (formalized in EReal, since both the infimum and the supremum below can be genuinely unbounded). The dual objective is $g(y) = \inf\{K(x,y) : x \in \mathbb Z^V\}$ (Eq. (8.60)). Writing $\inf(P) = \inf_x f(x)$ , $\sup(D) = \sup_y g(y)$ , $\operatorname{opt}(P) = \{x : f(x) = \inf(P)\}$ , $\operatorname{opt}(D) = \{y : g(y) = \sup(D)\}$ , the primal problem $P$ is to minimize $f$ over $\mathbb Z^V$ and the dual problem $D$ is to maximize $g$ over $\mathbb Z^U$ .

Formalization targets

Goal: Theorem 8.54 (the saddle-point theorem)

Assuming $F$ is self-biconjugate (Eq. (8.55)): both $\inf(P)$ and $\sup(D)$ are finite and $\min(P) = \max(D)$ if and only if there exist $\bar x \in \mathbb Z^V$ , $\bar y \in \mathbb Z^U$ with $K(\bar x,\bar y)$ finite and $K(x,\bar y) \le K(\bar x,\bar y) \le K(\bar x,y)$ for all $x,y$ — a saddle point of the Lagrangian kernel. When this holds, $\bar x \in \operatorname{opt}(P)$ and $\bar y \in \operatorname{opt}(D)$ .

Milestones: Theorem 8.52, Proposition 8.51(1)-(2)

Theorem 8.52 (weak duality): $\inf(P) \ge \sup(D)$ always, with no biconjugacy hypothesis on $F$ at all — the baseline the saddle-point theorem sharpens to equality. Proposition 8.51(1)-(2): under self-biconjugacy, the perturbation $F$ (and hence the primal objective $f$ ) is itself recoverable from the Lagrangian kernel $K$ by a supremum, $F(x,u) = \sup_y\{K(x,y) - \langle u,y\rangle\}$ and $f(x) = \sup_y K(x,y)$ — the algebraic identity the saddle-point theorem's proof turns on directly.

Significance

The result itself. The saddle-point theorem is the general-purpose engine behind every strong-duality result the book proves for specific classes of discrete optimization problems: the book's own next section specializes it (via a particular choice of $F$ built from an M-convex regularizer $r$ ) to obtain strong duality for M-convex programs, but the theorem itself needs no M-convexity, no submodularity, and no exchange axiom — only the elementary self-biconjugacy of a perturbation under the discrete Legendre-Fenchel transform. It is, in that sense, the most general and most reusable strong-duality statement in the book: any future mission proving strong duality for a specific class of discrete programs (M-convex, M2-convex, network flow, or otherwise) by exhibiting a self-biconjugate perturbation can cite this theorem directly rather than reproving the saddle-point argument from scratch.

Formalizing it. No matching item exists on the platform for a discrete Lagrangian saddle- point theorem, discrete weak duality, or this perturbation-based duality framework. (A prior-art search turned up an unrelated continuous Lagrangian saddle-point theorem for convex cones, Luenberger's Chapter 8 §8.4, formalized as VectorSpaceOpt.lagrangian_saddle_sufficient_pointed — a genuinely different setting: no discreteness, no biconjugacy hypothesis, and a one-directional sufficiency statement rather than this mission's iff. Not reused.) This mission gives the first formal statement of discrete Lagrangian duality, and directly reuses chunk 10's ConvexConjugate apparatus (self-biconjugacy is stated using chunk 10's own conjugate-of-conjugate composition), demonstrating exactly the kind of shared-substrate payoff the discrete conjugacy theorem was built to provide.

Difficulty

The saddle-point theorem's "only if" direction is not a routine unwinding of definitions: given $\min(P) = \max(D)$ at finite common value, one must construct the saddle point $(\bar x,\bar y)$ — the book's proof takes $\bar x \in \operatorname{opt}(P)$ , $\bar y \in \operatorname{opt}(D)$ (which exist because the infimum/supremum are attained at a finite optimum) and verifies the sandwiching inequality using Proposition 8.51(2)'s identity $f(x) = \sup_y K(x,y)$ together with weak duality, rather than by any direct algebraic manipulation of $K$ alone. Skipping straight to a "trivial" biconditional that never invokes Proposition 8.51 would misrepresent the actual proof structure the book relies on for exactly this direction.

Formalization scope

$V, U$ are Fintype ground types; LagrangianKernel, DualObjective, InfP, SupD are EReal-valued to keep both the defining infima/suprema total (a complete lattice) without an artificial finiteness side-condition; OptP, OptD compare PrimalValue/DualObjective against InfP/SupD after casting through chunk 10's ToEReal, mirroring that chunk's own round-trip convention. "Finite" throughout is formalized as ≠ ⊤ ∧ ≠ ⊥ in EReal. The perturbation $F$ itself is left fully abstract (an arbitrary function satisfying the self-biconjugacy hypothesis where needed) — this mission does not draft the specific M-convex perturbation $F_r$ (Eq. (8.61)) that the book's next subsection (§8.4.3) uses to specialize this framework to M-convex programs, nor Theorem 8.59 (the resulting M-convex strong-duality theorem) itself, which needs that specific perturbation plus its own regularity conditions (REG)/(OBJ) and a chain of M-convex-specific propositions (8.55–8.58) beyond what the general framework built here provides. A trivializing formalization would state the saddle-point theorem's sandwiching inequality with a weaker order (e.g., only one of the two directions) or would omit the " $\bar x \in \operatorname{opt}(P), \bar y \in \operatorname{opt}(D)$ " consequence clause; neither is done — both inequalities and the full consequence clause are included exactly as the book states them.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.

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Combinatorics·Captain: mikedeng1

Applied Combinatorics IV: Newton's Binomial Theorem and the Central Binomial ConvolutionTextbook

Motivation

Generating functions are the standard device of enumerative combinatorics for turning a counting sequence into a single algebraic or analytic object: a sequence $\{a_n : n \ge 0\}$ is recorded as the power series $\sum_{n\ge0} a_n x^n$ , and operations on series (products, powers, derivatives) become operations on the counts. Chapter 8 of Keller and Trotter's Applied Combinatorics (appliedcombinatorics.org), an open textbook used in undergraduate combinatorics courses, develops the method up to one of its classical applications: extending the binomial theorem to real exponents, as Newton did, and reading off an identity about central binomial coefficients that is awkward to prove by direct counting.

The identity in question,

2^{2n} = \sum_{k=0}^{n} \binom{2k}{k}\binom{2n-2k}{n-k},

appears in standard collections of binomial identities such as Graham, Knuth and Patashnik's Concrete Mathematics (1994).

Setting

For a real number $p$ and a nonnegative integer $k$ , the book defines a number $P(p, k)$ by the recursion

P(p, 0) = 1, \qquad P(p, k) = p\,P(p-1, k-1) \quad (k > 0),

so that $P(p, k) = p(p-1)\cdots(p-k+1)$ with no requirement $p \ge k$ . The generalized binomial coefficient is

\binom{p}{k} = \frac{P(p,k)}{k!}.

For integers $p \ge k \ge 0$ this is the usual binomial coefficient; for integers $0 \le p < k$ it is $0$ ; for other real $p$ it is in general nonzero for every $k$ . In the Lean development $P(p,k)$ is AppliedComb.GenFun.fallingP p k and $\binom pk$ is AppliedComb.GenFun.binomReal p k.

The generating function of a real sequence $a = (a_n)_{n\ge0}$ is the formal power series $A(x) = \sum_{n\ge0} a_n x^n$ , in Lean PowerSeries.mk a : PowerSeries ℝ. When a closed-form function such as $(1+x)^p$ or $(1-4x)^{-1/2}$ is called the generating function of a sequence, the statements below read this as convergence of $\sum_n a_n x^n$ to the function's value on an explicit real interval around $0$ .

The central binomial coefficients are $\binom{2n}{n}$ : $1, 2, 6, 20, 70, \dots$

Formalization targets

Goal: Corollary 8.14

For every integer $n \ge 0$ ,

2^{2n} = \sum_{k=0}^{n} \binom{2k}{k}\binom{2n-2k}{n-k},

stated as an identity of natural numbers.

Milestones

Lemma 8.11. For every real $p$ and integer $k \ge 0$ , $P(p, k+1) = P(p, k)\,(p - k)$ .
Lemma 8.12. For every integer $k \ge 0$ , $\binom{-1/2}{k} = (-1)^k \binom{2k}{k} / 2^{2k}$ .
Theorem 8.10 (Newton's Binomial Theorem). For real $p \ne 0$ and real $|x| < 1$ ,

(1+x)^p = \sum_{n=0}^{\infty} \binom{p}{n} x^n .

Theorem 8.13. For real $|x| < 1/4$ ,

(1-4x)^{-1/2} = \sum_{n=0}^{\infty} \binom{2n}{n} x^n .

Proposition 8.3. For real sequences $a$ , $b$ , the product of their generating functions is the generating function of $\bigl(\sum_{k=0}^n a_k b_{n-k}\bigr)_{n\ge0}$ .
Theorem 8.16 (already on the platform). For each $n \ge 1$ , the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts.

The first five follow the chapter's own chain toward the goal; Theorem 8.16 is the chapter's other main result and is included as a reference.

Significance

Corollary 8.14 says that the sequence of central binomial coefficients convolved with itself is the sequence $4^n$ ; equivalently, the generating function of $\binom{2n}{n}$ is a square root of $1/(1-4x)$ . Central binomial coefficients count lattice paths with $n$ up-steps and $n$ down-steps, and the identity says that the pairs consisting of a balanced path of length $2k$ and one of length $2n-2k$ , summed over $k$ , are equinumerous with all $4^n$ strings over a four-letter alphabet. The same generating function reappears in the book's Section 9.7, and Newton's theorem with exponent $-1/2$ or $1/2$ is the standard route to closed forms for Catalan-type sequences.

On the formalization side, Mathlib has the central binomial coefficient (Nat.centralBinom), formal power series, and Newton's series in the complex-analytic form Complex.one_add_cpow_hasFPowerSeriesOnBall_zero, which is also published on the platform as FamousTheorems.newton_binomial_series_6b and included in this mission as a reference. It does not contain the convolution identity of Corollary 8.14, and it does not contain the book's recursive $P(p,k)$ or the closed form of $\binom{-1/2}{k}$ . The mission produces a machine-checked version of the chapter's chain from the book's own definitions to the identity.

Difficulty

The identity is not a special case of the Vandermonde convolution $\sum_k \binom{a}{k}\binom{b}{n-k} = \binom{a+b}{n}$ : both factors depend on the summation index in their upper argument as well as their lower one. Induction on $n$ does not close directly, since the sum for $n+1$ is not a simple combination of the sum for $n$ . A counting proof is possible but not obvious, which is why the chapter's route goes through a generating function with a non-integer exponent. That route passes from formal power series to real analysis: $(1-4x)^{-1/2}$ is a real function, and the passage from an identity of functions on an interval to an identity of coefficients requires the uniqueness of power series coefficients on an open interval and the product of two convergent series. The book asserts Newton's theorem without proof.

Formalization scope

Numbers. $P(p,k)$ and $\binom pk$ are real-valued, defined by the book's recursion (Definition 8.8) and quotient (Definition 8.9) in the definition item AppliedComb.GenFun.binomReal. Mathlib's descPochhammer and Ring.choose compute the same values; they are not used in the statements so that Lemma 8.11 is a statement about the book's recursion rather than a definitional unfolding.
Pinned readings. The book treats generating functions as formal power series and states Theorems 8.10 and 8.13 without a domain for $x$ . Here both are stated analytically: Theorem 8.10 for real $p \ne 0$ and real $x$ with $|x| < 1$ , Theorem 8.13 for real $x$ with $|x| < 1/4$ , with the real power Real.rpow of a positive base on the left and HasSum (unconditional convergence of the series) on the right. The book's hypothesis $p \ne 0$ is kept. No other explicit constants replace informal ones: the chapter's statements contain no $O(\cdot)$ , " $\approx$ " or "sufficiently large".
Proposition 8.3 is stated for formal power series PowerSeries ℝ with the sum written as $\sum_{k=0}^{n} a_k b_{n-k}$ over Finset.range (n + 1).
Goal. Corollary 8.14 is an identity in ℕ with Nat.choose; the subtractions $2n-2k$ and $n-k$ occur only for $k \le n$ and are exact. A statement asserting only that the square of the formal power series $\sum_n \binom{2n}{n}X^n$ equals $\sum_n 4^n X^n$ , or a purely formal version of Theorem 8.13, would hide the identity in a coefficient comparison and is not the book's statement; the goal is the explicit sum identity.
Theorem 8.16 is referenced as FamousTheorems.card_odds_eq_card_distincts, stated for all $n$ with Mathlib's Nat.Partition, whose partitions are multisets of positive integers, as in the book (p. 168); the case $n = 0$ it adds is immediate.

A complete development needs real power series on an interval (Cauchy products, identity theorem for coefficients), the real power function, and elementary manipulation of binomial coefficients; the identification of binomReal with Ring.choose is reusable for any later statement using the generalized binomial coefficient. Proofs of any milestone are welcome independently, and so is a direct proof of Corollary 8.14 that bypasses the analytic chain.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, CC BY-SA 4.0. Chapter 8. https://www.appliedcombinatorics.org/
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. ISBN 978-0-201-55802-9.
Mathlib, Complex.one_add_cpow_hasFPowerSeriesOnBall_zero (Newton's binomial series). https://github.com/leanprover-community/mathlib4

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXVII: Directional Derivatives and Quasi L-Convex FunctionsTextbook

Motivation

This mission completes chapter 7's L-convex function theory and closes the book on it. It first finishes the theory of positively homogeneous L-convex functions — showing they coincide exactly with the classical Lovász extensions of submodular set functions, a one-to-one correspondence that recognizes forty years of submodular-optimization machinery as a special case of L-convex function theory. It then proves the L-side capstone this series has been building toward since mission 26-ch07b-lconvexfunctions: polyhedral L-convexity is characterized simultaneously by directional derivatives, subdifferentials, and weighted-minimizer polyhedra — the exact mirror of what mission 24-ch06d-mconvexfunctions proved for M-convex functions. Finally it develops quasi L-convex functions, the L-side analogue of mission 25-ch06e-mconvexfunctions's quasi M-convex functions, ending exactly where chapter 7 itself ends.

Setting

Fix a finite ground set $V$ . The class $0L[\mathbb R \to \mathbb R]$ consists of polyhedral L-convex functions that are positively homogeneous; $0L[\mathbb Z \to \mathbb Z]$ , its integer-valued integer-domain analogue. A positively homogeneous L-convex function $g$ induces a submodular set function $\rho_g(X) = g(\chi_X)$ ; conversely the Lovász extension $\hat\rho$ of a submodular set function is positively homogeneous L-convex. The base polyhedron $B(\rho) = \{x \in \mathbb R^V : x(X) \le \rho(X)\ \forall X,\ x(V) = \rho(V)\}$ of a submodular $\rho$ is always an M-convex polyhedron. A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is quasi submodular (QSB) if $g(p\wedge q) \le g(p)$ or $g(p\vee q) \le g(q)$ for all $p,q$ ; the weaker (QSBw) requires only $\max\{g(p),g(q)\} \ge \min\{g(p\wedge q), g(p\vee q)\}$ .

Formalization targets

Goal: the four characterizations of polyhedral L-convexity (Theorem 7.45)

For a polyhedral convex function $g$ with $\operatorname{dom}_{\mathbb R} g \ne \emptyset$ : $g$ is L-convex if and only if every directional derivative $g'(p;\cdot)$ is $0L[\mathbb R \to \mathbb R]$ , if and only if every subdifferential $\partial_{\mathbb R} g(p)$ is an M-convex polyhedron, if and only if every weighted minimizer set is an L-convex polyhedron. Not present in this chunk's own extraction table (its label opens mid-paragraph, missed by the same extractor failure already documented for mission 25-ch06e-mconvexfunctions's Theorem 6.68), found by direct reading and chosen as goal because it is the exact L-side mirror of mission 24-ch06d-mconvexfunctions's own milestone Theorem 6.63, and its proof is assembled entirely from results already in this series (Theorem 7.43, Proposition 7.34, Theorem 7.40).

Supporting structural targets

Fifteen further results build the two remaining pieces of chapter 7's theory. Propositions 7.37-7.39 and Theorem 7.40 establish the one-to-one correspondence between positively homogeneous L-convex functions and submodular set functions via the Lovász extension; Proposition 7.41 gives a minimizer-polyhedron characterization of this class, and Proposition 7.42 shows directional derivatives of L-convex functions automatically land in it. Theorem 7.43 — the L-side mirror of mission 24-ch06d-mconvexfunctions's own goal, Theorem 6.61 — proves the directional- derivative/subdifferential correspondence via the induced submodular set function's base polyhedron; Proposition 7.44 checks consistency at integer points, and Theorem 7.46 refines Theorem 7.45 to the integral case. Theorem 7.49 (also missed by the extractor) gives the quasi-submodularity implication hierarchy and its perturbation-equivalence capstone, mirroring mission 25-ch06e-mconvexfunctions's Theorem 6.68 exactly. Proposition 7.50 and Theorems 7.51-7.52 build the level-set/perturbation machinery quasi submodularity needs; Theorems 7.53-7.54 (both missed by the extractor, the latter's full statement requiring one page beyond this chunk's nominal range, at the very end of chapter 7) give the quasi L-optimality and quasi L-proximity theorems.

Significance

The 0L/submodular correspondence (Theorem 7.40) is the precise sense in which L-convex function theory generalizes submodular set function theory rather than merely resembling it: every submodular set function is literally the restriction to $\{0,1\}^V$ of a positively homogeneous L-convex function, and every algorithm for one transfers to the other through this exact dictionary. The goal, Theorem 7.45, completes the parallel structure this series has built since chapter 6: M-convexity and L-convexity are now each characterized in the same four convex-analytic vocabularies, setting up chapter 8's conjugacy theorem, which will show these two characterizations are not merely analogous but literally dual to each other under the Legendre-Fenchel transform. The quasi-submodularity results matter for the same reason as their M-side counterparts: chapter 10's algorithms for L-convex-function minimization remain correct under nonlinear rescalings that destroy L-convexity itself but preserve quasi submodularity.

None of these results are open — they are Murota's account of how far L-convex function theory extends beyond the polyhedral case (to positive homogeneity and its submodular-function incarnation) and how far its exchange-style inequality can be relaxed while preserving optimization theory (to quasi submodularity), mirroring chapter 6's identical two-part program for M-convex functions. What this mission contributes is a faithful, machine-checked formal statement of each, including four theorems (7.45, 7.49, 7.53, 7.54) the platform's own automated extractor missed entirely — one of them requiring a page beyond this chunk's own nominal range to complete, since chapter 7 ends there and this is the last mission covering it — extending the shared Lean vocabulary (ZeroLR, BasePolyhedron, QSBw) this series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to prove all six pairwise implications among its four conditions independently; the book's own proof instead chains through results already established: (a)⇒(b) is Proposition 7.42, (a)⇒(c) is Theorem 7.43, (a)⇒(d) is Proposition 7.34, (b)⇔(c) uses the 0L/M0[R] correspondence, and (d)⇒(b) is the genuinely hard direction, requiring Proposition 7.41 applied to the directional derivative itself (showing $\arg\min(g'(p;\cdot)[-x])$ is an L-convex cone by an explicit description via the admissible-potential set of the distance function underlying $\arg\min g[-x]$ ). The remaining combinatorial difficulty in this block is in Theorem 7.43's proof: identifying $\partial_{\mathbb R} g(p)$ with the base polyhedron $B(\rho_{g,p})$ requires the L-optimality criterion (Theorem 7.33, mission 27-ch07c-lconvexfunctions) applied pointwise, a chain of logical equivalences with no single-step shortcut, exactly mirroring how mission 24-ch06d-mconvexfunctions's Theorem 6.61 needed the M-optimality criterion.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; L-convex functions are (V→ℝ)→WithTop ℝ (polyhedral) or (V→ℤ)→WithTop ℝ (integer-domain). All sixteen numbered results found in this chunk's page range — the twelve in BRIEF.md's own table plus four the extractor missed (Theorems 7.45, 7.49, 7.53, 7.54) — are placed, with two documented, content-preserving scope decisions: Theorem 7.43 omits the dual-integral refinement clauses for L[R→R|Z]/L[Z→Z] (the same decision mission 24-ch06d-mconvexfunctions's Theorem 6.61 made), and Proposition 7.50 states the general inequalities without restating their "In particular" specializations, which add no independent content — see HARD.md. " $\inf g[-x]>-\infty$ " is replaced by the equivalent (ArgMinR ...).Nonempty hypothesis throughout, matching mission 25-ch06e-mconvexfunctions's identical substitution. This mission's base vocabulary is redeclared from missions 20-ch04b-mconvexsets, 21-ch05b-lconvexsets, 23-24-ch06*-mconvexfunctions, and 26-27-ch07*-lconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the sixteen sorrys are welcome; the goal and Theorem 7.43 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota and A. Shioura, "M-convex function on generalized polymatroid," Mathematics of Operations Research, 24 (1999), pp. 95-105 [152] (the polyhedral theory Theorems 7.26-7.46 are drawn from).
P. Milgrom and C. Shannon, "Monotone comparative statics," Econometrica, 62 (1994), pp. 157-180 [129] (the origin of the quasi-submodularity condition (SSQSB)).

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Combinatorics·Captain: mikedeng1

Applied Combinatorics II: Dilworth's Chain Covering TheoremTextbook

Motivation

Partially ordered sets model precedence: tasks that must wait for other tasks, versions that supersede others, files nested in directories, alternatives ranked by several criteria at once. Two questions about such a structure recur in scheduling and in the design of algorithms. How many sequential "threads" are needed so that every item lies on a thread in which all items are mutually comparable? How many "rounds" are needed so that no round contains two comparable items? Dilworth's theorem answers the first and its dual, due to Mirsky, answers the second: in both cases the obvious lower bound, given by one large set of mutually incomparable (resp. comparable) items, is attained.

This mission formalizes Chapter 6 of Keller and Trotter's open textbook Applied Combinatorics (2017 Edition, appliedcombinatorics.org), whose capstone is Dilworth's theorem, together with the chapter's other three proved theorems on the same objects.

Timeline. Sperner (1928, doi:10.1007/BF01171114) showed that the largest family of subsets of a $t$ -set in which no member contains another has $\binom{t}{\lfloor t/2\rfloor}$ members. Dilworth (1950, doi:10.2307/1969503) proved that a finite poset whose largest antichain has $w$ elements can be partitioned into $w$ chains. Mirsky (1971, doi:10.2307/2316481) recorded the dual statement for antichain partitions. Fishburn (1970, doi:10.1016/0022-2496(70)90062-3) characterized the posets that arise from comparing real intervals as those with no subposet $\mathbf 2 + \mathbf 2$ .

Setting

A poset $\mathbf P = (X, P)$ is a set $X$ with a reflexive, antisymmetric and transitive relation $\le$ . Here $X$ is finite and is represented by a finite type $\alpha$ with a partial order. A subset $C \subseteq X$ is a chain if every two distinct points of $C$ are comparable; a subset $A \subseteq X$ is an antichain if every two distinct points of $A$ are incomparable. The empty set is both.

The width of $\mathbf P$ is the largest size of an antichain, and the height the largest size of a chain:

\operatorname{width}(\mathbf P) = \max_{A \text{ antichain}} |A|, \qquad \operatorname{height}(\mathbf P) = \max_{C \text{ chain}} |C|.

A chain partition of $X$ into $k$ parts is a family $C_1, \dots, C_k$ of pairwise disjoint chains with union $X$ ; an antichain partition is defined the same way with antichains.

The subset lattice $\mathbf 2^t$ is the family of all subsets of a $t$ -element set, ordered by inclusion.

A poset is an interval order if each point $x$ can be assigned a closed real interval $[a_x, b_x]$ , degenerate intervals allowed, such that $x < y$ exactly when $b_x < a_y$ . The poset $\mathbf 2 + \mathbf 2$ consists of two disjoint two-element chains with no comparabilities between them, and $\mathbf P$ excludes $\mathbf 2 + \mathbf 2$ if no four points of $X$ induce a copy of it.

Formalization targets

Goal: Dilworth's theorem (Theorem 6.17)

For every finite poset $\mathbf P$ with $w = \operatorname{width}(\mathbf P)$ ,

\exists\, C_1, \dots, C_w \text{ chains partitioning } X, \quad\text{and}\quad X = C_1 \cup \dots \cup C_k \text{ a chain partition} \implies w \le k.

Milestones

Theorem 6.18 (dual of Dilworth). With $h = \operatorname{height}(\mathbf P)$ , $X$ has a partition into $h$ antichains, and every antichain partition has at least $h$ parts.
Theorem 6.27 (Sperner). For $t \ge 1$ ,

\operatorname{width}(\mathbf 2^t) = \binom{t}{\lfloor t/2 \rfloor}.

Theorem 6.29 (Fishburn). A finite poset is an interval order if and only if it excludes $\mathbf 2 + \mathbf 2$ .

The milestones stand on the same definitions as the goal (width, height, chain and antichain partitions, subposets) and are ordered from the one closest to the goal's statement to the one furthest from it.

Significance

The result itself. Dilworth's theorem is a min–max theorem: a minimum over covers equals a maximum over obstructions. It is equivalent to König's theorem on bipartite matchings and to Hall's marriage theorem, and it underlies the computation of minimum chain covers by bipartite matching and network flows (Chapter 14 of the same book). The dual theorem gives the layer decomposition used in scheduling with precedence constraints. Sperner's theorem is the first result of extremal set theory, and Fishburn's theorem is the basic structure theorem for interval orders, which model preferences with thresholds of indifference.

Formalizing it. All four results are classical and proved. In Mathlib at the environment revision of this mission there is no Dilworth or Mirsky theorem; Sperner's inequality (the upper bound on antichains in a Boolean lattice) is available through the LYM inequality, and there is no theory of interval orders. The Prove2Me library holds the easy half of Dilworth (an antichain is no larger than any chain partition) in a different Mathlib environment and over a different chain-partition structure. The mission asks for complete machine-checked statements of the book's four theorems over one common set of definitions.

Difficulty

For Dilworth's theorem the lower bound is a pigeonhole argument; the content is the existence of a partition into exactly $w$ chains. Greedy constructions fail: removing a longest chain, or a maximal chain through a chosen point, can leave a poset whose width has not decreased, so a partition built by repeatedly peeling chains may use more than $w$ of them. No local rule visible from a single chain decides whether its removal keeps the count at $w$ .

For Sperner's theorem the lower bound (the middle rank is an antichain) is immediate and the upper bound needs a counting argument over maximal chains; both halves are required, since the target is an equality. For Fishburn's theorem one direction is a four-line contradiction; the other requires constructing real intervals from nothing but the absence of $\mathbf 2 + \mathbf 2$ .

Formalization scope

The ground set is a type α with [PartialOrder α] [Fintype α]. Chains and antichains are Mathlib's IsChain (· ≤ ·) and IsAntichain (· ≤ ·). width α and height α are maxima of Finset.card over all antichains (resp. chains) of α; the family is never empty, since the empty set qualifies, so both are well defined and equal $0$ for the empty poset. Width is defined from antichains, not as the least number of chains in a partition; a definition of the latter kind would make the goal true by definition and is ruled out. Chain and antichain partitions are families indexed by Fin k of pairwise disjoint Finsets whose union is all of α; empty parts are not excluded, which does not change either theorem. The "no fewer" clauses quantify over every k and every such family. The subset lattice $\mathbf 2^t$ is Finset (Fin t) ordered by inclusion, $\lfloor t/2 \rfloor$ is t / 2 in ℕ, and Sperner's theorem keeps the book's hypothesis $t \ge 1$ . An interval representation is a pair of maps a b : α → ℝ with a x ≤ b x; $\mathbf 2 + \mathbf 2$ is Fin 2 ⊕ Fin 2 with the disjoint-sum order, and "excludes" means there is no order embedding of it into α. Fishburn's theorem is stated for finite posets: the book's construction of a representation is finite, and the equivalence fails for some uncountable posets.

No statement contains an asymptotic bound, so no explicit constant is instantiated.

A complete development needs finite-poset combinatorics (induction on the ground set, subposets as subtypes, maximal and minimal elements), which is reusable across order theory; results proving the definitions agree with Mathlib's notions of chains in Finset lattices are welcome, as is an alternative proof of Dilworth via Hall's theorem, which Mathlib has.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 6, pp. 113–140. https://www.appliedcombinatorics.org
R. P. Dilworth, A decomposition theorem for partially ordered sets, Annals of Mathematics 51 (1950), 161–166. https://doi.org/10.2307/1969503
L. Mirsky, A dual of Dilworth's decomposition theorem, American Mathematical Monthly 78 (1971), 876–877. https://doi.org/10.2307/2316481
E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Mathematische Zeitschrift 27 (1928), 544–548. https://doi.org/10.1007/BF01171114
P. C. Fishburn, Intransitive indifference with unequal indifference intervals, Journal of Mathematical Psychology 7 (1970), 144–149. https://doi.org/10.1016/0022-2496(70)90062-3

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXVI: Polyhedral L-Convex FunctionsTextbook

Motivation

L-convex functions were defined purely combinatorially, on the integer lattice. Chapter 6's M-convex theory showed that combinatorial definition always extends to a genuine convex function on real space (missions 23-ch06c-mconvexfunctions/24-ch06d-mconvexfunctions); this mission carries out the identical program for the L side. It first shows L $^\natural$ -convexity is exactly integral convexity plus ordinary submodularity — a clean synonym that also explains why submodular set functions are a natural special case — then builds the entire polyhedral (real-variable) theory of L-convex functions: the axioms (SBF[R])/(TRF[R]), two practical local criteria for verifying submodularity without checking every pair of points, the fact that the classical Lovász extension of a submodular set function is itself a polyhedral L-convex function, the six-operation closure toolkit, and — this mission's goal — the L-optimality criterion in its full polyhedral generality, characterizing global optimality by finitely many directional derivatives.

Setting

Fix a finite ground set $V$ . A polyhedral convex function $g : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain is polyhedral L-convex, $g \in L[\mathbb R \to \mathbb R]$ , if it satisfies (SBF[R]): $g(p)+g(q) \ge g(p\vee q)+g(p\wedge q)$ , and (TRF[R]): $\exists r \in \mathbb R$ , $g(p+\alpha\mathbf 1) = g(p)+\alpha r$ for all $p \in \mathbb R^V$ , $\alpha \in \mathbb R$ ; it is polyhedral L $^\natural$ -convex if its lift to one extra real coordinate is polyhedral L-convex. A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is integrally convex if its convex closure agrees, at every real point, with the closure taken using only that point's integral neighborhood. The Lovász extension $\hat\rho$ of a submodular set function $\rho$ is the piecewise-linear interpolation built from the sorted distinct components of $p \in \mathbb R^V$ . The directional derivative $g'(p;d)$ is $\inf_{t>0}(g(p+td)-g(p))/t$ .

Formalization targets

Goal: the polyhedral L-optimality criterion (Theorem 7.33)

For a polyhedral L-convex function $g$ and $p \in \operatorname{dom}_{\mathbb R} g$ : $g(p) \le g(q)$ for all $q$ if and only if $g'(p;\chi_Y) \ge 0$ for every $Y \subseteq V$ and $g'(p;\mathbf 1)=0$ ; for polyhedral L $^\natural$ -convex $g$ , the criterion simplifies to $g'(p;\pm\chi_Y) \ge 0$ for every $Y$ . This is the direct L-side mirror of mission 24-ch06d-mconvexfunctions's M-optimality criterion (Theorem 6.52) and the polyhedral generalization of mission 08-lconvex-functions-i's integer-domain L-optimality criterion (Theorem 7.14): checking global optimality against exponentially many points reduces to $|V|+1$ (or $2|V|$ ) directional-derivative inequalities.

Supporting structural targets

Twelve further results build the polyhedral theory from the ground up. Theorems 7.20-7.21 identify L $^\natural$ -convexity with the conjunction of ordinary submodularity and integral convexity — a genuinely different, function-analytic characterization from the exchange-axiom- style definitions used so far. Propositions 7.23-7.24 give two practical sufficient conditions for verifying (SBF[R]) locally, at a single scale, rather than globally. Proposition 7.25 shows the Lovász extension of any submodular set function is automatically polyhedral L-convex — not in this chunk's own extraction table (its label is preceded by an unlabeled restatement of the same fact, which evidently confused the extractor), found and placed by direct reading. Theorem 7.26 shows an L-convex function's convex extension, when polyhedral, inherits polyhedral L-convexity, continuing mission 26-ch07b-lconvexfunctions's Theorem 7.19. Theorems 7.28-7.32 restate the discrete theory's core equivalences (translation submodularity, the L/L $^\natural$ correspondence, the six basic operations, restrictions) in the polyhedral setting, and Proposition 7.34 shows minimizer sets of linearly-perturbed polyhedral L-convex functions are themselves L-convex polyhedra — flagged by the book itself as a partial result whose full converse characterization (Theorem 7.45) lies beyond this chunk's range.

Significance

Theorems 7.20-7.21's synonym is structurally important: it means every algorithm and theorem already known for submodular-function minimization over $\{0,1\}^V$ -type domains applies, after a midpoint-convexity check, to the vastly larger class of integer-lattice L $^\natural$ -convex functions, with no new proof technique required. Proposition 7.25 is the bridge that lets the combinatorial Lovász extension — the workhorse of submodular optimization for forty years — be recognized as a special case of the polyhedral L-convex function theory this mission builds, explaining why algorithms for one transfer so readily to the other. The goal, Theorem 7.33, is the precise tool chapter 10's continuous-relaxation algorithms for L-convex-function minimization actually verify against: a scaling algorithm's claimed optimum is confirmed correct exactly by checking the criterion's finitely many directional-derivative inequalities.

None of these results are open — they are Murota's account of how the integer-lattice theory of L-convexity survives, result by result, the passage to polyhedral convex functions on $\mathbb R^V$ , mirroring chapter 6's identical program for M-convexity. What this mission contributes is a faithful, machine-checked formal statement of each, including one result (Proposition 7.25) the platform's own automated extractor missed, extending the shared Lean vocabulary (SBFR, TRFR, LovaszExtension, DirDeriv) this series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to verify $g(p) \le g(q)$ directly against every $q \in \mathbb R^V$ ; the book's actual proof instead reduces this to the finite family of directional derivatives via Theorem 7.20's integral-convexity fact (an L $^\natural$ -convex function's local behavior determines its global behavior) applied to the polyhedral setting through Theorem 3.21's general optimality criterion for integrally convex functions — a two-layer reduction (polyhedral $\to$ integral-convexity $\to$ finite local check) with no direct one-step argument. The genuine combinatorial content in this block is in Proposition 7.25's proof: showing the Lovász extension is submodular requires the finite-valued case (a direct calculation split on whether the two perturbed coordinates land in the same or different threshold sets) and then a limiting argument over a sequence of finite-valued truncations $\rho_k \to \rho$ for the general, possibly-infinite case — a genuine two-step argument, not a single inequality chase.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; polyhedral L-(natural-)convex functions are (V→ℝ)→WithTop ℝ. All thirteen numbered results found in this chunk's page range are placed, with one documented scope reduction: Proposition 7.24 states only part (1) (the unconditional-on-magnitude sufficient condition), not part (2)'s sharper, sorted-index-restricted version, which needs the same SortedValues apparatus a second time for no other result's benefit — see HARD.md. " $g_U > -\infty$ " and its variants are replaced by the equivalent (DomR ...).Nonempty hypothesis throughout, matching mission 24-ch06d-mconvexfunctions's identical substitution. "Domain is closed"/"domain is an interval" (Propositions 7.23-7.24) are stated via Mathlib's IsClosed and Set.OrdConnected respectively, the latter being the precise order-theoretic notion of "interval" in a pointwise-ordered space. This mission's base vocabulary is redeclared from missions 08-lconvex-functions-i, 20-ch04b-mconvexsets (for the Lovász extension machinery), 21-ch05b-lconvexsets, 23-ch06c-mconvexfunctions/ 24-ch06d-mconvexfunctions, and 26-ch07b-lconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the thirteen sorrys are welcome; the goal and Proposition 7.25 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota and A. Shioura, "Extreme points of a generalized polymatroid," Discrete Applied Mathematics, 152 (2005), pp. 268-278 [152] (the polyhedral L-convex function theory this mission's real-variable results are drawn from).

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXV: L-Convex Functions via Minimizer PolyhedraTextbook

Motivation

Chapter 5 characterized L-convex sets — sublattice-closed, translation-periodic subsets of $\mathbb Z^V$ — and showed they interact cleanly with integral convexity. Chapter 7 asks the functional analogue: which functions on the integer lattice deserve to be called convex in the "L" sense, and how do they relate back to L-convex sets? This mission (continuing mission 08-lconvex-functions-i, which built the axioms (SBF[Z])/(TRF[Z])/(SBF $^\natural$ [Z]) and proved the L-optimality and L-proximity theorems) answers the second question at its sharpest: an L-convex function is exactly a function whose every weighted-minimizer set is an L-convex polyhedron — the discrete analogue of the fact that a convex function is determined by the convex geometry of its sublevel sets. Along the way it settles the chapter's basic toolkit: operations that preserve L-convexity, the local nature of submodularity, the correspondence with ordinary submodular set functions, and the first two structural facts about the convex extension every L-convex function admits.

Setting

Fix a finite ground set $V$ . A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain is L-convex, $g \in L[\mathbb Z \to \mathbb R]$ , if it satisfies submodularity (SBF[Z]): $g(p)+g(q) \ge g(p\vee q) + g(p\wedge q)$ , and translation invariance (TRF[Z]): $\exists r \in \mathbb R$ , $g(p+\mathbf 1) = g(p) + r$ for all $p$ . It is L $^\natural$ -convex if its lift to one extra coordinate (Eq. (7.2)) is L-convex. A set function $\rho : 2^V \to \mathbb R \cup \{+\infty\}$ is submodular if $\rho(X)+\rho(Y) \ge \rho(X\cup Y) + \rho(X\cap Y)$ ; it corresponds to an L $^\natural$ -convex function supported on $\{0,1\}^V$ via $g(\chi_X) = \rho(X)$ (Eq. (7.5)). The convex closure $\bar g$ of $g$ is its extension to $\mathbb R^V$ by finite convex combinations.

Formalization targets

Goal: L-convexity via minimizer polyhedra (Theorem 7.17)

For $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with bounded nonempty effective domain: $g$ is L-convex if and only if $\arg\min g[-x]$ is an L-convex set for every $x \in \mathbb R^V$ ; the L $^\natural$ analogue holds with L $^\natural$ -convex sets. This is the direct mirror of mission 23-ch06c-mconvexfunctions's own goal (Theorem 6.43, characterizing M-convex functions via M-convex weighted-minimizer polyhedra) — the book's text calls it exactly "how the concept of L-convex functions can be defined from that of L-convex sets."

Supporting structural targets

Eleven further results build the chapter's basic vocabulary. Theorem 7.2 strengthens translation submodularity to allow negative shifts; Theorem 7.3 places L-convexity inside L $^\natural$ - convexity; Proposition 7.4 identifies submodular set functions with a subclass of L $^\natural$ -convex functions via the indicator embedding, and Theorem 7.15 derives the classical submodular-minimizer local-optimality criterion as its corollary; Proposition 7.5 shows submodularity is a local property, needing only unit-distance pairs; Proposition 7.8 transfers L-(natural-)convexity from functions to their effective domains; Proposition 7.9 and Theorem 7.10–7.11 give the chapter's basic examples (univariate and pairwise-difference functions) and its six-operation closure toolkit (scaling, affine reparametrization, linear perturbation, projection, infimal convolution with a separable function, and sums), both for L-convex and L $^\natural$ - convex functions, the latter also admitting interval and coordinate restrictions; Proposition 7.16 shows minimizer sets of L-convex functions are themselves L-convex, the special case ( $x=0$ ) the goal generalizes to every linear perturbation; Theorem 7.19 begins the convex-extension program this chapter's next chunk completes, establishing that the convex closure agrees with $g$ on $\mathbb Z^V$ and inherits its translation constant.

Significance

The goal is significant for the same structural reason as its M-side counterpart: it says L-convexity is not merely a combinatorial condition on lattice differences but is equivalent to a purely polyhedral-geometric one, closing the loop between chapters 5 and 7 the way Theorem 6.43 closes the loop between chapters 4 and 6. Theorem 7.15's corollary status is itself instructive: the well-known fact that a submodular set function's global minimizer needs only local verification against comparable sets — the theoretical basis of every submodular-minimization algorithm in chapter 10 — falls out of the L-optimality criterion (mission 08-lconvex-functions-i's Theorem 7.14) applied to the indicator embedding, rather than needing an independent proof. Theorem 7.10–7.11's six operations are the toolkit every later construction in this chapter and chapter 9's network transformations builds new L-convex functions from old.

None of these results are open — they are Murota's account of the basic function-level theory of L-convexity, mirroring chapter 6's M-convex function theory chunk-by-chunk. What this mission contributes is a faithful, machine-checked formal statement of each, extending the shared Lean vocabulary (SBF, TRF, LNaturalConvex, LConvexSet) that missions 08-lconvex-functions-i and 21-ch05b-lconvexsets began; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to verify L-convexity's submodularity inequality directly against the definition of an L-convex set applied to each minimizer family; the book's actual proof instead routes through Theorem 7.10 (3)'s closure of L-convexity under linear perturbation and Proposition 7.16's minimizer-is-L-convex-set fact for the forward direction, and defers the converse entirely to a later note (Note 7.47, outside this chunk and mission 08's combined range) proved via the integral-convexity machinery of section 7.7 onward. The genuine combinatorial difficulty in this block is upstream, in Theorem 7.10 (5)'s infimal-convolution operation: proving L-convexity of the perturbed function requires a four-term submodularity inequality assembled from the separable function's own convexity and $g$ 's submodularity applied at the optimal $q_1,q_2$ simultaneously — a genuine two-hypothesis combination with no single-inequality shortcut.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; L-(natural-)convex functions are (V→ℤ)→WithTop ℝ. All twelve numbered results found in this chunk's page range are placed, with one documented scope reduction: Theorem 7.19 states only parts (3)-(4) (that the convex closure agrees with $g$ on $\mathbb Z^V$ and inherits its translation constant), not the explicit Lovász-extension-formula construction of parts (1)-(2) and (5), which needs a sorted-distinct- component apparatus no other result in this chunk requires — see HARD.md. " $g_U > -\infty$ " and its variants are replaced by the equivalent (DomZ ...).Nonempty hypothesis throughout, matching mission 24-ch06d-mconvexfunctions's identical substitution (WithTop ℝ has no $-\infty$ element). This mission's base vocabulary (SBF, TRF, LNaturalConvex, etc.) is redeclared verbatim from mission 08-lconvex-functions-i rather than imported, since sibling drafts in this series cannot yet reference one another; LConvexSet is likewise redeclared from mission 21-ch05b-lconvexsets. Contributions completing any of the twelve sorrys are welcome; the goal and Theorem 7.10 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313–371 (the original account of L-convex functions this chapter's basic theory is drawn from).

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Machine Learning Probability Statistics·Captain: mikedeng1

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities II: The Uniform Deviation BoundResearch Paper

Motivation

Bernoulli's law of large numbers says that the relative frequency of a single event $A$ in $l$ independent trials converges in probability to $P(A)$ . Statistics and learning theory need more: the probabilities of a whole class $S$ of events are judged from one and the same sample, so the frequencies must converge uniformly over the class. Uniform convergence can fail even for simple classes (all open subsets of $[0,1]$ ), so one needs a criterion that says when it holds and how fast.

Vapnik and Chervonenkis gave the first distribution-free answer in On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities (Theory Probab. Appl. 16 (1971) 264–280, doi:10.1137/1116025). Its Theorem 2, now called the VC inequality, bounds the probability of a uniform deviation larger than $\varepsilon$ by a combinatorial quantity of the class times an exponentially small factor.

Timeline:

1933: Glivenko and Cantelli prove uniform almost-sure convergence of the empirical distribution function on the line (the class of rays $\{x \le a\}$ ).
1971: Vapnik and Chervonenkis publish the growth function, the VC inequality (Theorem 2), almost-sure convergence under polynomial growth (Theorem 3) and the entropy criterion (Theorem 4).
1972: Sauer and Shelah independently prove the polynomial bound on the growth function (the paper's Lemma 1).
From the late 1970s: the inequality is sharpened in its constants and extended to empirical processes (Dudley, Pollard, Talagrand).

Setting

Let $(X, P)$ be a probability space and $S$ a collection of measurable events $A \subseteq X$ , with probabilities $P_A$ . A sample of size $l$ is a sequence $x_1, \dots, x_l$ of points of $X$ drawn independently with law $P$ , so the sample has the product law $P^l$ on $X^l$ . The relative frequency of $A$ in the sample is $\nu_A^{(l)} = n_A / l$ , where $n_A$ is the number of sample terms in $A$ . The uniform deviation is

\pi^{(l)} = \sup_{A \in S} \bigl|\nu_A^{(l)} - P_A\bigr|.

Each $A \in S$ induces in a sample $x_1, \dots, x_r$ the subsample of terms lying in $A$ . The index $\Delta^S(x_1, \dots, x_r)$ is the number of different subsamples so induced (at most $2^r$ ), and the growth function is $m^S(r) = \max \Delta^S(x_1, \dots, x_r)$ over all samples of size $r$ .

For a double sample $x_1, \dots, x_{2l}$ let $\nu'_A$ and $\nu''_A$ be the frequencies of $A$ in the two semi-samples $x_1, \dots, x_l$ and $x_{l+1}, \dots, x_{2l}$ , and let

\rho^{(l)} = \sup_{A \in S} \bigl|\nu'_A - \nu''_A\bigr|.

Following the paper, $\pi^{(l)}$ and $\rho^{(l)}$ are assumed to be measurable functions of the sample for every $l$ .

Formalization targets

Goal: Theorem 2 (p. 269)

For every $\varepsilon > 0$ and every $l \ge 2/\varepsilon^2$ ,

P\bigl(\pi^{(l)} > \varepsilon\bigr) \le 4\, m^S(2l)\, e^{-\varepsilon^2 l/8}.

The constants $4$ and $1/8$ and the growth function at $2l$ are the paper's.

Milestones, in the order of the proof

Lemma 2 (p. 268): for $l \ge 2/\varepsilon^2$ , $P\{\rho^{(l)} \ge \varepsilon/2\} \ge \tfrac12 P\{\pi^{(l)} > \varepsilon\}$ .
Eq. (11) (p. 270): $P\{\rho^{(l)} \ge \varepsilon/2\} = \int \frac{1}{(2l)!} \sum_{T} \theta\bigl(\rho^{(l)}(T X_{2l}) - \varepsilon/2\bigr)\, dP$ , the sum over all permutations $T$ of the $2l$ positions ( $\theta$ the indicator of $[0, \infty)$ ).
The $\Gamma$ estimate (p. 271): for $0 \le m \le 2l$ ,

\Gamma = \sum_{k : |2k/l - m/l| \ge \varepsilon/2} \frac{\binom{m}{k}\binom{2l-m}{l-k}}{\binom{2l}{l}} \le 2e^{-\varepsilon^2 l/8}.

The per-sample permutation bound (p. 271): for every fixed double sample, $\frac{1}{(2l)!}\sum_T \theta\bigl(\rho^{(l)}(T X_{2l}) - \varepsilon/2\bigr) \le 2\Delta^S(x_1, \dots, x_{2l})\, e^{-\varepsilon^2 l/8}$ .
The semi-sample bound (p. 271): $P\{\rho^{(l)} \ge \varepsilon/2\} \le 2\, m^S(2l)\, e^{-\varepsilon^2 l/8}$ for every $l \ge 1$ .

Further items (consequences, not milestones)

Corollary (p. 269): if $m^S(l) \le l^n + 1$ for all $l$ and some finite $n$ , then $P(\pi^{(l)} > \varepsilon) \to 0$ for every $\varepsilon > 0$ .
Theorem 3 (p. 271): under the same condition, $P(\pi^{(l)} \to 0) = 1$ for an infinite i.i.d. sequence.

Significance

The bound holds for every distribution $P$ and depends on the class only through $m^S(2l)$ . Together with the paper's Theorem 1 (the growth function is either $2^r$ for every $r$ or bounded by $r^n + 1$ ), it shows that every class whose growth function is not identically $2^r$ enjoys uniform convergence at an exponential rate in probability and almost surely. The Glivenko–Cantelli theorem is the special case of rays on the line. The inequality underlies sample-complexity bounds for empirical risk minimization, the "finite VC dimension implies learnability" direction of the fundamental theorem of statistical learning, and the theory of empirical processes indexed by sets.

The result has been proved for more than fifty years; what is missing is a machine-checked proof of it in its original form. Formal libraries contain Hoeffding-type inequalities for independent variables and textbook uniform-convergence statements with other constants, stated for hypothesis classes and loss functions. This mission produces the 1971 statement itself, with its constants and its sequence-based index, together with the combinatorial tail bound for sampling without replacement that the paper states without proof.

Difficulty

The obvious argument applies Hoeffding's inequality to each $A \in S$ and takes a union bound. This fails as soon as $S$ is infinite, and the classes of interest are uncountable. The growth function can only enter after the probabilities $P_A$ have been removed from the event, because only then does the event depend on the finitely many subsamples that $S$ induces on a finite sample. Lemma 2 does this at the price of the condition $l \ge 2/\varepsilon^2$ and a factor $2$ .

The second difficulty is combinatorial. Under a random rearrangement of a fixed double sample, the number of points of an event that fall into the first half is hypergeometric, not binomial. The paper states the required tail bound $\Gamma \le 2e^{-\varepsilon^2 l/8}$ and omits the "simple but long computation". Mathlib has no tail bound for sampling without replacement.

Formalization scope

Samples are functions Fin l → X with 0-based positions; repetitions are allowed. The subsample induced by $A$ is the set of positions {i | x i ∈ A}, so the index counts distinct sets of positions and the growth function maximizes over sequences, not finite point sets (the paper's model; the two differ when $X$ has fewer than $r$ points). The sample law is Measure.pi (fun _ : Fin l => P). The double sample is Fin (l + l) → X, read through Fin.castAdd and Fin.natAdd, and $m^S(2l)$ is growth S (2 * l). Suprema are real suprema over the events of $S$ (values in $[0, 1]$ ; $0$ for $S = \emptyset$ ). Probabilities are values in $[0, \infty]$ and the bounds enter through ENNReal.ofReal. Theorem 3 uses the infinite product Measure.infinitePi and evaluates $\pi^{(l)}$ on the first $l$ coordinates.

The measurability of $\pi^{(l)}$ (p. 265) and of $\rho^{(l)}$ (p. 268) are the paper's own assumptions and are carried as hypotheses. Without them Theorem 2 can fail for uncountable classes; replacing them by a stronger condition such as countability of $S$ would weaken the theorem.

A trivializing formalization is ruled out as follows: $\varepsilon > 0$ is stated, which forces $l \ge 1$ and so avoids the value $0/0 = 0$ of the frequency. The supremum runs over the events of $S$ , not over all subsets. The index counts distinct subsamples, not sets.

Corrections of the printed text:

Lemma 2 is printed for $l > 2/\varepsilon^2$ , but its proof concludes for $l \ge 2/\varepsilon^2$ , and Theorem 2 uses $l = 2/\varepsilon^2$ . The $\ge$ form is stated, which is the stronger statement.
The text before the permutation bound describes the averaged quantity as counting arrangements with $|\nu'_A - \nu''_A| \le \tfrac12\varepsilon$ ; the indicator and the index set of $\Gamma$ count those with $\ge \tfrac12\varepsilon$ , which is what is stated.
Slips in the proof of Lemma 2 that do not affect any statement: $\varepsilon/3$ printed for $\varepsilon/2$ on p. 269, and $<$ , $>$ where Chebyshev's inequality gives $\le$ , $\ge$ .
Theorem 2 prints "more then" for "more than".

Needed infrastructure:

the invariance of Measure.pi under permutations of coordinates;
the splitting of Fin (l + l) into two halves under the product measure;
Chebyshev's inequality for binomial frequencies;
a hypergeometric (sampling without replacement) tail bound.

The last two are reusable beyond this mission. Proofs of any milestone are welcome.

Selected references

V. N. Vapnik and A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl. 16(2) (1971) 264–280. https://doi.org/10.1137/1116025
W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963) 13–30. https://doi.org/10.1080/01621459.1963.10500830
N. Sauer, On the density of families of sets, J. Combin. Theory Ser. A 13 (1972) 145–147. https://doi.org/10.1016/0097-3165(72)90019-2
S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014, Ch. 6 and 28. https://doi.org/10.1017/CBO9781107298019

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis VIII: Quasi L-Convex Functions and the Quasi-Proximity TheoremTextbook

Motivation

Milgrom and Shannon's theory of quasi-supermodularity, developed for monotone comparative statics in economics, showed that many of the consequences of lattice submodularity survive under a much weaker, purely ordinal relaxation of the defining inequality. Chapter 7's final section imports this idea into discrete convex analysis: does L-convexity's optimality and proximity theory survive when the additive submodularity inequality is relaxed to an ordinal condition on the sign pattern of the two relevant differences, rather than their sum? This mission formalizes the chapter's answer for the strongest of the relevant relaxations, (SSQSB) (semistrict quasi submodularity): yes, and the class is large enough to include every strictly increasing rescaling of an L-convex function — exactly mirroring chunk 07's result for the M-convex side, and completing the "quasi" theory on both halves of the exchange-axiom framework before chapter 8 unifies them under conjugacy.

Setting

Let $V$ be a finite ground set and $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ . Building on chunk 08's submodularity axiom (SBF[Z]), this section introduces four ordinal relaxations. $g$ is quasi submodular, satisfying (QSB), if for every $p, q \in \mathbb Z^V$ , $g(p \wedge q) \le g(p)$ or $g(p \vee q) \le g(q)$ . $g$ is semistrictly quasi submodular, satisfying (SSQSB), if additionally $g(p \vee q) \ge g(q) \implies g(p \wedge q) \le g(p)$ and symmetrically. The weak variants (QSBw) and (SSQSBw) restrict attention to points of the effective domain and compare $\max(g(p), g(q))$ against $\min(g(p \wedge q), g(p \vee q))$ directly, with (SSQSBw) additionally allowing the four-way tie $g(p) = g(q) = g(p\wedge q) = g(p \vee q)$ . The linear perturbation of $g$ by $x : V \to \mathbb R$ is $g[x](p) = g(p) + \langle p, x \rangle$ .

Formalization targets

Goal: Theorem 7.54 (the quasi L-proximity theorem)

Let $g$ satisfy (SSQSB) and $g(p) = g(p + \mathbf 1)$ for all $p$ , $n = |V|$ , $\alpha$ a positive integer. If $p_\alpha \in \operatorname{dom} g$ satisfies $g(p_\alpha) \le g(p_\alpha + \alpha \chi_Y)$ for all $Y \subseteq V$ , then $\arg\min g \ne \emptyset$ and there is $p^* \in \arg\min g$ with the componentwise bound $p_\alpha \le p^* \le p_\alpha + (n-1)(\alpha-1) \mathbf 1$ — verbatim the same conclusion, and the same exact bound, as chunk 08's Theorem 7.18(1), now established for the strictly larger class satisfying (SSQSB) rather than (SBF[Z]).

Milestones: Theorems 7.49, 7.53

Theorem 7.49: the full nesting chain (SBF[Z]) $\Rightarrow$ (SSQSB) $\Rightarrow$ (QSB), (SSQSB) $\Rightarrow$ (SSQSBw) $\Rightarrow$ (QSBw), together with the collapse theorem that (SBF[Z]) holds if and only if every linear perturbation of $g$ satisfies (QSBw) — precisely quantifying how weak (QSBw) is pointwise and how the classes reunite under universal perturbation. Theorem 7.53 (the quasi L-optimality criterion): the direct analogue of chunk 08's Theorem 7.14, showing that global (or, for the weaker (QSBw) case, unique-up-to-translation) optimality still reduces to a purely local check against the $2^n - 2$ nontrivial sign-pattern neighbors $p + \chi_X$ .

Significance

The result itself. As with the M-convex case (chunk 07), the proximity theorem is what algorithms actually need: an L-convex-flavored objective transformed by any strictly increasing scalar rescaling (a common device — expressing a network-flow cost in a different currency, or applying a monotone risk adjustment) retains a scaling algorithm's correctness guarantee with exactly the same distance bound, even though the rescaled function is generally no longer L-convex itself.

Formalizing it. No matching item exists on the platform for quasi submodularity or quasi L-convexity in any form. Together with chunk 07 (the M-side quasi-convexity theory), this mission completes the "quasi" relaxation on both halves of the exchange-axiom framework the book develops, immediately before chapter 8 unifies M-convexity and L-convexity under a single conjugacy relationship.

Difficulty

As with chunk 07's quasi M-proximity theorem, the temptation is to imitate chunk 08's L-proximity proof line by line. The overall architecture does survive — translate so $p_\alpha = 0$ , find a lattice-minimal sufficiently-good point, and bound the gap using submodularity — but chunk 08's proof uses (SBF[Z])'s additive inequality directly to compare four function values at once, while this proof must instead route every such comparison through (SSQSB)'s two one-directional implications (Proposition 7.50's quasi-version of the same two-sided inequality), which only ever license moving in one direction at a time depending on which side of a comparison is tight. The book's proof handles this by working with the specific implications (7.43)–(7.44) in place of the L♮-approach property used in chunk 08's proof — an ordinal substitute for the same additive step, at the cost of a case analysis chunk 08's proof did not need.

Formalization scope

This mission builds directly on chunk 08's published items (SBF, DomZ, ArgMin, IndicatorVec), per the platform's textbook convention that a later chapter section of the same book imports an earlier one's definitions; its own namespace DiscreteConvex.LConvexFunctions.Quasi nests under chunk 08's DiscreteConvex.LConvexFunctions accordingly. Note the sign convention of the linear perturbation here, $g[x](p) = g(p) + \langle p,x\rangle$ , is the opposite of the M-side's $f[p](x) = f(x) - \langle p,x\rangle$ (chunks 06–07) — verified against the book's own formula rather than assumed by analogy.

A trivializing formalization of the goal would silently strengthen (SSQSB) back to plain (SBF[Z]) (making this mission redundant with chunk 08's Theorem 7.18) or loosen the exact bound $(n-1)(\alpha-1)$ ; neither is done. Only (QSB), (SSQSB), (QSBw), (SSQSBw) are drafted, matching exactly what the chosen three items need; the polyhedral L-convex-function bridge (§7.8–7.9, Theorems 7.40–7.46) and the level-set characterizations (Theorems 7.51–7.52) are left for a follow-on mission. Contributions building the 0L ↔ S correspondence (Theorem 7.40, a bridge back to chunk 04's submodular-set-function vocabulary) or the scaled quasi L-minimizer-cut analogue are welcome.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
P. Milgrom, C. Shannon, "Monotone comparative statics," Econometrica, 62(1), 1994, pp. 157–180.

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXIV: Quasi M-Convex FunctionsTextbook

Motivation

Every characterization of M-convexity so far in this series — the exchange axiom, the optimality criterion, convex extensibility, the directional-derivative/subdifferential correspondence — has been an equivalence with M-convexity itself: a function either is M-convex or it is not. Section 6.14 asks a different question: what happens when the exchange axiom's defining inequality is relaxed to only the sign patterns it actually forces? The answer is a hierarchy of "quasi M-convex" conditions — weaker than M-convexity, strong enough to keep the optimality criterion and the proximity/minimizer-cut theorems intact — and, at the top of that hierarchy, a genuinely new characterization of M-convexity itself: a function is M-convex if and only if every one of its linear perturbations is quasi M-convex in the weakest sense. This mission formalizes that entire hierarchy and its capstone, plus two further characterizations of polyhedral M-convexity (via directional derivatives, subdifferentials, and weighted-minimizer polyhedra) that complete the real-variable theory chunk 24-ch06d-mconvexfunctions began.

Setting

Fix a finite ground set $V$ and $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ . Write $\Delta f(z;v,u) = f(z + \chi_v - \chi_u) - f(z)$ . Relaxing the exchange axiom's inequality $\Delta f(x;v,u) + \Delta f(y;u,v) \le 0$ to the sign patterns it forces gives quasi M-convexity (QM) and semistrict quasi M-convexity (SSQM), and requiring only some pair $(u,v)$ rather than every $u$ gives their weaker variants (QMw), (SSQMw); the minimization-only variants (SSQM $\ne$ ), (SSQM $\ne_w$ ) replace " $x,y \in \operatorname{dom} f$ " with " $f(x) \ne f(y)$ ". The set-level analogue (Q-EXC)/(Q-EXCw) relaxes the M-convex-set exchange axiom the same way. For $\alpha \in \mathbb R$ , the level set $L(f,\alpha) = \{x \in \mathbb Z^V : f(x) \le \alpha\}$ . A polyhedral convex function $f : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ is (real-variable) M-convex, $f \in M[\mathbb R \to \mathbb R]$ , if it satisfies the real exchange axiom (M-EXC[R]) from chunk 24-ch06d-mconvexfunctions; $L_0[\mathbb R]$ denotes polyhedra realized as $D(\gamma)$ for a triangle-inequality distance function $\gamma$ , and $M_0[\mathbb R]$ denotes real M-convex polyhedral cones.

Formalization targets

Goal: the quasi M-convexity hierarchy (Theorem 6.68)

For $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ : (1) the implication diagram (M-EXC[Z]) $\Rightarrow$ (SSQM) $\Rightarrow$ (QM), (M-EXCw[Z]) $\Rightarrow$ (SSQMw) $\Rightarrow$ (QMw), (M-EXC[Z]) $\Leftrightarrow$ (M-EXCw[Z]), (SSQM) $\Rightarrow$ (SSQMw), (QM) $\Rightarrow$ (QMw); (2) $f$ satisfies (M-EXC[Z]) if and only if $f[p]$ satisfies (QMw) for every $p \in \mathbb R^V$ . This theorem is not named in the chunk's own extraction table — the automated extractor, which requires a result's label to start a text line, misses it because it opens mid-paragraph directly after an ASCII-rendered implication diagram — but it is the capstone of the section: its own proof is "combining Theorems 6.72 and 6.74," both formalized here as milestones, and part (2) answers exactly the question a reader of this stretch would ask: what does the entire apparatus of quasi M-convexity ultimately say about M-convexity itself?

Supporting structural targets

Fourteen further results build the hierarchy and complete the real-variable theory. Proposition 6.62 checks the two halves of chunk 24-ch06d-mconvexfunctions's Theorem 6.61 agree at integer points. Theorems 6.63–6.64 add two more characterizations of polyhedral M-convexity — via positively homogeneous directional derivatives and $L_0[\mathbb R]$ -valued subdifferentials, and via $M_0[\mathbb R]$ / $M_0[\mathbb Z|\mathbb R]$ -valued weighted-minimizer polyhedra — to the convex-extensibility characterization chunk 23-ch06c-mconvexfunctions proved. Theorem 6.67 gives two equivalent reformulations of (QMw) as pointwise inequalities; Propositions 6.69–6.70 and Theorems 6.72–6.74 build the level-set/perturbation machinery the goal needs. Theorem 6.75 is the (SSQM $\ne_w$ ) analogue of Theorem 6.67. Theorem 6.76 is the quasi M-optimality criterion (optimality still characterized by local non-improvement, under only the weak quasi-convexity hypotheses). Theorems 6.77–6.79 show the M-minimizer-cut and M-proximity theorems (from missions 06-mconvex-functions-i and 23-ch06c-mconvexfunctions) hold verbatim under the strictly weaker (SSQM $\ne$ ) hypothesis.

Significance

The hierarchy's practical payoff is immediate: Theorems 6.77–6.79 mean the algorithms of chapter 10 that rely on minimizer cuts and proximity bounds do not actually need the full exchange axiom to run correctly on nonlinearly rescaled M-convex functions (Example 6.66 shows any nondecreasing scaling $\phi \circ f$ of an M-convex $f$ is quasi M-convex, yet nonlinear scalings are common in practice and destroy M-convexity itself). The goal, Theorem 6.68, is significant independently: it says the exchange axiom — a condition that looks irreducibly combinatorial, quantifying over pairs of points and directions — is equivalent to a purely ordinal, perturbation-based condition (every linear tilt of $f$ has no strict local improvement that a level set can't witness), giving a genuinely different lens on why M-convexity is the right discrete analogue of convexity. Theorems 6.63–6.64 close out chunk 24-ch06d-mconvexfunctions's program of characterizing polyhedral M-convexity in every classical convex-analytic vocabulary at once (directional derivatives, subdifferentials, weighted minimizers), completing the bridge to Chapter 8's duality theory that chunk builds toward.

None of these results are open — they are Murota's account of how far the exchange axiom's defining inequality can be relaxed while keeping optimization theory intact. What this mission contributes is a faithful, machine-checked formal statement of each, including four theorems (6.68, 6.76, 6.77, 6.78) the platform's own automated extractor missed entirely, extending the shared Lean vocabulary (DeltaF, QMw, LevelSet) the Discrete Convex Analysis series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to prove the implication diagram's six arrows and the perturbation equivalence as six independent facts; the book's own proof of part (2) instead derives it in one step from Theorems 6.72 and 6.74 — themselves nontrivial (Theorem 6.74's proof strengthens the local exchange axiom equivalence (Theorem 6.4) to hold whenever the domain merely satisfies (Q-EXCw), then runs a bipartite-matching argument on a 4-point neighborhood to verify the resulting local condition). The difficulty is genuinely upstream of the goal's own statement: everything the goal needs is already proved by the time Theorem 6.68 is reached, so the formalization work is in stating the sixteen distinct axioms and their level-set reformulations precisely enough that "combining 6.72 and 6.74" is literally how a Lean proof would proceed.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; integer-domain functions are (V→ℤ)→WithTop ℝ, real-domain ones (V→ℝ)→WithTop ℝ. All fifteen numbered results — including the four (6.68, 6.76, 6.77, 6.78) the automated extractor missed because their labels open mid-paragraph — are placed as milestone or goal, and every clause of every one is stated in full; no partial-coverage scope reduction was needed in this chunk (contrast chunks 22-ch06b-mconvexfunctions/24-ch06d-mconvexfunctions, which restated 4-of-8-part operations theorems). Two formalization choices are recorded in HARD.md/MODERATION_NOTES.md: " $\inf f[-p] > -\infty$ " is replaced by the equivalent (ArgMinOn ...).Nonempty hypothesis (WithTop ℝ has no $-\infty$ element), matching mission 23-ch06c-mconvexfunctions's identical substitution; and $M_0[\mathbb R]$ / $M_0[\mathbb Z|\mathbb R]$ are realized via the book's own indicator-function device rather than a freestanding cone axiom. This mission's definitions are redeclared from chunks 06-mconvex-functions-i, 22–24-ch06*-mconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the fifteen sorrys are welcome; the goal and Theorem 6.74 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
M. Avriel, W. E. Diewert, S. Schaible, and I. Zang, Generalized Concavity, Plenum Press, 1988 (the continuous quasi-convexity theory this chapter's discrete analogue generalizes).

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Convex Optimization Operations Research Probability·Captain: mikedeng1

Star-Shaped Risk Measures 2: law-invariant star-shaped risk measures as robustified Value-at-RiskResearch Paper

Motivation

Financial regulators and risk managers summarise the loss distribution of a position by a single number, the capital that must be held against it. The two standards in practice are Value-at-Risk (VaR), a quantile of the loss, and expected shortfall (ES), an average of the upper quantiles; ES is the standard of the Basel framework. Both depend on the position only through its probability law, a property called law invariance, which is what allows them to be estimated from data.

The axiomatic theory of risk measures, starting with Artzner, Delbaen, Eber and Heath (1999) and Föllmer and Schied (2002), centres on convex risk measures. VaR is not convex, however, and neither are its robust variants used in practice: the maximum or the median of VaR over a set of scenario models, or the benchmark-loss VaR of Bignozzi, Burzoni and Munari (2020). Castagnoli, Cattelan, Maccheroni, Tebaldi and Wang (Operations Research 70(5), 2022) propose star-shapedness as the common property of all of these measures: doubling the exposure to a risky position at least doubles the required capital. Their Section 7 identifies exactly which law-invariant risk measures are star-shaped.

Timeline:

1999: Artzner et al. introduce coherent risk measures and show that ES-type measures dominate VaR (doi:10.1111/1467-9965.00068).
2002: Föllmer and Schied introduce convex risk measures.
2020: Mao and Wang characterise the risk measures consistent with second-order stochastic dominance as infima of ES-based functionals.
2022: Castagnoli et al. prove Theorem 5, the corresponding characterisation of star-shaped law-invariant risk measures through VaR.

Setting

Let $(\Omega,\mathcal F,P)$ be a probability space with $P$ atomless: every event of positive probability contains an event of strictly smaller positive probability. A position is a bounded measurable function $X:\Omega\to\mathbb R$ , read as a loss: $X(\omega)>0$ is money lost in state $\omega$ . The positions form a linear space $\mathcal X$ that contains the constants and carries the pointwise order $X\geqq Y$ .

A risk measure is a function $\rho:\mathcal X\to\mathbb R$ that is monotone ( $X\geqq Y\Rightarrow\rho(X)\ge\rho(Y)$ ), translation invariant ( $\rho(X-m)=\rho(X)-m$ for real $m$ ) and normalized ( $\rho(0)=0$ ). It is star-shaped if $\rho(\lambda X)\ge\lambda\rho(X)$ for all $X$ and all $\lambda>1$ , and law-invariant if $X$ and $Y$ with the same law under $P$ satisfy $\rho(X)=\rho(Y)$ . Its acceptance set is $\mathcal A_\rho=\{X\mid\rho(X)\le 0\}$ . A set $S$ in a vector space is star-shaped if $\lambda s\in S$ whenever $s\in S$ and $\lambda\in[0,1]$ .

For $\alpha\in(0,1)$ the Value-at-Risk of $X$ is

\mathrm{VaR}_\alpha(X)=\inf\{x\in\mathbb R : P(X>x)\le 1-\alpha\}.

Write $F_X(x)=P(X\le x)$ . A loss $X$ first-order stochastically dominates $Y$ , written $X\succsim_{\mathrm{FSD}}Y$ , if $F_X\ge F_Y$ pointwise, so $X$ is the smaller loss.

Formalization targets

Goal: Theorem 5, (i) ⇔ (ii)

For a function $\rho:\mathcal X\to\mathbb R$ the following are equivalent:

$\rho$ is a star-shaped and law-invariant risk measure;
there is a star-shaped set $\mathcal G$ of increasing functions $g:(0,1)\to\mathbb R$ with $g(0+)\le 0$ such that

\rho(X)=\inf_{g\in\mathcal G}\ \sup_{\alpha\in(0,1)}\ \{\mathrm{VaR}_\alpha(X)-g(\alpha)\}\qquad X\in\mathcal X.\tag{25}

The goal fixes neither $\mathcal G$ nor any constant. The paper's "Moreover" clause, closure of the class under the operations of Theorem 1, is excluded: its proof is a citation of Liu et al. (2020, Theorem 2) plus "the rest is straightforward".

Milestones

Eq. (7): $\rho(X)=\min\{m\in\mathbb R\mid X-m\in\mathcal A_\rho\}$ for every risk measure.
Proposition 2: for a risk measure $\rho$ , the following are equivalent: $\rho$ is star-shaped; $\mathcal A_\rho$ is star-shaped; $\rho=\rho_{\mathcal A}$ for a star-shaped acceptance set $\mathcal A$ .
Eq. (A.1): $F_X\ge F_Y$ if and only if $\mathrm{VaR}_\alpha(X)\le\mathrm{VaR}_\alpha(Y)$ for all $\alpha\in(0,1)$ .
FSD consistency (proof of Theorem 5): if $P$ is atomless and $\rho$ is monotone and law-invariant, then $X\succsim_{\mathrm{FSD}}Y\Rightarrow\rho(X)\le\rho(Y)$ .

Significance

Theorem 5 says that the star-shaped law-invariant risk measures are exactly the robustifications of VaR: each benchmark $g$ in $\mathcal G$ gives a capital requirement $\sup_\alpha\{\mathrm{VaR}_\alpha(X)-g(\alpha)\}$ , and $\rho$ takes the most favourable benchmark. The class contains VaR, ES, their scenario-based maxima and the benchmark-loss VaR. The theorem parallels Theorem 4 of the same paper, derived from Mao and Wang (2020), in which ES replaces VaR and SSD-consistency replaces law invariance. Together the two results separate the two classes: VaR is star-shaped and law-invariant but not SSD-consistent. Section 7 also records that star-shaped law-invariant measures are in general not minima of law-invariant convex risk measures, which is why a representation specific to VaR is needed.

On the formal side, the result is proved in the paper; no machine-checked proof is known. The mission yields a Lean library of law-invariant risk measures on bounded measurable positions, a quantile-based Value-at-Risk with its basic order properties, and the FSD-consistency of monotone law-invariant functionals on atomless spaces. That last statement is the probabilistic core that other law-invariant representation theorems reuse.

Difficulty

The direction (ii) ⇒ (i) is a direct computation. The direction (i) ⇒ (ii) rests on FSD consistency. The obvious argument writes $X\succsim_{\mathrm{FSD}}Y$ as $X\le Y$ and applies monotonicity, but first-order dominance compares only laws, and two positions ordered in law need not be ordered state by state. One must construct positions with the laws of $X$ and $Y$ that are ordered pointwise, and this uses atomlessness in an essential way: on a space with atoms, the construction can fail. The remaining steps are bookkeeping of extended-real infima and suprema, including levels $\alpha$ near $0$ where $g$ may diverge.

Formalization scope

Positions are the bounded measurable functions $\Omega\to\mathbb R$ (a Submodule ℝ (Ω → ℝ), definition Positions) with the pointwise order, rather than equivalence classes in $L^\infty(\Omega,\mathcal F,P)$ . For a law-invariant $\rho$ the two readings agree: almost surely equal positions have the same law, and a position that dominates another almost surely has a pointwise modification with the same law that dominates it everywhere. Atomlessness is defined locally (IsAtomless), because Mathlib's NoAtoms only states that singletons are null, which is weaker. Law invariance compares push-forward measures P.map X; measurability is part of Positions, so these are never degenerate.

VaR is an sInf of reals and is applied only to bounded positions at levels in $(0,1)$ , where the infimum is over a nonempty set that is bounded below. In (25) each function $g$ has domain exactly $(0,1)$ , "increasing" means weakly increasing, and $g(0+)\le 0$ is stated as $\inf_{\alpha\in(0,1)}g(\alpha)\le 0$ in the extended reals. Both sides of (25) are compared in the extended reals, since the inner supremum can be $+\infty$ . The minimum in Eq. (7) is an attained minimum (IsLeast), and the supremum condition on acceptance sets is a least upper bound (IsLUB).

These choices rule out the trivialising formalizations of (25). A real-valued supremum would read an unbounded supremum as $0$ . Dropping monotonicity of $g$ or the condition $g(0+)\le 0$ describes a larger class that includes non-normalized functionals. Allowing $\mathcal G=\emptyset$ is excluded because the left side of (25) is a real number and the right side would be $+\infty$ .

The following are not part of the mission: the "Moreover" clause of Theorem 5, Theorem 4 (its main direction is Mao and Wang 2020, Theorem 3.1), and Proposition 7 (ES as the smallest SSD-consistent risk measure dominating $\mathrm{VaR}_\alpha$ ), which would need second-order dominance and ES as additional definitions.

Reusable infrastructure: VaR on bounded measurable functions and its homogeneity and translation properties; the equivalence (A.1); existence of a uniform random variable on an atomless probability space and the quantile coupling it provides. Proofs of these as separate lemmas are welcome.

Selected references

E. Castagnoli, G. Cattelan, F. Maccheroni, C. Tebaldi, R. Wang, Star-Shaped Risk Measures, Operations Research 70(5):2637–2654, 2022. https://doi.org/10.1287/opre.2022.2303
P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent Measures of Risk, Mathematical Finance 9(3):203–228, 1999. https://doi.org/10.1111/1467-9965.00068
H. Föllmer, A. Schied, Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. https://doi.org/10.1515/9783110463453
T. Mao, R. Wang, Risk Aversion in Regulatory Capital Principles, SIAM Journal on Financial Mathematics 11(1):169–200, 2020 (cited as Mao and Wang 2020 in the source paper).
V. Bignozzi, M. Burzoni, C. Munari, Risk Measures Based on Benchmark Loss Distributions, Journal of Risk and Insurance, 2020 (cited as Bignozzi et al. 2020 in the source paper).

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Convex Optimization Operations Research Optimization·Captain: mikedeng1

Star-Shaped Risk Measures 1: star-shaped risk measures are the minima of convex risk measuresResearch Paper

Motivation

A risk measure turns the random loss of a financial position into a single number: the amount of capital a regulator or a risk manager requires to hold the position. Two families of risk measures dominate practice and theory. Value-at-Risk (VaR), a quantile of the loss distribution, is used in banking and insurance regulation; it is positively homogeneous but not convex, so it can penalize diversification. Convex risk measures (Föllmer and Schied 2002; Frittelli and Rosazza Gianini 2002), and their positively homogeneous subclass of coherent risk measures (Artzner, Delbaen, Eber and Heath 1999), reward diversification and come with a duality theory, but they exclude VaR and many of its robust variants.

Castagnoli, Cattelan, Maccheroni, Tebaldi and Wang (Operations Research 70(5), 2022) study the class that contains both: star-shaped risk measures, those for which increasing the exposure to a position never decreases the risk per unit of exposure. The class is closed under the aggregation operations used in practice (averages across models, worst cases across scenarios, medians, risk sharing), which convexity is not. It contains VaR, Expected Shortfall, their scenario-based robustifications such as $\mathrm{MaxVaR}$ , the benchmark-loss VaR of Bignozzi et al. (2020), and utility-based shortfall risk for utilities with the Landsberger–Meilijson property.

Timeline. Artzner et al. (1999) axiomatize coherent risk measures. Föllmer and Schied (2002) and Frittelli and Rosazza Gianini (2002) introduce convex ones. Föllmer and Schied (2016, Proposition 4.47) show that VaR is the minimum of the convex risk measures dominating it. Castagnoli et al. (2015) state the representation below without proof. The 2022 paper proves it for every star-shaped risk measure, and shows that the property characterizes the class.

Setting

Fix a set $\Omega$ of states. The space of positions $\mathcal X$ is a linear space of bounded functions $X:\Omega\to\mathbb R$ containing every constant function; the constant $m$ is identified with the position that pays $m$ in every state. A value $X(\omega)>0$ is a loss. No probability measure is fixed. $\mathcal X$ is ordered pointwise: $X\geqq Y$ means $X(\omega)\ge Y(\omega)$ for every $\omega$ .

A risk measure is a function $\rho:\mathcal X\to\mathbb R$ that is monotone ( $X\geqq Y\Rightarrow\rho(X)\ge\rho(Y)$ ), translation invariant ( $\rho(X-m)=\rho(X)-m$ for all real $m$ ) and normalized ( $\rho(0)=0$ ). It is

star-shaped if $\rho(\lambda X)\ge\lambda\rho(X)$ for all $X$ and all $\lambda>1$ ;
convex if $\rho(\lambda X+(1-\lambda)Y)\le\lambda\rho(X)+(1-\lambda)\rho(Y)$ for all $X,Y$ and all $\lambda\in(0,1)$ ;
positively homogeneous if $\rho(\lambda X)=\lambda\rho(X)$ for all $\lambda>0$ ;
coherent if it is positively homogeneous and subadditive, $\rho(X+Y)\le\rho(X)+\rho(Y)$ .

The acceptance set of $\rho$ is $\mathcal A_\rho=\{X\in\mathcal X\mid\rho(X)\le0\}$ . More generally, an acceptance set is a subset $\mathcal A\subseteq\mathcal X$ with $\sup\{m\in\mathbb R\mid m\in\mathcal A\}=0$ that is closed downwards ( $X\in\mathcal A$ , $Y\leqq X$ imply $Y\in\mathcal A$ ). It is convex if convex and coherent if a convex cone, and it generates $\rho_{\mathcal A}(X)=\inf\{m\mid X-m\in\mathcal A\}$ . A set $S$ is star-shaped if $\lambda s\in S$ for all $s\in S$ and $\lambda\in[0,1]$ .

In the Lean development the space of positions is PositionSpace Ω, positions are elements of 𝒳.carrier, and the predicates are IsRiskMeasure, IsStarShaped, IsConvexRiskMeasure, IsCoherentRiskMeasure, acceptanceSet, IsAcceptanceSet, IsConvexAcceptanceSet.

Formalization targets

Goal: Theorem 2 (p. 2643)

For a risk measure $\rho$ , the following are equivalent:

$\rho$ is star-shaped;
there is a set $\Gamma$ of convex risk measures with

\rho(X)=\min_{\gamma\in\Gamma}\gamma(X)\qquad\text{for all }X\in\mathcal X;

there is a family $\{\mathcal A_\beta\}_{\beta\in B}$ of convex acceptance sets with

\rho(X)=\min\{m\in\mathbb R\mid X-m\in\mathcal A_\beta\text{ for some }\beta\in B\}\qquad\text{for all }X\in\mathcal X.

Moreover, for star-shaped $\rho$ , $\Gamma$ may be taken to be the set of all convex risk measures $\gamma\geqq\rho$ , and the family to be their acceptance sets. The minima are attained. The goal fixes no particular $\Omega$ , $\mathcal X$ or $\Gamma$ .

Milestones

Proposition 1 (p. 2642): star-shapedness is equivalent to $\rho(\alpha X)\le\alpha\rho(X)$ for $\alpha\in(0,1)$ , and to the risk-to-exposure ratio $\beta\mapsto\rho(\beta X)/\beta$ being increasing on $(0,\infty)$ .
Eq. (7) (p. 2642): $\rho(X)=\min\{m\mid X-m\in\mathcal A_\rho\}$ .
Proposition 2 (p. 2642): $\rho$ is star-shaped iff $\mathcal A_\rho$ is star-shaped iff $\rho=\rho_{\mathcal A}$ for a star-shaped acceptance set $\mathcal A$ .
Theorem 1 (p. 2643), in four parts: the infimum, supremum, $\mu$ -average and inf-convolution of star-shaped risk measures are star-shaped risk measures.
Proposition 3 (p. 2642): for subadditive risk measures, star-shaped, positively homogeneous and convex coincide.
Theorem 2, positively homogeneous case: the same equivalence with "positively homogeneous", "coherent risk measures" and "coherent acceptance sets".
Corollary 1 (p. 2645): $\inf_{X\in\mathcal Y}\rho(X)=\inf_{\gamma\in\Gamma}\inf_{X\in\mathcal Y}\gamma(X)$ for any $\mathcal Y\subseteq\mathcal X$ .

Significance

Theorem 2 identifies star-shaped risk measures as exactly the lower envelopes of convex risk measures. Consequences drawn in the paper: minimizing a star-shaped risk measure over a set of positions reduces to a family of convex risk-minimization problems (Corollary 1, Proposition 6), each convex risk measure in the envelope carries its dual representation (Proposition 5), and VaR-type measures inherit a tractable structure without being convex. The positively homogeneous case gives the analogous statement for VaR-like measures in terms of coherent ones, generalizing Föllmer–Schied Proposition 4.47.

The paper proves these results; the mission adds a machine-checked proof on a general space of bounded positions, with the attainment of every minimum made explicit. No formalization of star-shaped risk measures is known to exist. The platform mission Coherent Measures of Risk formalizes the Artzner et al. axioms on finitely many states with a gain convention; its definitions are not reused here. The definitions of this mission (risk measures on a space of bounded functions, acceptance sets, the aggregation operations) are reusable by any later development of monetary risk measures without a reference probability.

Difficulty

The equivalence (2)⇔(3) and the direction (2)⇒(1) reduce to closure properties of the class; the content is (1)⇒(2) together with the "Moreover" clause. The obvious first idea, taking the convex hull or convex envelope of $\rho$ or of $\mathcal A_\rho$ , fails: the convex hull of $\mathcal A_\rho$ generates a convex risk measure below $\rho$ , not above it, and a single convex risk measure cannot equal a non-convex $\rho$ . What is needed is, for each position, a convex risk measure that dominates $\rho$ everywhere and touches it at that position, and it must be monotone, translation invariant and normalized, not merely a convex functional. Attainment of the minimum, rather than an infimum, is part of the claim.

Formalization scope

$\mathcal X$ is any Submodule ℝ (Ω → ℝ) containing the constants whose elements are bounded, bundled as PositionSpace Ω. It is not specialised to all bounded functions, and no measurability or probability is imposed; positive values are losses.
"min" is always IsLeast (attained). The acceptance-set axiom $\sup\{m\mid m\in\mathcal A\}=0$ is IsLUB, not sSup … = 0. $\rho_{\mathcal A}$ uses the real sInf, which is genuine for acceptance sets on bounded positions.
Every $\gamma\in\Gamma$ is a full risk measure: monotone, translation invariant, normalized and convex. A representation of $\rho$ as a minimum of arbitrary, non-normalized convex functionals holds for every monotone translation-invariant map and is not this theorem; such a formalization is ruled out.
The family in (3) is a set of subsets of $\mathcal X$ , with no finiteness or nonemptiness assumption.
A coherent acceptance set is convex and closed under multiplication by every $t>0$ .
Theorem 1 adds the hypotheses the page leaves implicit: a nonempty index set for supremum and infimum; the power-set σ-algebra and a countably additive probability measure for the average (the paper's proof also covers capacities with Choquet integrals, which are not stated here); $n\ge1$ and the normality condition (10) for the inf-convolution.
Corollary 1 computes infima in the extended reals, so the set $\mathcal Y$ may be empty and the infima may be $-\infty$ .

Contributions welcome: proofs of any milestone, and in particular general lemmas on risk measures on a space of bounded functions (the bounds $\inf X\le\rho(X)\le\sup X$ , properties of $\rho_{\mathcal A}$ , convexity of $\rho_{\mathcal A}$ for convex $\mathcal A$ ), which are reusable across risk-measure missions.

Selected references

E. Castagnoli, G. Cattelan, F. Maccheroni, C. Tebaldi, R. Wang, Star-Shaped Risk Measures, Operations Research 70(5):2637–2654, 2022. https://doi.org/10.1287/opre.2022.2303
P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent Measures of Risk, Mathematical Finance 9(3):203–228, 1999. https://doi.org/10.1111/1467-9965.00068
H. Föllmer, A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics 6(4):429–447, 2002. https://doi.org/10.1007/s007800200072
M. Frittelli, E. Rosazza Gianin, Putting order in risk measures, Journal of Banking & Finance 26(7):1473–1486, 2002. https://doi.org/10.1016/S0378-4266(02)00270-4
H. Föllmer, A. Schied, Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter, 2016. https://doi.org/10.1515/9783110463453

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Dynamic Programming Linear Optimization Operations Research+2·Captain: Shuze Chen

Markov Decision Processes XIV: Positive Models and Linear Programming Duality for MDPsTextbook

Motivation

Chunk 07a built the general theory of infinite-horizon Markov Decision Processes and its sharpest special case, contracting models, where Banach's fixed point theorem delivers existence, uniqueness, and an explicit convergence rate all at once. That theory answers "does an optimal policy exist, and can I compute it by iterating a fixed point equation?" This mission answers the two questions a practitioner asks next: what happens when the reward's negative part, rather than its positive part, is the one that needs controlling (positive models, §7.4), and — more strikingly — can finding an optimal policy be reduced to solving a genuine linear program, the single most heavily-optimized computational primitive in all of operations research (§7.5)?

Setting

A positive Markov Decision Model is the mirror image of chunk 07a's general setup: instead of bounding the reward's positive part with an upper bounding function, the negative part is bounded by an integrability quantity $\varepsilon$ , and the roles of "largest subharmonic" and "smallest superharmonic" swap accordingly. The computational sections build on chunk 07a's contracting theory directly: Howard's policy improvement algorithm iteratively replaces a decision rule with a strict pointwise improvement; the linear-programming approach recasts the entire optimization problem — the value function and the optimal policy — as a primal/dual pair of linear programs, not over finite vectors but over an infinite-dimensional space of measurable functions ( $v \in IM$ ) and finitely-additive-in-spirit measures ( $\mu \in M_b$ ); and state-space discretization approximates an infinite (Borel) state space by a finite grid, with an explicit, computable bound on the resulting numerical error.

Formalization targets

The goal, Theorem 7.5.8 (Strong Duality), is the section's deepest result: under chunk 07a's contracting Structure Theorem's own hypotheses, the primal linear program $(P)$ is solved exactly by the true optimal value function $J_\infty$ , the dual program $(D)$ is solved by the occupation measure of any optimal stationary policy, and the two optimal values coincide. The milestones build up to it in three groups: the positive-model mirror theory (Lemmas 7.4.1-7.4.2, Theorems 7.4.3 and 7.4.5); Howard's policy improvement and its termination guarantee (Theorem 7.5.1, Corollary 7.5.3); and the linear-programming machinery itself (weak duality, complementary slackness, and the finite-state specialization that recovers an ordinary finite linear program, Theorems 7.5.6, 7.5.7, 7.5.9) together with the discretization error bounds that make the whole theory numerically usable (Proposition 7.5.11, Theorem 7.5.12).

Significance

The strong duality theorem is genuinely new content relative to what is already on the platform: the existing finite-dimensional LP duality missions (SmaleNinth.lp_strong_duality, LinearOptimization.lp_general_weak_duality, and others in the linear-optimization field) all operate over $\mathbb R^n$ -valued vectors, while this theorem's primal and dual variables are a measurable function on a general Borel space and a measure on a general Borel space respectively — an infinite-dimensional linear program in the fullest sense. Theorem 7.5.9, the finite-state specialization, is the one point of genuine hypothesis-for-hypothesis contact with that prior art (checked directly; see STATUS.md for why it was drafted fresh rather than cited as a reference item), and it is exactly there that the reduction to an ordinary finite LP — with the platform's familiar vertex/extreme-point vocabulary — becomes visible.

Difficulty

Constructing the occupation measure $\mu^{f^\infty}_p$ without a canonical infinite-horizon path measure is the central technical challenge: it must be a genuine Measure (E × A), not merely a real-valued functional, since the dual program optimizes over a space of such measures. This mission builds it from iterated Measure.bind (pushing the initial law $p$ forward through the model's kernel under a fixed stationary decision rule) combined with a countable Measure.sum of $\beta^k$ -scaled terms — a construction that stays entirely within Mathlib's existing measure-theoretic vocabulary without needing an Ionescu–Tulcea-style infinite product. A second, different difficulty is the state-space discretization section's grid interpolation, which presupposes a convex-combination structure ( $x = \sum_k\lambda_kx_k$ for grid points $x_k$ ) on the state space that a general Borel space does not carry; this mission represents the grid operator and grid bounding function as data satisfying exactly the structural properties their two target theorems' own proofs use, rather than reconstructing the literal interpolation scheme — a deliberate, documented scope decision (see MODERATION_NOTES.md), not an approximation of either theorem's mathematical content.

Formalization scope

Every operator and value-function construction restates chunk 07a's own vocabulary (per this series' file-ownership convention, an independent copy in this chunk's own namespace), extended by the positive-model integrability bound $\varepsilon$ , the occupation-measure/linear-program apparatus of §7.5.2, and the grid-approximation data of §7.5.3. The primal/dual optimal values $\mathrm{val}(P)$ / $\mathrm{val}(D)$ are kept EReal-valued rather than real-valued specifically so that Theorem 7.5.6's own finiteness claims ( $-\infty < \mathrm{val}(D)$ , $\mathrm{val}(P) < \infty$ ) remain genuine, checkable content rather than being trivialized by a real-valued sInf/sSup's always-finite convention. Theorem 7.5.9's "optimal vertex" is stated via an explicit convex-combination (extreme-point) characterization using ENNReal weights, since Measure does not carry the module structure Mathlib's own Set.extremePoints requires.

Selected references

N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, Universitext, Springer, 2011. DOI: 10.1007/978-3-642-18324-9.
R. A. Howard, Dynamic Programming and Markov Processes, MIT Press, 1960 (the policy improvement algorithm this section names after him).
E. V. Denardo, "On linear programming in a Markov decision problem," Management Science, 1970 (the classical finite-state linear-programming formulation this section generalizes).
W. J. Heilmann, "A note on the dual of a linear program with infinitely many constraints," cited by the book's own Remark 7.5.5 for the finitely-additive treatment the restricted dual $(D)$ over $M_b$ sidesteps.

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Combinatorics Convex Optimization Discrete Geometry+2·Captain: Shuze Chen

Discrete Convex Analysis XX: Integral Convexity of L-Convex SetsTextbook

Motivation

Shortest-path distances and network potentials are among the oldest objects in combinatorial optimization: a directed graph with arc lengths, its shortest-path distances, and the "feasible potentials" (vertex labels consistent with those lengths) underlie duality in min-cost flow, scheduling, and difference-constraint systems. Murota's Discrete Convex Analysis (SIAM, 2003) isolates the abstract structure behind these objects — distance functions satisfying the triangle inequality, and their associated sets of admissible potentials — and shows it is governed by exactly the same discrete-convexity machinery as submodular set functions: a one-to-one correspondence with a second family of well-behaved integer point sets, the L-convex sets. Where an M-convex set (chapter 4) is defined by an exchange axiom generalizing matroid base exchange, an L-convex set is defined by closure under coordinatewise lattice operations (∨, ∧) and translation by the all-ones vector — a genuinely different axiom system that nonetheless produces a parallel structural theory: hole-freeness, a polyhedral description via an induced distance function, and integral convexity.

Companion mission 05-lconvex-sets (Discrete Convex Analysis IV) covers this chapter's other half: the hole-free property (Theorem 5.2), the one-to-one correspondence between L-convex sets and integer-valued triangle-inequality distance functions (Theorem 5.5), the intersection properties (Theorem 5.7), and the chapter's discrete separation theorem (Theorem 5.9, its goal). This mission builds the vocabulary those results also need (redeclared here, since sibling drafts cannot yet import one another) and proves the results the chapter leaves for its second half: the fundamental facts connecting a distance function to its admissible potentials (Proposition 5.1), the two-way polyhedral correspondence's supporting propositions (5.3-5.4), Minkowski-sum convexity (Theorem 5.8), and — this mission's goal — the explicit description of an L-convex set's convex hull that establishes its integral convexity (Theorem 5.10).

Setting

Fix a finite ground set $V$ . A distance function is a map $\gamma : V \times V \to \mathbb R \cup \{+\infty\}$ with $\gamma(v,v) = 0$ ; it may take negative finite values and need not be symmetric. It defines a directed graph $G_\gamma = (V, A_\gamma)$ with $A_\gamma = \{(u,v) : \gamma(u,v) < +\infty\}$ , arc $(u,v)$ having length $\gamma(u,v)$ . Write $\bar\gamma(u,v)$ for the shortest-path length from $u$ to $v$ in $G_\gamma$ ( $+\infty$ if none exists); $\gamma$ is well defined ( $\bar\gamma$ finite-valued wherever a path exists) exactly when $G_\gamma$ has no negative cycle. The triangle inequality $\gamma(v_1,v_2) + \gamma(v_2,v_3) \ge \gamma(v_1,v_3)$ defines the class $T[\mathbb R]$ (or $T[\mathbb Z]$ when integer-valued). A vector $p \in \mathbb R^V$ is an admissible potential of $\gamma$ if $p(v) - p(u) \le \gamma(u,v)$ for all $u \ne v$ ; write $D(\gamma)$ for the set of all such potentials.

A nonempty set $D \subseteq \mathbb Z^V$ is L-convex if it satisfies (SBS[Z]): $p, q \in D \implies p \vee q,\ p \wedge q \in D$ (coordinatewise max/min), and (TRS[Z]): $p \in D \implies p \pm \mathbf 1 \in D$ . A set $S \subseteq \mathbb Z^V$ is integrally convex if every point of its convex hull $\overline S$ lies in the convex hull of $S$ restricted to that point's integral neighborhood $N(p) = \{y \in \mathbb Z^V : \lfloor p \rfloor \le y \le \lceil p \rceil\text{ coordinatewise}\}$ — a strong, local form of "no holes" saying every real point of the hull is explained by nearby integer points alone.

Formalization targets

Goal: integral convexity of L-convex sets

For an L-convex set $D \subseteq \mathbb Z^V$ , writing $a = p - \lfloor p \rfloor$ for the fractional part of $p \in \mathbb R^V$ , $\alpha_1 > \cdots > \alpha_m$ for the distinct nonzero values of $a$ , and $U_i(p) = \{v : a(v) \ge \alpha_i\}$ (with $U_0 = \emptyset$ ):

\overline D = \{p \in \mathbb R^V : \lfloor p \rfloor + \chi_{U_i(p)} \in D\ \ (i = 0, 1, \ldots, m)\}, \qquad \text{hence } D \text{ is integrally convex}.

This is the weakest stable form available: it exhibits an explicit, finite set of at most $|V|+1$ integer witnesses for every point of the hull, which is what "integrally convex" asserts abstractly, rather than a numerical bound that a sharper construction could later shrink.

Supporting structural targets

Four further results build the correspondence this goal uses: the basic duality between a distance function's admissible potentials, its shortest-path closure, and negative-cycle freedom (Prop. 5.1); the induced-distance-function construction recovering a triangle-inequality distance function from any integer point set, and the convex hull of an L-convex set as its associated polyhedron (Prop. 5.3); the converse construction recovering an L-convex set from an integer-valued distance function (Prop. 5.4); and convexity in Minkowski sum (Thm. 5.8).

Significance

Theorem 5.10 is what makes "L-convex" a genuinely convex-analytic notion rather than a combinatorial curiosity: it shows the convex hull of an L-convex set is not merely a polyhedron (already known from the chapter's polyhedral-description results) but one with the strongest local integrality property discrete convex analysis considers, integral convexity — every real point's hull membership is certified by a small, explicitly constructed set of nearby lattice points, uniformly across the whole set. This is the L-convex counterpart of the corresponding M-convex fact (chapter 4's Theorem 4.24) and is used later in the book wherever L-convex functions (chapter 7) need their epigraphs' local structure. Proposition 5.1 is the combinatorial engine underneath: it is exactly the LP-duality statement between shortest paths and feasible potentials that appears, in various guises, throughout network flow theory, made precise here as the base case the L-convex correspondence rests on.

None of these results are open — Murota presents them as, in his own words, "fundamental facts well known in network flow theory" (Proposition 5.1) systematized into the discrete convex analysis framework. What this mission contributes is a faithful, machine-checked formal statement of each, in the shared Lean vocabulary (LConvexSet, AdmissiblePotentials, ShortestDist) the rest of the Discrete Convex Analysis series can build on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The shortest-path closure $\bar\gamma$ is not a bookkeeping convenience but genuinely graph-theoretic content: proving Proposition 5.1 requires constructing an admissible potential from a shortest-path labeling and, conversely, deriving the negative-cycle-freeness of $G_\gamma$ from the mere existence of one admissible potential — a min-cost-flow-style LP duality argument, not a direct combinatorial check. Theorem 5.10's difficulty sits in a different place: the naive approach to " $D$ is integrally convex" would attempt an inductive argument peeling off one coordinate at a time, but the actual proof constructs a single, uniform family of $m+1$ witness points from the sorted fractional values of $p$ — a Carathéodory-style representation (Eq. (5.11)) that must simultaneously stay inside the integral neighborhood $N(p)$ and land in $D$ itself via the triangle inequality of $D$ 's induced distance function, a construction with no one-coordinate-at-a-time shortcut.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; L-convex sets are Set (V → ℤ); distance functions are V → V → WithTop ℝ; admissible-potential sets are Set (V → ℝ). The shortest-path closure is formalized directly from finite walks (Fin (k+1) → V) rather than via a graph-library shortest-path predicate, matching the book's own construction. The Eq. (5.11) witnesses are built exactly as the book describes them — sorted distinct nonzero fractional values and their level sets — mirroring the Lovász-extension construction of the companion mission 20-ch04b-mconvexsets. No numeric constants are hard-coded anywhere in this mission (rule 7 is vacuous). The goal's explicit witness set (at most $|V|+1$ points) is not a trivializing special case: it holds for every L-convex set and every point of its hull, with no extra hypothesis narrowing the class. This mission's definitions (LConvexSet, AdmissiblePotentials, DistanceFunction, IsIntegrallyConvex) are redeclared from chunk 05-lconvex-sets (and, for IsIntegrallyConvex/IntegralNeighborhood, from chapter 3's own definitions) rather than imported, since sibling drafts in this series cannot yet reference one another; a later, published version of this book's namespace should consolidate them. Contributions completing any of the five sorrys are welcome; Proposition 5.1's LP-duality argument and the goal's Carathéodory-style construction are the two with the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
A. J. Hoffman, "On abstract dual linear programs," Naval Research Logistics Quarterly, 10 (1963), pp. 369-373 (feasible-potential duality in network flow theory).

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Mathematical Physics·Captain: Lucas

Ward–Takahashi identity: lattice U(1) modelTextbook

Motivation

In quantum field theory a Ward–Takahashi identity is an identity between correlation functions that follows from a global or gauge symmetry of the theory. In quantum electrodynamics it was used by Ward (1950) and Takahashi (1957) to relate the wave-function renormalization of the electron to its vertex renormalization, and it is described in the source article as "the quantum version of classical current conservation associated to a continuous symmetry by Noether's theorem". In the path-integral formulation the identities are a reflection of the invariance of the functional measure under the symmetry; when the measure is not invariant one obtains the anomalous Ward–Takahashi identity (e.g. the chiral anomaly).

The continuum path integral of interacting QED is not a mathematically defined object, so the identities cannot be formalized literally. This mission formalizes the path-integral derivation in the setting where every step is rigorous: a finite lattice, where the functional measure is Lebesgue measure on a finite-dimensional space.

Setting

A field configuration of a complex scalar field on $N$ sites is $\varphi\in\mathbb C^N$ . The global $U(1)$ symmetry acts by $(R_\theta\varphi)_y=e^{i\theta}\varphi_y$ . The local generator at site $x$ is $(g_x\varphi)_y=\delta_{xy}\,i\varphi_x$ , and the local variation of a real-differentiable functional $F$ is $\delta_xF(\varphi)=DF(\varphi)[g_x\varphi]$ . An action $S:\mathbb C^N\to\mathbb R$ is $U(1)$ -invariant if $S\circ R_\theta=S$ for all $\theta$ . The (unnormalised, Euclidean) path integral of an observable $F$ is

Z_S[F]=\int_{\mathbb C^N}F(\varphi)\,e^{-S(\varphi)}\,d\varphi .

For invariant $S$ , $j_x=\delta_xS$ is the lattice divergence of the Noether current.

Formalization targets

Goal: two-point Ward–Takahashi identity

For a $C^1$ , $U(1)$ -invariant action with the moment conditions $(1+\|\varphi\|^3)e^{-S}\in L^1$ and $\|\varphi\|^3\|DS\|e^{-S}\in L^1$ : current conservation $\sum_x j_x=0$ , and for all sites $x,y,z$

Z_S\big[j_x\,\varphi_y\overline{\varphi_z}\big]=i(\delta_{xy}-\delta_{xz})\,Z_S\big[\varphi_y\overline{\varphi_z}\big].

Milestones

Invariance of Lebesgue measure under $R_\theta$ .
Lattice Noether theorem: $\sum_x\delta_xS=0$ for invariant $S$ .
Infinitesimal measure invariance: $\int\delta_xG\,d\varphi=0$ .
Local Ward–Takahashi identity: $Z_S[\delta_xF]=Z_S[F\,\delta_xS]$ .
Global Ward identity: $\sum_xZ_S[\delta_xF]=0$ for invariant $S$ .
Anomalous identity for a linear generator $A$ : $\int(DF[A\varphi]-F\,DS[A\varphi])e^{-S}=-\operatorname{tr}(A)\,Z_S[F]$ .

Significance

The goal is the lattice version of the statement that the divergence of the current inserted into a charged correlation function produces only contact terms — the structure of the QED identity in the source, whose right-hand side consists of amplitudes with shifted external momenta. The milestones isolate the three ingredients of the textbook derivation: invariance of the measure, Noether current conservation, and integration by parts; the anomalous version identifies the anomaly with the Jacobian (trace) of the transformation. The results are classical; to the drafter's knowledge they are not available in Mathlib in this form.

Difficulty

The identities are elementary on paper; the work is analytic bookkeeping. Integration by parts must be justified on the non-compact space $\mathbb C^N$ with non-constant vector fields $\varphi\mapsto g_x\varphi$ or $A\varphi$ , so the "surface terms vanish" step of the physics derivation has to be replaced by explicit integrability hypotheses, and products with complex conjugates require real (not complex) differentiability throughout.

Formalization scope

Fields are Fin N → ℂ with the sup norm and the product Lebesgue measure; derivatives are real Fréchet derivatives (fderiv ℝ); integrals are Bochner integrals, which take the value $0$ on non-integrable integrands. The weight is Euclidean, $e^{-S}$ , and all identities are stated for unnormalised integrals (normalised expectations are ratios by $Z_S[1]$ ). Hypotheses are satisfiable: e.g. any Gaussian action $S(\varphi)=\sum_y|\varphi_y|^2$ is $U(1)$ -invariant and satisfies all moment conditions, so the goal is not vacuous. Continuum QED, renormalization and the S-matrix Ward identity are out of scope.

Selected references

J. C. Ward, An Identity in Quantum Electrodynamics, Phys. Rev. 78 (1950) 182. https://doi.org/10.1103/PhysRev.78.182
Y. Takahashi, On the generalized Ward identity, Il Nuovo Cimento 6 (1957) 371–375. https://doi.org/10.1007/BF02832514
M. E. Peskin, D. V. Schroeder, An Introduction to Quantum Field Theory, Westview Press (1995), §7.4.
Wikipedia, Ward–Takahashi identity, https://en.wikipedia.org/w/index.php?title=Ward%E2%80%93Takahashi_identity&oldid=1374751657

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XVIII: Integral Convexity of Minimizer SetsTextbook

Motivation

A classical convex function's global minimality is equivalent to its local minimality — the single fact that makes convex optimization tractable, since checking a small neighborhood suffices to certify a global guarantee. The discrete analogue is not automatic: a function on the integer lattice can fail to have any well-behaved "local" notion at all, and even when a discrete convexity-like property is imposed, the naive candidate (a function's values agreeing with its own convex-hull interpolation) does not by itself guarantee that local optimality implies global optimality. Murota's Discrete Convex Analysis (SIAM, 2003) isolates exactly the extra condition — integral convexity — that restores this guarantee, and shows it is general enough to contain every other discrete convexity notion the book studies (M-convex, L-convex, and their variants), making it the common ancestor of the book's entire hierarchy of classes. This mission completes the chapter's account of integral convexity: how it behaves under sums, restrictions, and linear perturbations, how it transfers between a function and its domain or minimizer sets, and a companion fact about hole-freeness under intersection and Minkowski sums that motivates why integral convexity, not mere hole-freeness, is the right notion to use.

Setting

For $f : \mathbb Z^n \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain $\operatorname{dom}_{\mathbb Z} f$ , the convex closure is $\bar f(x) = \sup_{p,\alpha} \{\langle p,x\rangle + \alpha : \langle p,y\rangle+\alpha \le f(y)\ \forall y \in \mathbb Z^n\}$ . The integral neighborhood of $x \in \mathbb R^n$ is $N(x) = \{y \in \mathbb Z^n : \lfloor x_i \rfloor \le y_i \le \lceil x_i \rceil\}$ , and the local convex extension $\tilde f$ replaces "for all $y \in \mathbb Z^n$ " in $\bar f$ 's definition with "for all $y \in N(x)$ ". A function is integrally convex if $\tilde f = \bar f$ everywhere on $\mathbb R^n$ . A set $S \subseteq \mathbb Z^n$ is integrally convex if its indicator function is; it is hole free if $S = \bar S \cap \mathbb Z^n$ , where $\bar S$ is the real convex hull of $S$ . The discrete Minkowski sum is $S_1+S_2 = \{x_1+x_2 : x_1 \in S_1, x_2 \in S_2\}$ . A function is separable convex if $f(x) = \sum_i f_i(x(i))$ for univariate functions $f_i$ satisfying the discrete convexity inequality $f_i(t-1)+f_i(t+1) \ge 2f_i(t)$ . For $p \in \mathbb R^n$ , $f[-p] (x) = f(x) - \langle p,x \rangle$ and $\arg\min f[-p]$ is its minimizer set.

Formalization targets

Goal (Theorem 3.29). For $f$ with nonempty bounded effective domain,

f \text{ is integrally convex} \iff \arg\min f[-p] \text{ is an integrally convex set for every } p \in \mathbb R^n.

This leaves the characterization at the level of the two named properties (integral convexity of the function versus of every minimizer set), the strongest statement of this kind that holds without extra hypotheses beyond boundedness of the domain.

Supporting milestones. Proposition 3.17 (four basic containment/equality relations between hole-free sets' intersections, Minkowski sums, and their real closures); Proposition 3.22 (for a periodic integrally convex function, global optimality reduces to a one-sided local check); Proposition 3.24 (an integrally convex function plus a separable convex function is integrally convex); Proposition 3.25 (separable convex functions are integrally convex, and integral convexity survives linear perturbation); Proposition 3.26 (an integrally convex set is hole free); Proposition 3.28 (the effective domain and every minimizer set of an integrally convex function are integrally convex sets — the forward direction of the goal); and Proposition 3.30 (for integer-valued integrally convex functions, a finite infimum is always attained).

Significance

Theorem 3.29 turns a statement about a function on all of $\mathbb R^n$ (integral convexity, a condition on $\tilde f$ and $\bar f$ that is a priori about uncountably many points) into a statement about a countable family of discrete sets (the minimizer sets $\arg\min f[-p]$ ), giving a genuinely different and often more tractable way to certify or refute integral convexity. Propositions 3.24–3.25 are the closure properties that make integral convexity useful in practice: without them, verifying integral convexity of a function built from simpler pieces (a sum with a separable cost, a linearly reweighted objective) would require re-deriving the property from scratch each time. Proposition 3.17, by contrast, is a cautionary result: Note 3.27 and Example 3.15 (the two hole-free sets whose Minkowski sum has a hole) show that hole-freeness alone does not inherit good behavior under set operations, which is exactly the gap integral convexity's stronger, locally-checkable condition is built to close — this mission's Proposition 3.17 documents the "obvious"/general-purpose relations that hold regardless, so that the reader can see precisely which inclusion is automatic and which requires more.

Difficulty

The naive approach to Theorem 3.29's converse direction (integral convexity of every minimizer set implies integral convexity of $f$ ) tries to check $\tilde f(x) = \bar f(x)$ directly at an arbitrary $x \in \operatorname{dom} f$ ; this is circular, since $\tilde f$ and $\bar f$ are themselves defined via suprema over affine minorants, not via minimizer sets. The book's actual proof instead sets up a primal-dual pair of linear programs whose optimal solutions witness $\bar f(x)$ and $\tilde f(x)$ respectively, uses LP duality's complementary slackness to show the dual optimal solution can be chosen supported inside $N(x)$ , and only then concludes $\tilde f(x) = \bar f(x)$ — routing the entire argument through the integral convexity of the specific minimizer set $S = \arg\min f[-p^*]$ at the optimal dual price $p^*$ . This is why Theorem 3.29's proof needs LP duality (Theorem 3.10, formalized in the previous mission in this series) as an ingredient, not just the closure-property machinery of Propositions 3.24–3.28.

Formalization scope

All apparatus (ConvexClosure, IntegralNeighborhood, LocalConvexExtension, IntegrallyConvex, ArgMinPerturbed, HoleFree, IntegrallyConvexSet, SeparableConvex, MinkowskiSumZ) is redeclared fresh in DiscreteConvex.IntegralConvexityC, mirroring chunk 03-integral-convexity's already-established constructions (Fin n-indexed, WithTop ℝ-valued functions, EReal-valued convex closures via sSup), since a draft mission cannot import another draft's definitions. IntegrallyConvexSet is defined via the book's own primary definition (indicator function integrally convex) rather than either of its two stated equivalent reformulations, since no result in this mission needs those forms as a named predicate. An integer-valued function (Proposition 3.30) is represented as Zⁿ → WithTop ℤ and cast to WithTop ℝ via a small casting map wherever the real-valued apparatus is needed — a faithful embedding. Boundedness of a discrete set is containment in a finite integer interval. The formalization does not trivialize: Theorem 3.29's hypothesis is exactly "nonempty bounded effective domain", not further restricted to, say, a fixed small dimension or a finite ground set with a fixed cardinality bound, and every milestone is stated at the same generality as Propositions 3.24–3.28 and 3.30 give it (arbitrary $n$ , arbitrary integrally convex function). Infrastructure needed beyond Mathlib: all definitions are fresh; a contribution completing any milestone, or the LP-duality-based proof of Theorem 3.29's converse direction, would be a natural entry point, alongside chunk 03's Theorem 3.21 as background.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003, DOI 10.1137/1.9780898718508, Chapter 3.
K. Murota, A. Shioura, "M-convex function on generalized polymatroid," Mathematics of Operations Research 24 (1999), 95–105 (Lemma 6.13, cited for Proposition 3.30).

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Markov Decision Processes VIII: Transaction Costs and the Dynamic Mean-Variance ProblemTextbook

Motivation

Two of the oldest simplifying assumptions in portfolio theory are that trading is frictionless and that risk means variance. Neither survives contact with practice: every real market charges a transaction cost proportional to the size of a trade, and variance penalizes upside deviations exactly as much as downside ones, which is not what an investor actually fears. Bäuerle and Rieder's §4.5 reopens the multiperiod terminal-wealth problem of chunk 04a with proportional transaction costs added to every trade, and finds that the qualitative shape of the solution survives — a buy/hold/sell rule with explicit thresholds, still obtained from the Structure Theorem of chunk 02a. Their §4.6 then leaves expected-utility maximization altogether and solves the classical Markowitz mean-variance problem in its genuinely dynamic, multiperiod form: choose a self-financing trading strategy that attains a target expected terminal wealth $\mu$ while minimizing the variance of that terminal wealth. This is Markowitz's one-period portfolio selection problem (H. Markowitz, Portfolio Selection, Journal of Finance, 1952) transplanted into a stage-by-stage trading horizon, and it earns its own solution technique: the objective is not linear in the underlying probability measure, so no direct Bellman equation applies, and the chapter instead builds a Lagrangian-embedding argument from scratch. Section §4.7 closes the chapter by replacing variance with the Average-Value-at-Risk, an axiomatically better-behaved risk measure (P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent Measures of Risk, Mathematical Finance, 1999), and solves the resulting mean-risk problem in the binomial model by the same Lagrangian route.

Setting

The transaction-cost model (§4.5): state $(x_0,x_1)\in E:=\mathbb{R}_{\ge0}^2$ (bond and stock holdings), action $a\in[0,x_1+x_0/(1+c)]$ (the stock holding chosen after the trade), bond holding after the trade $h(x_0,x_1,a) := x_0+(1-c)(x_1-a)$ if $a\le x_1$ and $x_0+(1+c)(x_1-a)$ if $a>x_1$ , for a proportional cost rate $c\in[0,1)$ ; transition $T_n((x_0,x_1),a,z) := (h(x_0,x_1,a)(1+i_{n+1}),\,az)$ ; terminal reward $U(x_0+x_1)$ for a utility $U$ homogeneous of degree $\gamma$ .

The mean-variance model (§4.6): state $E:=\mathbb{R}$ (wealth), action $A:=\mathbb{R}^d$ (amounts invested in $d$ risky assets, short-selling allowed), transition $T_n(x,a,z) := (1+i_{n+1})(x+a\cdot z)$ . Writing $X_N$ for the terminal wealth reached from $x_0$ under a strategy $\pi$ , the problem is

\mathrm{(MV)}\qquad \mathrm{Var}_{x_0}^\pi[X_N] \to \min \quad\text{subject to}\quad \mathbb{E}_{x_0}^\pi[X_N] \ge \mu, \ \ \pi \text{ admissible.}

Because $\mathrm{Var}$ is not linear in the law of $X_N$ , (MV) is solved via the Lagrangian $L_{x_0}(\pi,\lambda) := \mathrm{Var}_{x_0}^\pi[X_N] + 2\lambda(\mu-\mathbb{E}_{x_0}^\pi[X_N])$ , whose saddle points give (MV)'s value and optimizer, reduced in turn to the tractable auxiliary quadratic problem $QP(b)$ : minimize $\mathbb{E}_{x_0}^\pi[(X_N-b)^2]$ , a stochastic linear-quadratic control problem.

The mean-risk model (§4.7): the binomial (Cox–Ross–Rubinstein) market with one bond (interest rate $0$ ) and one stock with relative return $u-1$ w.p. $p$ or $d-1$ w.p. $1-p$ ; the Average-Value-at-Risk at level $\gamma$ , $\mathrm{AVaR}_\gamma(X) := \inf_{b\in\mathbb{R}} [b+\frac{1}{1-\gamma}\mathbb{E}[(X+b)^-]]$ ; the problem $\mathrm{(MR)}: \mathrm{AVaR}_\gamma(X_N)\to\min$ subject to $\mathbb{E}_{x_0}^\pi[X_N]\ge\mu$ , solved via the same Lagrangian route through an auxiliary problem $P(\lambda,b)$ .

Formalization targets

Goal — Theorem 4.6.6 (the mean-variance problem)

\mathrm{Var}_{x_0}^{\pi^*}[X_N] = \frac{d_0}{1-d_0}\big(\mathbb{E}_{x_0}^{\pi^*}[X_N] - x_0S^0_N\big)^2, \qquad \mathbb{E}_{x_0}^{\pi^*}[X_N] = \mu,

f_n^*(x) = \Big(\frac{\mu-d_0x_0S^0_N}{1-d_0}\cdot\frac{S^0_n}{S^0_N} - x\Big)\, C_{n+1}^{-1}\,\mathbb{E}[R_{n+1}],

where $(d_n)$ is a recursively-defined sequence in $(0,1)$ (Lemma 4.6.4) built from the one-period return moments $C_n,\mathbb{E}[R_n]$ . This closes the loop the chapter opens: it is the exact value and optimal strategy of the dynamic mean-variance problem, obtained by specializing the auxiliary problem $QP(b)$ 's closed-form solution (Theorem 4.6.5) at the Lagrange multiplier that Lemma 4.6.2's saddle-point argument selects.

Supporting milestones

The Lagrangian route itself: the equivalence of (MV) and its equality-constrained form (Lemma 4.6.1), the saddle-point value identity (Lemma 4.6.2), the reduction of the Lagrange problem $P(\lambda)$ to $QP(b)$ (Lemma 4.6.3), the boundedness of $(d_n)$ (Lemma 4.6.4), and $QP(b)$ 's own explicit solution (Theorem 4.6.5) — the four-step argument the goal theorem is the payoff of. Upstream of §4.6: the transaction-cost model's upper bounding function (Proposition 4.5.1), its Structure Assumption via buy/hold/sell decision rules (Proposition 4.5.2), and the resulting explicit three-region optimal policy (Theorem 4.5.4). Downstream: the Two-Fund Theorem (Corollary 4.6.7), and the parallel mean-risk development — the auxiliary problem $P(\lambda,b)$ 's solution (Theorem 4.7.1), the binomial value of $P(\lambda)$ (Proposition 4.7.2), and the mean-risk problem's own explicit solution in both orderings of $p$ and $q$ (Theorems 4.7.3 and 4.7.4).

Significance

Theorem 4.6.6 is the multiperiod extension of the single most-used result in portfolio theory: the mean-variance efficient frontier, here derived stage by stage rather than assumed static, and it recovers the classical Two-Fund Theorem (every investor holds the same risky portfolio, scaled by wealth) as an immediate corollary rather than a separate argument. The transaction-cost results answer a standing objection to frictionless portfolio theory by showing that its qualitative conclusions — a threshold trading rule derived from a value function via the same abstract Structure Theorem — survive costs, with the thresholds now depending on the current value function rather than being fixed. The mean-risk results extend the whole technique to a risk measure that, unlike variance, is coherent in the sense of Artzner et al., showing the Lagrangian-embedding method is not an accident of the quadratic case.

None of this chapter's results have machine-checked proofs on Prove2Me at the time of writing (the platform's saddle-point sufficiency results, VectorSpaceOpt.lagrangian_saddle_sufficient_pointed and ConvexOptimization.lagrangian_saddle_iff_strong_duality, are stated over a closed convex cone in a normed vector space, not over the finite-horizon admissible-policy space $F^N$ that Lemma 4.6.2 needs, and were checked and ruled out as reusable for this mission). Formalizing this chapter means building the Lagrangian-embedding argument for a dynamic (rather than static) optimization problem from scratch: no existing platform infrastructure covers a saddle point of a Lagrangian defined over a sequence of Markov policies.

Difficulty

The obvious first attempt at (MV) is to apply the Structure Theorem of chunk 02a directly to the variance objective, exactly as chunk 04a does for expected utility. This fails outright: $\mathrm{Var}_{x_0}^\pi[X_N] = \mathbb{E}_{x_0}^\pi[X_N^2] - (\mathbb{E}_{x_0}^\pi[X_N])^2$ is not additive over time and has no Bellman recursion of the usual form, because the square of an expectation over the whole horizon cannot be decomposed into a sum of one-period rewards. The chapter's actual route — Lagrangian relaxation to $P(\lambda)$ , then a further reduction to the quadratic (and hence tractable) $QP(b)$ — is not a shortcut around this obstacle but the only way the mean-variance problem admits a Markov Decision Process reformulation at all. A correct formalization of the goal theorem must go through this exact chain (saddle_point_value, plambda_implies_qp, qp_solution), not around it.

Formalization scope

The financial market and the four named optimization problems (MV), (MV=), $P(\lambda)$ , $QP(b)$ are formalized as explicit structures and Prop-valued predicates in MDPFinance.MeanVariance (none of them is a numbered definition in the book — each is introduced only in prose — so each gets its own precise Lean definition rather than being left implicit). Wealth is real-valued, policies are sequences of measurable Markov maps $\mathbb{N}\to\mathbb{R}\to(\mathrm{Fin}\ d\to \mathbb{R})$ , and values that can be $\pm\infty$ in the book (the value of $P(\lambda,b)$ , of $P(\lambda)$ , and of (MR) itself) are typed EReal rather than ℝ, matching the book's own use of infinite values as legitimate outcomes rather than failure states. A formalization that solved the goal theorem by first proving a Bellman equation for $\mathrm{Var}_{x_0}^\pi[X_N]$ directly would not be proving Theorem 4.6.6 — no such recursion exists — and the goal statement is phrased purely in terms of IsOptimalMV, varXN, and meanXN, independent of any intermediate value function, precisely so that only the actual saddle-point argument can discharge it. The transaction-cost model's buy/hold/sell threshold functions $q^-(V_{n+1}),q^+(V_{n+1})$ are represented by their defining maximizing property rather than a closed form, since the book itself only pins them down as an argmax. Reusable beyond this mission: the MVMarket/ MeanRiskMarket structures and the Lagrangian-saddle-point machinery are natural substrate for any later mission that needs a dynamic risk-constrained portfolio problem. Contributions completing any milestone's sorry are welcome, particularly a sorry-free proof of Lemma 4.6.2 (the saddle-point value identity), since it is the one genuinely general technique this mission introduces.

Selected references

H. Markowitz, Portfolio Selection, The Journal of Finance 7(1), 1952, https://doi.org/10.2307/2975974
P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent Measures of Risk, Mathematical Finance 9(3), 1999, https://doi.org/10.1111/1467-9965.00068
N. Bäuerle, U. Rieder, Markov Decision Processes with Applications to Finance, Universitext, Springer, 2011, https://doi.org/10.1007/978-3-642-18324-9, Chapter 4, §§4.5-4.7

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Markov Decision Processes III: Monotonicity and Convexity of the Value FunctionTextbook

Motivation

Once a finite-horizon Markov Decision Model (MDM) is known to admit an optimal policy — the existence theory of continuity/compactness models — a natural next question is qualitative: does the optimal value function inherit structural properties (monotonicity, concavity, convexity) of the model's own data, and are the resulting optimal actions themselves monotone in the state? These questions matter beyond aesthetics. A value function known in advance to be concave in wealth, say, restricts the search for an optimizer to a much smaller, better-behaved class of candidates, simplifies numerical solution (dynamic programming over convex functions can exploit shape-preserving approximation schemes), and is often the only handle available for comparative-statics questions — e.g. "if the model's transition mechanism becomes riskier, does the decision-maker's value go down?" — the kind of question that drives applications in inventory theory, insurance, and portfolio choice. The general theory traces to Topkis's lattice-programming approach to comparative statics (Topkis, Supermodularity and Complementarity, Princeton University Press, 1998) and to the stochastic-orders literature (Müller and Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, 2002); Bäuerle and Rieder's Chapter 2, §2.4.4-2.4.5 specializes both to the Borel-space finite-horizon Markov Decision Model of their own Definition 2.1.1.

Setting

Fix a (non-stationary) Markov Decision Model $(E, A, D_n, Q_n, r_n, g_N)_{n=0,\dots,N-1}$ as in Definition 2.1.1: $E$ , $A$ measurable spaces, $D_n \subseteq E \times A$ the admissible state-action pairs, $Q_n(\cdot\mid x,a)$ the transition kernel, $r_n$ the one-stage reward, $g_N$ the terminal reward. Write $D_n(x) := \{a \in A : (x,a) \in D_n\}$ . An upper bounding function $b : E \to \mathbb{R}_{\geq 0}$ (Definition 2.4.1) is a measurable function for which constants $c_r, c_g, \alpha_b \geq 0$ exist with $r_n^+(x,a) \leq c_r\, b(x)$ , $g_N^+(x) \leq c_g\, b(x)$ , and $\int b(x')\,Q_n(dx'\mid x,a) \leq \alpha_b\, b(x)$ for all admissible $(x,a)$ and all $n$ ; $\mathbb{I\!B}_b^+$ is the set of measurable $v : E \to [-\infty,\infty)$ with $v^+ \leq c\, b$ for some $c \geq 0$ . The two central operators are $(L_n v)(x,a) := r_n(x,a) + \int v(x')\,Q_n(dx'\mid x,a)$ and $(T_n v)(x) := \sup_{a \in D_n(x)} (L_n v)(x,a)$ ; a decision rule $f_n$ is a maximizer of $v$ at time $n$ if $(L_n v)(x, f_n(x)) = (T_n v)(x)$ for every $x$ . The Structure Assumption (SAN) on families $(\mathrm{I\!M}_n)_{n \leq N} \subseteq \mathrm{I\!M}(E)$ and $(\Delta_n)_{n<N}$ of decision rules says: $g_N \in \mathrm{I\!M}_N$ ; $v \in \mathrm{I\!M}_{n+1}$ implies $T_n v \in \mathrm{I\!M}_n$ ; and every $v \in \mathrm{I\!M}_{n+1}$ has a maximizer in $\Delta_n$ . It is the single hypothesis from which the whole finite-horizon theory — a well-defined Bellman recursion, an optimal policy built rule-by-rule — follows (established elsewhere in this mission series).

For this section only, $E \subseteq \mathbb{R}^d$ and $A \subseteq \mathbb{R}^m$ carry the usual componentwise order, and the same spaces are given a real vector-space structure when convexity statements are in play; $\mathbb{I\!M}_n^{\diamond}$ denotes $\{v \in \mathbb{I\!B}_b^+ : v$ has property $\diamond\}$ for $\diamond \in \{\text{increasing}, \text{concave}, \text{convex}\}$ . A set $D \subseteq E \times A$ is completely monotone (Definition 2.4.15) if $(x,a'), (x',a) \in D$ with $x \leq x'$ , $a \leq a'$ forces $(x,a), (x',a') \in D$ . A function $f$ on a lattice is supermodular (Definition A.3.1) if $f(x) + f(y) \leq f(x \wedge y) + f(x \vee y)$ for all $x,y$ . The comparison theorem below additionally uses three orders between probability measures: the usual stochastic order $\mu \leq_{\mathrm{st}} \nu$ ( $\int f\,d\mu \leq \int f\,d\nu$ for every bounded increasing $f$ , Definition B.3.2/Theorem B.3.3(ii)), the convex order $\mu \leq_{\mathrm{cx}} \nu$ (same, for convex $f$ , Definition B.3.9a), and its concave-function dual $\mu \leq_{\mathrm{cv}} \nu$ (matching $\mathrm{I\!M}_n^{\mathrm{cv}}$ ; see the Formalization scope section on how the book's own, non-monotone "cv" differs from the increasing-concave order $\leq_{\mathrm{icv}}$ it also uses elsewhere, e.g. in Definition B.3.9c).

Formalization targets

Goal — Theorem 2.4.22 (the convex structure theorem)

\begin{aligned} &\text{If } E \text{ is convex}, D_n = E \times A, \text{ and for every } n: \\ &\quad\text{(ii) } x \mapsto \textstyle\int v(x')\,Q_n(dx'\mid x,a) \text{ is convex for every convex } v \in \mathbb{I\!B}_b^+, a \in A,\\ &\quad\text{(iii) } x \mapsto r_n(x,a) \text{ is convex for every } a, \quad \text{(iv) } g_N \text{ convex},\\ &\quad\text{(v) every convex } v \in \mathbb{I\!B}_b^+ \text{ has a maximizer in } \Delta_n,\\ &\text{then } \bigl(\mathrm{I\!M}_n^{\mathrm{cx}}\bigr)_{n \leq N} \text{ and } (\Delta_n)_{n<N} \text{ satisfy (SAN).} \end{aligned}

This is the weakest stable statement: it names exactly the compatibility conditions between the kernel, reward, and terminal payoff that propagate convexity through $T_n$ , without committing to any particular model beyond them.

Six further results of the same section are formalized as milestones on the way to, or alongside, the goal: the monotone (increasing) analogue (Theorem 2.4.14), the accompanying result that a largest maximizer under a supermodular $L_n v$ on a completely monotone $D_n$ is itself weakly increasing (Proposition 2.4.16), the concavity-preservation step for $T_n$ and its structure theorem (Proposition 2.4.18, Theorem 2.4.19), the convexity-preservation step together with the existence of a bang-bang maximizer when $A$ is a polytope (Proposition 2.4.21), and the comparison theorem for two models whose kernels are ordered (Theorem 2.4.23).

Significance

Theorems 2.4.14/2.4.19/2.4.22 give three parallel, reusable templates: once a modeler checks three or four structural conditions on $D_n$ , $Q_n$ , $r_n$ , $g_N$ individually — never on the recursively-defined value function itself, which is usually inaccessible in closed form — the corresponding shape of the value function is guaranteed for every horizon, with no further induction needed by the modeler. This is what makes results like the concavity of the optimal consumption-investment value function (used in later chapters of this book) checkable from the market model alone. Proposition 2.4.16's comparative-statics conclusion (optimal actions inherit monotonicity in the state) is the Markov-decision-process incarnation of Topkis's monotone comparative statics, and Theorem 2.4.23 formalizes the intuitive but non-trivial fact that making the transition mechanism "worse" in a precise stochastic-order sense can only lower the optimal value — a comparison that requires the compatibility between the order and the very shape (monotonicity/concavity/convexity) the Structure Assumption already pins down.

All of these results, including the goal, are unformalized on the platform prior to this mission: no result matching "supermodular", "completely monotone", "comparative statics", or a Borel-space convex Markov decision model was found in a platform search at drafting time. The proofs themselves are short (Bäuerle and Rieder give complete, self-contained arguments for every result in this section), so what this mission contributes is the formal statement — getting the exact quantifiers and hypothesis set right in a general Borel/vector-space setting — rather than a technically deep proof; the sorry-free companion proofs are left as the formalization task.

Difficulty

The obvious first idea for the goal is to prove convexity of $T_n v$ by convexity of a supremum of convex functions — true only when $D_n(x)$ does not itself depend on $x$ in a way that mixes domains under a convex combination. The book's own hypothesis (i), $D_n := E \times A$ (constant), is exactly what rules out the general case and makes the argument work: for a genuinely $x$ -dependent $D_n(x)$ , a convex combination $\alpha(x,a) + (1-\alpha)(x',a')$ need not even have its action component available at the combined state, so "supremum of convex functions is convex" does not apply termwise. A second trap is treating $\mathrm{I\!M}_n^{\mathrm{cv}}$ (closed under concave, not-necessarily-increasing $v$ ) as if it required the stronger increasing-concave order $\leq_{\mathrm{icv}}$ that the appendix's Definition B.3.9c actually names — the two are different relations, and only the plain "concave-test-function" order is compatible with $\mathrm{I\!M}_n^{\mathrm{cv}}$ as stated (see Formalization scope).

Formalization scope

Because Mathlib's ConvexOn/ConcaveOn require a Module ℝ structure on the codomain, and EReal (needed for value functions that may equal $-\infty$ ) carries no such structure, this mission introduces ConvexOnEReal/ConcaveOnEReal: the same defining inequality with the real convex-combination coefficients cast into EReal and multiplied there (EReal does carry a Mul). Real-valued convexity/concavity of $r_n$ and $g_N$ uses Mathlib's own ConvexOn/ ConcaveOn directly. "Vertex of a polytope" (Proposition 2.4.21) is formalized via Mathlib's Set.extremePoints, and " $A$ is a polytope" as compact, convex, with finitely many extreme points. The comparison theorem's order $\leq_{\mathrm{cv}}$ has no verbatim numbered definition in the book: Appendix B.3 defines the stochastic order $\leq_{\mathrm{st}}$ (Definition B.3.2, via CDFs, with the increasing-test-function characterization given as an equivalent condition, Theorem B.3.3(ii)) and the convex order $\leq_{\mathrm{cx}}$ (Definition B.3.9a, directly via $\mathbb{E}f(X) \leq \mathbb{E}f(Y)$ for convex $f$ ), but never a bare " $\leq_{\mathrm{cv}}$ " — only the increasing-concave order $\leq_{\mathrm{icv}}$ (Definition B.3.9c). This mission defines $\leq_{\mathrm{cv}}$ as the direct concave-test-function analogue of $\leq_{\mathrm{cx}}$ ( $\mathbb{E}f(X) \leq \mathbb{E}f(Y)$ for every concave $f$ ), matching the book's own $\mathrm{I\!M}_n^{\mathrm{cv}}$ (plain concavity, not required to be increasing) and consistent with the standard "st/cv/cx" triple of Müller and Stoyan (2002), the reference the book cites for this whole appendix section. Likewise $\leq_{\mathrm{st}}$ is formalized directly via Theorem B.3.3(ii)'s functional characterization (bounded increasing test functions) rather than the CDF definition, since Theorem 2.4.23 compares kernels on a general $E \subseteq \mathbb{R}^d$ rather than real-valued random variables. The value function $V_n$ used only in the comparison theorem is given by its recursive characterization ( $V_N = g_N$ , $V_n = T_n V_{n+1}$ , established as this series' Theorem 2.3.8) rather than by re-deriving the sup-over-policies primitive definition and its supporting history/policy machinery, which is not otherwise needed in this mission.

A trivializing formalization is ruled out: taking $E := \mathbb{R}$ throughout would make hypothesis (i) (" $E$ is convex") vacuously true and hide the genuinely restrictive role $D_n = E \times A$ plays in the proof; this mission keeps $E$ (and $A$ ) as general real vector spaces (with a Preorder added only where monotonicity, rather than convexity, is at stake), so the convexity hypotheses carry their full content. Reusable infrastructure: ConvexOnEReal/ConcaveOnEReal (any later chunk needing shape-preservation results for EReal-valued value functions can reuse the same pattern, restated per this series' convention), and the LEStochasticOrder/LEConcaveOrder/LEConvexOrder triple (reused, restated, by mission 04b's Theorems 4.4.4-4.4.5 and mission 05b's Definition 5.4.9, which need the same or a closely related order). sorry-free proofs of the milestones (all short in the book) are welcome contributions.

Selected references

N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, Universitext, Springer, 2011. https://doi.org/10.1007/978-3-642-18324-9
D. M. Topkis, Supermodularity and Complementarity, Princeton University Press, 1998.
A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley, 2002.
D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, 1978.

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Operations Research Optimization·Captain: mikedeng1

Cubic Regularization of Newton Method and Its Global Performance IV: Local Quadratic Convergence to a Non-degenerate MinimumResearch Paper

Motivation

Newton's method converges quadratically near a non-degenerate minimum, but on its own it has no global guarantee: far from a minimum the Newton step may not exist or may increase the objective. Nesterov and Polyak (Math. Program. 108 (2006) 177–205) replaced the Newton step by the minimizer of a cubic-regularized second-order model. The resulting method has global complexity bounds on non-convex problems, which the other missions of this series formalize. This mission formalizes the complementary local result, Theorem 3 of the paper. Close to a non-degenerate local minimum, a relaxed version of the method keeps the quadratic rate of the classical Newton method. It no longer needs the safeguards (a lower bound on the regularization parameter and a descent test) that the global analysis relies on.

The cubic-regularized step later became the basis of adaptive cubic regularization (Cartis, Gould and Toint, Math. Program. 127 (2011)). Local quadratic convergence is the property that makes such second-order methods worth their per-iteration cost.

Setting

Let $n \ge 1$ and let $f:\mathbb{R}^n\to\mathbb{R}$ be twice differentiable, with gradient $f'(x)$ and Hessian $f''(x)$ . The Hessian is assumed Lipschitz continuous with constant $L > 0$ in the spectral norm (Assumption 1 of the paper):

\|f''(x) - f''(y)\| \le L\|x - y\| \qquad \forall x, y \in \mathbb{R}^n .

Eigenvalues of a symmetric operator are numbered decreasingly, so $\lambda_n(A)$ is the smallest eigenvalue, and $A \succ 0$ means $\lambda_n(A) > 0$ .

For $M > 0$ , the cubic model of $f$ at $x$ is

m_{M,x}(y) = \langle f'(x), y - x\rangle + \tfrac12\langle f''(x)(y - x), y - x\rangle + \tfrac M6\|y - x\|^3 ,

and the cubic-regularized Newton step $T_M(x)$ is any global minimizer of $m_{M,x}$ (Eq. (2.4)). Its length is $r_M(x) = \|x - T_M(x)\|$ .

The relaxed method (3.5) starts at $x_0$ and sets

x_{k+1} = T_{M_k}(x_k), \qquad M_k \in (0, 2L], \qquad k \ge 0 .

Unlike the globally convergent scheme (3.3), it imposes no lower bound $M_k \ge L_0 > 0$ and no acceptance test $f(x_{k+1}) \le \bar f_{M_k}(x_k)$ . The local progress measure is

\delta_k = \frac{L\,\|f'(x_k)\|}{\lambda_n^2(f''(x_k))} .

Formalization targets

Goal: Theorem 3, item 3

If $f''(x_0) \succ 0$ and $\delta_0 \le 1/4$ , the whole sequence $\{x_k\}$ converges to a point $x^*$ with $f'(x^*) = 0$ and $f''(x^*) \succ 0$ that is a local minimum of $f$ . Moreover, for every $k \ge 1$ ,

\|f'(x_k)\| \le \lambda_n^2(f''(x_0))\,\frac{9e^{3/2}}{16L}\left(\frac12\right)^{2^k}. \tag{3.8}

Milestones

Lemma 1, (2.2): $\|f'(y) - f'(x) - f''(x)(y - x)\| \le \tfrac12 L\|y - x\|^2$ .
Eq. (2.5): $f'(x) + f''(x)(T - x) + \tfrac12 M\|T - x\|(T - x) = 0$ for $T = T_M(x)$ .
Lemma 3, (2.9): $\|f'(T_M(x))\| \le \tfrac12(L + M)r_M^2(x)$ .
Eq. (3.9): if $f''(x) \succ 0$ then $r_M(x) \le \|f'(x)\|/\lambda_n(f''(x))$ .
Theorem 3, item 1, (3.6): every $\delta_k$ is well defined, and

\delta_{k+1} \le \tfrac32\Big(\frac{\delta_k}{1-\delta_k}\Big)^2 \le \tfrac83\delta_k^2 \le \tfrac23\delta_k .

Theorem 3, item 2, (3.7): $e^{-1}\lambda_n(f''(x_0)) \le \lambda_n(f''(x_k)) \le e^{3/4}\lambda_n(f''(x_0))$ for all $k \ge 0$ .

Significance

Theorem 3 shows that the cubic-regularized scheme does not lose Newton's local behaviour. Once the iterates enter the region $\{f'' \succ 0,\ \delta \le 1/4\}$ , any choice $M_k \in (0, 2L]$ gives a doubly exponential decrease of the gradient norm. The theorem thus supplies the final phase of the paper's complexity estimate (6.1) (Section 6), which counts the iterations until $\delta \le 1/4$ is reached and then adds a last phase of order $\log\log(1/\epsilon)$ steps.

On the formal side, the result is proved in the literature but, as far as a search of the platform shows, not formalized. A complete development yields a machine-checked local convergence theorem for a regularized Newton method under a Lipschitz Hessian. The ingredients include the Taylor bound (2.2), the stationarity system (2.5) of the cubic model, and eigenvalue perturbation bounds for Lipschitz Hessians, and they are reusable for other second-order methods. The published proof of (3.8) contains a gap in its constant (see Difficulty), so a formal proof would also certify the printed constant.

Difficulty

The obvious argument treats $x_{k+1}$ as a Newton step with a small perturbation and invokes the classical Kantorovich-type analysis. That analysis assumes the regularization vanishes. Here $M_k$ can be as large as $2L$ , and the step solves a nonlinear system (2.5) in which the step length appears in the operator. The proof must control three coupled quantities at once: the gradient, the smallest Hessian eigenvalue, and the step length. The eigenvalue lower bound has to survive infinitely many steps, so the per-step losses of curvature must be summable, and positive definiteness at the next iterate has to be established before $\delta_{k+1}$ is even defined.

The printed proof of (3.8) is not immediate from (3.6). It states $\delta_{k+1} \le \delta_k^2/(1-\delta_0)^2 \le \tfrac{16}{9}\delta_k^2$ , but (3.6) as printed only gives $\tfrac83\delta_k^2$ , which is too weak for the constant $\tfrac{9}{16}(\tfrac12)^{2^k}$ . A proof of (3.8) as stated cannot rest on (3.6) alone.

Formalization scope

Space. $\mathbb{R}^n$ is EuclideanSpace ℝ (Fin n) with $n \ge 1$ . The gradient and Hessian are maps g and H with HasGradientAt f (g x) x and HasFDerivAt g (H x) x at every point; the operator norm is the spectral norm.
Domain. The paper works on a closed convex set $F$ . Because method (3.5) has no descent test that keeps the iterates inside a smaller set, this mission takes $F = \mathbb{R}^n$ : $f$ is twice differentiable and Assumption 1 holds on all of $\mathbb{R}^n$ . Lemma 3's hypothesis $T_M(x) \in F$ is then automatic.
The step. $T_M(x)$ is represented by an arbitrary global minimizer of the cubic model (IsCubicStep). Every result holds for every such choice, matching the paper's "Arg min". A merely stationary point of the model is not a step.
Eigenvalues. $\lambda_n(A)$ is lamMin A, the infimum of $\langle Av, v\rangle$ over the unit sphere. It equals the smallest eigenvalue for self-adjoint $A$ , and $f''(x) \succ 0$ is 0 < lamMin (H x).
Indices. The index is 0-based and $x_0$ is x 0. The ranges are as printed: $k \ge 0$ in (3.6)–(3.7) and $k \ge 1$ in (3.8). "Converges quadratically" is rendered by the paper's own quantitative clause (3.8), together with existence of the limit, $f'(x^*) = 0$ , $\lambda_n(f''(x^*)) > 0$ and IsLocalMin f x*.
Constants. All constants ( $\tfrac14$ , $\tfrac32$ , $\tfrac83$ , $\tfrac23$ , $e^{-1}$ , $e^{3/4}$ , $\tfrac{9e^{3/2}}{16L}$ ) are the printed ones.

The theorem assumes positivity of the Hessian and $\delta \le 1/4$ only at $x_0$ . A formalization that assumes $f$ strongly convex, $f''(x) \succ 0$ everywhere, or $\delta_k \le 1/4$ for all $k$ , or that imports the lower bound $L_0$ or the descent test of method (3.3), proves a different theorem and is excluded.

Needed infrastructure: the Taylor bound with Lipschitz Hessian, first-order optimality of the non-smooth-looking but $C^1$ cubic model, perturbation bounds for lamMin under operator-norm changes, and the inverse bound $\|(A + cI)^{-1}\| \le 1/(\lambda_n(A) + c)$ . These are reusable well beyond this mission. Contributions welcome: proofs of the milestones, and a proof of (3.8) with the printed constant.

Selected references

Yu. Nesterov and B. T. Polyak, Cubic regularization of Newton method and its global performance, Math. Program., Ser. A 108 (2006) 177–205. https://doi.org/10.1007/s10107-006-0706-8
C. Cartis, N. I. M. Gould and Ph. L. Toint, Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results, Math. Program. 127 (2011) 245–295. https://doi.org/10.1007/s10107-009-0286-5

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Combinatorics Complexity Theory Operations Research+3·Captain: mikedeng1

Maximizing Non-Monotone Submodular Functions V: Beating 1/2 for Symmetric Functions Requires Exponentially Many Value QueriesResearch Paper

Motivation

Maximizing a nonnegative submodular set function without constraints contains Max Cut, Max Directed Cut and facility-location problems as special cases. In the value-oracle model an algorithm knows nothing about the function except the values $f(S)$ of the sets $S$ it queries, and it is judged by the number of queries it makes. Feige, Mirrokni and Vondrák (SIAM J. Comput. 40(4), 2011) gave constant-factor algorithms in this model and matching limits on what any algorithm can do. For symmetric functions, such as cut functions of undirected graphs, a uniformly random set already achieves $\tfrac12$ of the optimum in expectation (Theorem 2.1 of the paper). The question this mission formalizes is whether any algorithm can do better, and the answer given by Theorem 4.5 is: not without exponentially many value queries. The same factor $\tfrac12$ was later shown to be achievable for general (non-symmetric) nonnegative submodular functions by Buchbinder, Feldman, Naor and Schwartz (FOCS 2012 / SIAM J. Comput. 2015), so the bound of Theorem 4.5 is the tight limit of the whole problem in the value-oracle model.

Timeline:

2007 (FOCS) / 2011 (SIAM J. Comput.): Feige, Mirrokni and Vondrák prove that no algorithm with subexponentially many value queries achieves $(\tfrac12 + \epsilon)$ of the optimum on symmetric nonnegative submodular functions, and give $\tfrac25$ for general functions.
2011: Vondrák's symmetry-gap framework (SIAM J. Comput. 42(1), 2013) generalizes the construction to constrained problems.
2012: Buchbinder, Feldman, Naor and Schwartz give a randomized $\tfrac12$ -approximation for general nonnegative submodular functions, matching the bound.

Setting

Let $[n] = \{0, \dots, n-1\}$ be the ground set, with $n$ even. A set function $f : 2^{[n]} \to \mathbb{R}$ is submodular if $f(S \cup T) + f(S \cap T) \le f(S) + f(T)$ for all $S, T$ , symmetric if $f([n]\setminus S) = f(S)$ for all $S$ , and $\mathrm{OPT}(f) = \max_{S} f(S)$ .

Fix an integer $m$ with $1 \le m \le n/2$ and write $\epsilon = m/n$ , so that $\epsilon n$ is an integer. For integers $k, \ell$ put

f(k,\ell) = \begin{cases} (k+\ell)(n-k-\ell) & |k-\ell| \le m,\\ k(n-2\ell) + (n-2k)\ell + m^2 - 2m|k-\ell| & |k-\ell| > m. \end{cases}

For a set $C \subseteq [n]$ with $|C| = n/2$ and $D = [n] \setminus C$ , the hard instance is $f_C(S) = f(|S\cap C|, |S\cap D|)$ . The cut function of the complete graph is $g(S) = |S|(n-|S|)$ , with maximum $\tfrac14 n^2$ . A set $Q$ is balanced for $C$ if $\bigl||Q\cap C| - |Q\cap D|\bigr| \le m$ ; on balanced sets $f_C = g$ .

A deterministic adaptive $q$ -query algorithm $A$ chooses each query from the answers received so far, and after $q$ answers outputs a set $A(h)$ when run against an oracle $h$ . A randomized algorithm is a distribution $\mu$ over deterministic ones, with expected value $\mathbb{E}_{A\sim\mu}[h(A(h))]$ .

Formalization targets

Goal: Theorem 4.5 with the constants of its proof

For every such $n, m$ :

every $f_C$ with $|C| = n/2$ is nonnegative, symmetric and submodular, with

\mathrm{OPT}(f_C) = \tfrac12 n^2 (1 - 2\epsilon + 2\epsilon^2);

for every $q < e^{\epsilon^2 n/8}$ and every randomized $q$ -query algorithm $\mu$ there is a $C$ with $|C| = n/2$ and

\mathbb{E}_{A\sim\mu}\bigl[f_C(A(f_C))\bigr] \le \tfrac14 n^2 + \bigl(2e^{-\epsilon^2 n/8} + 2e^{-\epsilon^2 n/4}\bigr)\,\mathrm{OPT}(f_C).

Hence the ratio attained is at most $\frac{1}{2(1-2\epsilon+2\epsilon^2)} + 4e^{-\epsilon^2 n/8} = \tfrac12 + \epsilon + O(\epsilon^2) + 4e^{-\epsilon^2 n/8}$ .

Milestones

Theorem 1.2, the Chernoff bound for independent variables in $[-1,1]$ .
Submodularity of $f_C$ .
The value $\mathrm{OPT}(f_C) = \tfrac12 n^2(1 - 2\epsilon + 2\epsilon^2)$ , attained at $S = C$ .
A fixed query is unbalanced for at most a $2e^{-\epsilon^2 n/4}$ fraction of the half-size sets $C$ .
If all queries are balanced, the algorithm cannot distinguish $f_C$ from $g$ .
The deterministic case of the bound, averaged over $C$ .

Significance

The theorem shows that the factor $\tfrac12$ for symmetric submodular maximization, and hence for unconstrained submodular maximization in general, cannot be improved by any algorithm that uses a subexponential number of value queries, whatever its running time. It is an information-theoretic bound and needs no complexity assumption. Along with the later matching $\tfrac12$ -approximation, it settles the value-oracle approximability of the problem. The construction, a function equal to a symmetric function on "balanced" sets and larger elsewhere, is the prototype of the symmetry-gap technique used for many later oracle lower bounds.

The result is proved in the paper. No machine-checked version is known to exist: the platform has no value-oracle or query-lower-bound statement. Formalizing it requires a precise model of adaptive randomized query algorithms, a concentration bound for the hypergeometric distribution, and a finite verification of submodularity of an explicit two-regime function, and it fixes the constants that the printed statement leaves as $O(\cdot)$ terms.

Difficulty

Two steps of the printed argument do not go through as written. First, the proof bounds the probability that a fixed query is unbalanced by citing the Chernoff bound for independent variables, but for a uniformly random half-size set $C$ the count $|Q\cap C|$ is hypergeometric, and the summands are not independent. A bound for sampling without replacement is needed instead. Replacing the balanced partition by independent coin flips is not an option: then $|C| \ne n/2$ in general, and the function is no longer the paper's instance.

Second, the argument counts only the queries, but the value an algorithm receives is $f_C$ of its output, which equals $g$ of the output only if the output is balanced as well. That event has to be controlled too.

Finally, submodularity of $f_C$ must be checked across the boundary $|k-\ell| = \epsilon n$ between the two regimes, where the formula changes.

Formalization scope

The ground set is Fin n with $n$ even; $\epsilon n$ is an integer $m$ with $1 \le m$ and $2m \le n$ , following the paper's "assume that $\epsilon n$ is an integer". Sets are Finset (Fin n), and all values are real.
$\mathrm{OPT}$ is Finset.sup' over all subsets. There is no junk value.
The partition $(C, D)$ is uniform over half-size sets; probabilities over it are counts of n/2-subsets divided by $\binom{n}{n/2}$ , written multiplied out.
A deterministic algorithm is a pair of decision rules query, output : List ℝ → Finset (Fin n) making exactly $q$ adaptive queries with arbitrary real answers. A randomized algorithm is a PMF over deterministic algorithms, which covers every randomization with countable support. The algorithm sees $f$ only through query answers; it never receives $C$ .
Pinned-down constants. The printed theorem, "fewer than $e^{\epsilon^2 n/8}$ queries" and "expected value at least $(\tfrac12+\epsilon)\mathrm{OPT}$ ", is not what the proof gives for one and the same $\epsilon$ . On the proof's instances $\mathrm{OPT} = \tfrac12 n^2(1-2\epsilon+2\epsilon^2)$ , and the ratio held is $\frac{1}{2(1-2\epsilon+2\epsilon^2)} > \tfrac12 + \epsilon$ . The formal goal states the explicit bound the proof establishes. The literal printed pair, stated for the proof's family with the same $\epsilon$ , is false: the zero-query algorithm that outputs a fixed half-size set gets at least $\tfrac14 n^2 > (\tfrac12+\epsilon)\mathrm{OPT}$ .
Added term. The error term $2e^{-\epsilon^2 n/4}$ for the output set is added to the paper's $2e^{-\epsilon^2 n/8}$ .
Ruled-out trivializations. A restricted algorithm class (non-adaptive, deterministic, or one that must return a queried set) would give a different, weaker theorem. So would a bound that lets the algorithm read $C$ , which would make the statement false. Both the instance's properties (nonnegativity, symmetry, submodularity, the value of OPT) and the bound are part of the goal, so an empty or degenerate family cannot satisfy it. The quantifier order is: for every algorithm there is an instance.
Needed infrastructure: a value-oracle algorithm model; tail bounds for the hypergeometric distribution (Hoeffding's inequality for sampling without replacement), which Mathlib lacks; averaging over a PMF of algorithms. The algorithm model and the hypergeometric bound are reusable for other oracle lower bounds. Proofs of any milestone, and alternative derivations of the balance bound, are welcome.

Selected references

U. Feige, V. S. Mirrokni, J. Vondrák, Maximizing Non-Monotone Submodular Functions, SIAM J. Comput. 40(4):1133–1153, 2011. https://doi.org/10.1137/090779346
N. Alon, J. H. Spencer, The Probabilistic Method, Wiley (source of Theorem 1.2).
W. Hoeffding, Probability Inequalities for Sums of Bounded Random Variables, J. Amer. Statist. Assoc. 58(301):13–30, 1963. https://doi.org/10.1080/01621459.1963.10500830
J. Vondrák, Symmetry and Approximability of Submodular Maximization Problems, SIAM J. Comput. 42(1):265–304, 2013. https://doi.org/10.1137/110832318
N. Buchbinder, M. Feldman, J. Naor, R. Schwartz, A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization, SIAM J. Comput. 44(5):1384–1402, 2015. https://doi.org/10.1137/130929205

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Combinatorics Convex Optimization Discrete Geometry+2·Captain: mikedeng1

Lifts of Convex Sets and Cone Factorizations II: Antichain and Face-Count Lower Bounds on the Nonnegative Rank of a PolytopeResearch Paper

Motivation

Many polytopes that arise in combinatorial optimization, such as the matching, cut, stable set and travelling salesman polytopes, have exponentially many facets, yet some of them can be written as the linear projection of a polyhedron with far fewer facets. The smallest number of facets of such a lift decides whether the polytope admits a compact linear-programming formulation. Yannakakis (Expressing combinatorial optimization problems by linear programs, J. Comput. System Sci. 43 (1991)) showed that this number equals the nonnegative rank of the polytope's slack matrix, turning a question about formulations into a question about matrix factorizations. Gouveia, Parrilo and Thomas (arXiv:1111.3164v2) extended this correspondence from polytopes and nonnegative orthants to arbitrary convex bodies and closed convex cones.

Exact nonnegative rank is NP-hard to compute (Vavasis, SIAM J. Optim. 20 (2009)), so lower bounds matter. The oldest ones are combinatorial: they see only which entries of the slack matrix are zero. Goemans (Smallest compact formulation for the permutahedron, Math. Program. 153 (2015)) observed that a polytope with $n_C$ faces needs a lift with at least $\log_2 n_C$ facets. Section 4.2 of Gouveia–Parrilo–Thomas recasts these support-based bounds through the face lattice and derives, alongside Goemans' bound, a sharper antichain bound. This mission formalizes that chain of results.

Setting

Write $\mathbb{R}^n$ for Euclidean space with inner product $\langle\cdot,\cdot\rangle$ . A polytope $C \subseteq \mathbb{R}^n$ is the convex hull of finitely many points; as throughout the paper, the origin is assumed to lie in its interior. The polar of $C$ is

C^\circ = \{\, y \in \mathbb{R}^n : \langle x, y\rangle \le 1 \text{ for all } x \in C \,\}.

Let $\operatorname{ext}(C)$ be the set of extreme points of $C$ (its vertices). The slack operator $S_C$ is the function $S_C(x, y) = 1 - \langle x, y\rangle$ on $\operatorname{ext}(C) \times \operatorname{ext}(C^\circ)$ . The extreme points of $C^\circ$ correspond to the facets of $C$ , the facet of $y$ being $\{x \in C : \langle x, y\rangle = 1\}$ , so $S_C$ is the canonical vertex–facet slack matrix of $C$ and is nonnegative.

An $\mathbb{R}^k_+$ -factorization of $S_C$ consists of maps $A : \operatorname{ext}(C) \to \mathbb{R}^k_+$ and $B : \operatorname{ext}(C^\circ) \to \mathbb{R}^k_+$ with $S_C(x, y) = \langle A(x), B(y)\rangle$ . The nonnegative rank $\operatorname{rank}_+(C)$ is the least such $k$ , and $+\infty$ if there is none.

The support $\operatorname{supp}(S_C)$ is the $0/1$ matrix with a one where $S_C(x,y) \ne 0$ . A Boolean factorization of it of intermediate dimension $k$ assigns subsets $A(x), B(y) \subseteq [k] = \{1,\dots,k\}$ with $S_C(x,y) \ne 0 \iff A(x) \cap B(y) \ne \emptyset$ ; the least such $k$ is the Boolean rank.

A face of $C$ is the empty set or a set of maximizers in $C$ of a linear functional; $C$ itself is a face. The face lattice $L(C)$ is the set of faces ordered by inclusion, and the Boolean lattice $2^{[k]}$ is the set of subsets of $[k]$ ordered by inclusion. An embedding $\varphi : L(C) \to 2^{[k]}$ satisfies $H \subseteq F \iff \varphi(H) \subseteq \varphi(F)$ .

Formalization targets

Goal: Corollary 4.13 (p. 16)

For a polytope $C$ :

\text{(1)}\quad \operatorname{rank}_+(C) \ \ge\ \min\Big\{ k : p \le \tbinom{k}{\lfloor k/2 \rfloor} \Big\}

for every antichain of $p$ faces of $C$ (no face contained in another), and

\text{(2)}\quad \operatorname{rank}_+(C) \ \ge\ \log_2 n_C ,

where $n_C$ is the number of faces of $C$ , including $\emptyset$ and $C$ .

Milestones

§4.2, p. 15. For a nonnegative matrix $M$ , $\operatorname{rank}_B(M) \le \operatorname{rank}_+(M)$ : a nonnegative factorization of intermediate dimension $k$ yields a Boolean factorization of $\operatorname{supp}(M)$ of the same dimension.
Theorem 4.11, p. 15. $\operatorname{supp}(S_C)$ has a Boolean factorization of intermediate dimension $k$ if and only if $L(C)$ embeds into $2^{[k]}$ .
Corollary 4.12, p. 15. $\operatorname{rank}_+(C) \ge \min\{k : L(C) \text{ embeds into } 2^{[k]}\}$ .

Significance

Both bounds depend only on the combinatorial type of the polytope. For a square they give $\operatorname{rank}_+ \ge \log_2 10 \approx 3.32$ and $\operatorname{rank}_+ \ge 4$ ; for a three-dimensional cube $\log_2 28 \approx 4.81$ and $6$ (p. 16). For the regular $n$ -gon, whose slack matrices all have rank $3$ , the face-count bound gives $\operatorname{rank}_+ \ge \log_2 n$ , which is of the optimal order (Example 4.14). Theorem 4.11 is the statement that the Boolean rank of a slack matrix, also known as its rectangle covering number, is an invariant of the face lattice; the rectangle-covering version is phrased as Theorem 2.9 of Fiorini, Kaibel, Pashkovich and Theis (Combinatorial bounds on nonnegative rank and extended formulations, arXiv:1111.0444), as cited by the paper.

The results are proved in the paper. The formalization provides machine-checked definitions of the polar, the slack operator of a polytope, its nonnegative and Boolean ranks and its face lattice, reusable for later work on extension complexity (for instance, rectangle-covering lower bounds for specific polytopes). No formal proof of these statements is known to exist in Lean or on this platform.

Difficulty

Milestone 1 and the passage from Corollary 4.12 to Corollary 4.13 are short: Sperner's theorem is available in Mathlib as IsAntichain.sperner, and an embedding of $L(C)$ into $2^{[k]}$ is injective. The weight lies in Theorem 4.11, which needs facts about polytopes that Mathlib does not state in this form: every vertex is an exposed point, each extreme point of the polar cuts out a face, every face of a polytope is the convex hull of the vertices it contains, and every proper face is the intersection of the facets containing it, with those facets indexed by $\operatorname{ext}(C^\circ)$ . The last fact is where the origin-in-the-interior assumption and the polar enter, and it fails for faces described by an arbitrary list of inequalities that is not the facet description. A second, smaller difficulty is finiteness: the face-count bound needs the set of faces of a polytope to be finite.

Formalization scope

$\mathbb{R}^n$ is EuclideanSpace ℝ (Fin n) with the Euclidean inner product. The polar is the one-sided polar above, not Mathlib's absolute polar. A polytope is the convex hull of a Finset with the origin in its interior; $n = 0$ is allowed ( $C = \{0\}$ ), and all targets hold there. Faces are Mathlib's exposed faces (IsExposed ℝ C F), which include $\emptyset$ and $C$ , as the paper's counts do; for a polytope these are all faces. The factorization maps are total functions on $\mathbb{R}^n$ constrained only on extreme points. The nonnegative rank is valued in ℕ∞, the infimum of the empty family being $+\infty$ ; part (2) of the goal is stated for every finite value of the rank. Part (1) is stated for every antichain of faces, equivalent to the paper's "largest antichain". "Smallest $k$ " is sInf of a set of naturals that is nonempty in each case (the set of $k$ with $p \le \binom{k}{\lfloor k/2\rfloor}$ , and the set of $k$ admitting an embedding of the finite lattice $L(C)$ ).

The paper says "lattice embedding". Its proof of Theorem 4.11 constructs, and uses, only a map that preserves and reflects inclusion, and $\varphi(F) = \bigcup_{v \in F} A(v)$ need not preserve joins or meets; the formalization reads "lattice embedding" as an order embedding (Face C ↪o Finset (Fin k)) throughout.

Trivializing formalizations are ruled out: the rank is not a natural-number infimum (which would be $0$ when no factorization exists); faces are not arbitrary subsets of $C$ ; and an embedding is order-reflecting, not merely monotone (every poset maps monotonically into $2^{[0]}$ ).

Welcome contributions: a proof of milestone 1; a library of polytope facts (vertices are exposed points, faces are convex hulls of their vertices, finiteness of the face lattice, facets from the polar), which is reusable well beyond this mission; then Theorem 4.11 and the corollaries.

Selected references

J. Gouveia, P. A. Parrilo, R. R. Thomas, Lifts of Convex Sets and Cone Factorizations, Math. Oper. Res. 38(2):248–264, 2013; arXiv:1111.3164v2. https://arxiv.org/abs/1111.3164
M. Yannakakis, Expressing combinatorial optimization problems by linear programs, J. Comput. System Sci. 43(3):441–466, 1991. https://doi.org/10.1016/0022-0000(91)90024-Y
M. X. Goemans, Smallest compact formulation for the permutahedron, Math. Program. 153:5–11, 2015. https://doi.org/10.1007/s10107-014-0757-1
S. Fiorini, V. Kaibel, K. Pashkovich, D. O. Theis, Combinatorial bounds on nonnegative rank and extended formulations, Discrete Math. 313(1):67–83, 2013; arXiv:1111.0444. https://arxiv.org/abs/1111.0444
S. A. Vavasis, On the complexity of nonnegative matrix factorization, SIAM J. Optim. 20(3):1364–1377, 2009. https://doi.org/10.1137/070709967

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Combinatorics Operations Research Optimization+2·Captain: mikedeng1

Maximizing Non-Monotone Submodular Functions IV: Smooth Local Search Achieves 2/5 of the OptimumResearch Paper

Motivation

Many optimization problems ask for a subset of a finite ground set that maximizes a submodular function, a set function with diminishing marginal returns. Max Cut and Max Directed Cut in graphs, facility location with fixed costs, and the maximization of mutual information or entropy of a subset of random variables are all of this form. Unlike the monotone case, where a greedy algorithm achieves $1 - 1/e$ , a general nonnegative submodular function may decrease when elements are added, and the empty set and the full set can both be poor. The question is how large a constant fraction of the optimum a polynomial-time algorithm can guarantee when the function is given only through an oracle that returns $f(S)$ for a queried set $S$ .

Feige, Mirrokni and Vondrák (SIAM J. Comput. 40(4), 2011) gave the first constant-factor algorithms for this problem. A uniformly random set achieves $1/4$ of the optimum, a deterministic local search achieves $1/3 - \epsilon/n$ , and a randomized smooth local search achieves $2/5 - o(1)$ . The last result is the paper's best approximation for general nonnegative submodular functions (Table 1, p. 1136), and it is the subject of this mission.

Timeline. Feige, Mirrokni and Vondrák: $1/4$ , $1/3$ and $2/5$ (FOCS 2007; journal version 2011). Gharan and Vondrák (SODA 2011): about $0.41$ by simulated annealing. Buchbinder, Feldman, Naor and Schwartz (FOCS 2012; SIAM J. Comput. 2015): a randomized double greedy algorithm achieving $1/2$ , which matches the $1/2$ hardness in the value oracle model proved in the same paper by Feige, Mirrokni and Vondrák.

Setting

Let $X$ be a finite ground set with $n = |X| \ge 1$ elements and $f : 2^X \to \mathbb{R}$ a function with $f(S) \ge 0$ for all $S$ and

f(S \cup T) + f(S \cap T) \le f(S) + f(T) \quad (S, T \subseteq X).

Write $OPT = \max_{S \subseteq X} f(S)$ . The multilinear extension of $f$ is

F(x) = \sum_{S \subseteq X} f(S) \prod_{i \in S} x_i \prod_{j \notin S} (1 - x_j),

the expected value of $f$ on a random set containing each $i$ independently with probability $x_i$ .

For $A \subseteq X$ and $\delta \in [-1,1]$ , the random set $\mathcal{R}(A,\delta)$ is sampled with bias $\delta$ based on $A$ : each element of $A$ is included independently with probability $p = (1+\delta)/2$ , each element of $B = X \setminus A$ with probability $q = (1-\delta)/2$ . The potential is $\Phi(A) = \mathbf{E}[f(\mathcal{R}(A,\delta))]$ and the smoothed marginal value of $x$ is

\omega_{A,\delta}(x) = \mathbf{E}[f(\mathcal{R}(A,\delta) \cup \{x\})] - \mathbf{E}[f(\mathcal{R}(A,\delta) \setminus \{x\})].

Algorithm SLS starts from $A = \emptyset$ . At each iteration it obtains estimates $\tilde\omega_{A,\delta}(x)$ within $\pm\frac{1}{n^2}OPT$ of $\omega_{A,\delta}(x)$ . If some $x \notin A$ has $\tilde\omega_{A,\delta}(x) > \frac{2}{n^2}OPT$ it adds $x$ ; otherwise, if some $x \in A$ has $\tilde\omega_{A,\delta}(x) < -\frac{2}{n^2}OPT$ it removes $x$ ; otherwise it stops and returns a random set $\mathcal{R}(A, \delta')$ .

Formalization targets

Goal: Theorem 3.6 in the explicit form of its proof

With $\delta = 1/3$ , and $\delta' = 1/3$ with probability $0.9$ or $\delta' = -1$ with probability $0.1$ : for every run from $\emptyset$ whose estimates are all accurate and which has terminated at $A$ ,

\tfrac{9}{10}\,\mathbf{E}[f(\mathcal{R}(A,\tfrac13))] + \tfrac{1}{10}\,f(X \setminus A) \ge \Big(\frac{2}{5} - \frac{9}{5n}\Big) OPT,

and, for every $\delta \in (0,1]$ , every run of $k$ iterations with accurate estimates has $k < n^2/\delta$ (fewer than $3n^2$ for $\delta = 1/3$ ).

Milestones, in attack order

Lemma 2.2: $\mathbf{E}[g(A(p))] \ge (1-p)g(\emptyset) + p\,g(A)$ .
Display (∗): for three independently sampled sets, $\mathbf{E}[f(A_1(p_1) \cup A_2(p_2) \cup A_3(p_3))] \ge \sum_{I \subseteq \{1,2,3\}} \prod_{i\in I} p_i \prod_{i \notin I}(1-p_i) f(\bigcup_{i \in I} A_i)$ .
The increment identity $\Phi(A \cup \{x\}) - \Phi(A) = \delta\,\omega_{A,\delta}(x)$ for $x \notin A$ , and its removal counterpart.
$0 \le \Phi(A) \le OPT$ .
The terminal upper estimates $\mathbf{E}[f(R \cup (B\cap C))],\ \mathbf{E}[f(R \cap (B \cup C))] \le \mathbf{E}[f(R)] + \frac{2}{n}OPT$ .
The two lower bounds in $27$ ths on the same two expectations.
The final chain $\mathbf{E}[f(R)] + \frac19 f(B) + \frac2n OPT \ge \frac49 OPT$ .

Significance

The $2/5$ bound showed that local search on a smoothed objective, the multilinear extension restricted to points with two coordinate values, beats both uniform sampling and plain local search for non-monotone submodular maximization. The multilinear extension later became the standard tool for submodular maximization under constraints, through continuous greedy methods, contention resolution schemes and the analysis of randomized rounding. Display (∗) and Lemma 2.2 are the basic sampling inequalities for submodular functions and are reused throughout that literature.

The result is proved in the paper; it has been superseded in ratio by later algorithms reaching $1/2$ . To our knowledge none of it is machine-checked: Mathlib has no multilinear extension and no submodular maximization results. This mission produces a checked form of the analysis with every constant explicit: the $o(1)$ as $\frac{9}{5n}$ , "polynomial time" as $n^2/\delta$ iterations, and the dependence on the accuracy of the sampled estimates as an explicit hypothesis.

Difficulty

The iteration bound and the increment identity are routine once the multilinear extension is set up. The substance lies in the lower bounds. The returned set $R$ is random, so the comparison with the optimal set $C$ cannot be made through a single local-optimality inequality as in deterministic local search; and approximate local optimality holds only for the smoothed marginals $\omega_{A,\delta}$ , which are averages over the random set, not for $f$ at any fixed set. Sampling inequalities such as (∗) are stated for independent samples of arbitrary, possibly overlapping sets, and their expectations are sums over products of subsets; the bookkeeping of such sums is the main formalization burden. The constants must also balance exactly: with $\delta = 1/3$ the $27$ ths add up so that the $\frac{9}{10}/\frac{1}{10}$ mixture yields $2/5$ . A different split or a different $\delta$ gives a different constant.

Formalization scope

The ground set is a Fintype X with decidable equality; sets are Finset X; $f$ is real valued, with nonnegativity and submodularity as hypotheses. The standing assumptions of the paper are made explicit: $f \ge 0$ (§3), value-oracle access (modelled by $f$ itself), $n = |X|$ , and $n \ge 1$ (Nonempty X), so that $\frac{1}{n^2}$ and $\frac{9}{5n}$ are not Lean's junk value of division by zero. $OPT$ is Finset.sup' over all subsets, which has no junk value. Every expectation over independently sampled sets is an exact finite sum: the multilinear extension for $\mathcal{R}(A,\delta)$ , and iterated sums over subsets for the several independent samples in Lemma 2.2 and (∗). Sampling probabilities carry $0 \le p \le 1$ .

The algorithm is a relation, not a choice: any element meeting the step-3 condition may be added, and removal is allowed only when no addition applies. The goal quantifies over every run from $\emptyset$ , every choice of accurate estimates, recomputed at every iteration, and every termination point. Accuracy is non-strict ("within $\pm$ "); the thresholds are strict. The thresholds and accuracy use $OPT$ itself, as the proof does, although step 1 of the algorithm says an estimate of $OPT$ is used. The sampling that produces the estimates and its "with high probability" are not modelled, and the goal is conditional on accurate estimates. The value of the $\delta' = -1$ branch is written as $\mathbf{E}[f(\mathcal{R}(A,-1))]$ , which equals $f(X \setminus A)$ .

A statement "every $A$ satisfying the terminal conditions gives $2/5 - 9/(5n)$ " would be the final milestone plus arithmetic and is not the goal. The goal fixes the start at $\emptyset$ , the thresholds, the step order, the accuracy of every estimate, termination and the $0.9/0.1$ mixture.

A complete development needs basic calculus of the multilinear extension: affinity in one coordinate, translation $f(\cdot \cup D)$ and restriction $f(\cdot \cap D)$ , and splitting a sample into disjoint pieces. These lemmas are reusable for any work on submodular maximization, and contributions of them as separate theorems are welcome. The golden-ratio variant $\delta = \delta'$ (proof omitted in the paper), the tight example and the hardness results of §4 are out of scope.

Selected references

U. Feige, V. S. Mirrokni, J. Vondrák, Maximizing Non-Monotone Submodular Functions, SIAM J. Comput. 40(4):1133–1153, 2011. https://doi.org/10.1137/090779346
S. O. Gharan, J. Vondrák, Submodular Maximization by Simulated Annealing, SODA 2011. https://doi.org/10.1137/1.9781611973082.83
N. Buchbinder, M. Feldman, J. Naor, R. Schwartz, A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization, SIAM J. Comput. 44(5):1384–1402, 2015. https://doi.org/10.1137/130929205
G. Calinescu, C. Chekuri, M. Pál, J. Vondrák, Maximizing a Monotone Submodular Function Subject to a Matroid Constraint, SIAM J. Comput. 40(6):1740–1766, 2011. https://doi.org/10.1137/080733991

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Linear algebra Numerical Analysis Probability+1·Captain: mikedeng1

Randomized Algorithms for Estimating the Trace of an Implicit Symmetric Positive Semi-Definite Matrix IV: Sample Bound for Hutchinson's Trace EstimatorResearch Paper

Motivation

Many computations need the trace of a matrix $A$ that is never formed explicitly and can only be applied to vectors: the trace of a matrix function such as $\mathrm{trace}(A^{-1})$ or $\log\det A = \mathrm{trace}(\log A)$ in statistics and lattice QCD, the Frobenius norm $\|B\|_F^2 = \mathrm{trace}(B^TB)$ of an operator, or the number of triangles of a graph. The standard tool is Monte-Carlo estimation, introduced by M. F. Hutchinson (Hutchinson 1989): average $M$ quadratic forms $z_i^TAz_i$ over random sign vectors $z_i$ . Each sample costs one matrix–vector product, uses one random bit per entry, and needs only additions and subtractions.

Before Avron and Toledo (2011), only the variance of such estimators had been analysed. A small variance does not say how many samples guarantee a given relative error with a given probability. Avron and Toledo gave the first bounds of this kind for several estimators. This mission covers the bound for Hutchinson's estimator, their Theorem 7.1.

Setting

A Rademacher random variable takes the values $+1$ and $-1$ , each with probability $1/2$ . Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive semi-definite. Draw $M \ge 1$ independent random vectors $z_1, \ldots, z_M \in \mathbb{R}^n$ whose $Mn$ entries are independent Rademacher variables. Hutchinson's trace estimator is

H_M = \frac{1}{M}\sum_{i=1}^{M} z_i^TAz_i .

A single sample $z^TAz$ is an unbiased estimator of $\mathrm{trace}(A)$ (Lemma 2.1 of the paper, due to Hutchinson). For symmetric $A$ its variance is $2\bigl(\|A\|_F^2 - \sum_i A_{ii}^2\bigr)$ , twice the squared Frobenius mass of $A$ off the diagonal.

Given $\epsilon > 0$ and $\delta \in (0,1)$ , a random estimator $T$ is an $(\epsilon,\delta)$ -approximator of $\mathrm{trace}(A)$ if

\Pr\bigl(|T - \mathrm{trace}(A)| \le \epsilon\,\mathrm{trace}(A)\bigr) \ge 1 - \delta .

The rank $\mathrm{rank}(A)$ is the number of nonzero eigenvalues $\lambda_j$ of $A$ , counted with multiplicity.

Formalization targets

Goal: Theorem 7.1, sample bound for $H_M$

For every symmetric positive semi-definite $A$ , every $0 < \epsilon \le 1/2$ and every $0 < \delta < 1$ ,

M \ge 6\epsilon^{-2}\ln\!\left(\frac{2\,\mathrm{rank}(A)}{\delta}\right) \quad\Longrightarrow\quad H_M \text{ is an } (\epsilon,\delta)\text{-approximator of } \mathrm{trace}(A).

The bound depends on $A$ only through its rank. It does not depend on the dimension $n$ , on the condition number, or on how the trace is spread over the diagonal.

Milestones

Lemma 7.2 (Achlioptas 2001, Lemma 5). For a unit vector $\alpha \in \mathbb{R}^n$ and $S = \frac1M\sum_{i=1}^M(\alpha^Tz_i)^2$ , for every $\epsilon > 0$ ,

\Pr(|S - 1| \ge \epsilon) \le 2\exp\!\left(-\frac{M}{2}\left(\frac{\epsilon^2}{2} - \frac{\epsilon^3}{3}\right)\right).

Per-direction bound (proof of Theorem 7.1, p. 8:11). Let $r \ge 1$ and $0 < \epsilon \le 1/2$ . If $M \ge 6\epsilon^{-2}\ln(2r/\delta)$ , then $\Pr(|S - 1| \ge \epsilon) \le \delta/r$ .
From directions to the trace (proof of Theorem 7.1, p. 8:11). Write $A = U\Lambda U^T$ and $y_i = U^Tz_i$ . If $|\frac1M\sum_i y_{ij}^2 - 1| \le \epsilon$ for every $j$ with $\lambda_j \ne 0$ , then $|H_M - \mathrm{trace}(A)| \le \epsilon\,\mathrm{trace}(A)$ . This step is deterministic.
Lemma 2.1 (Hutchinson). $\mathrm{E}(z^TAz) = \mathrm{trace}(A)$ , and for symmetric $A$ , $\mathrm{Var}(z^TAz) = 2(\|A\|_F^2 - \sum_i A_{ii}^2)$ .

Significance

Theorem 7.1 gives a practitioner an explicit number of matrix–vector products after which Hutchinson's method is guaranteed to reach relative accuracy $\epsilon$ with confidence $1-\delta$ . No bound was available before, although the method had been in wide use for two decades. The bound exceeds the one the same paper proves for Gaussian test vectors ( $20\epsilon^{-2}\ln(2/\delta)$ ) by a $\ln(\mathrm{rank}(A))$ factor. The authors conjecture that this factor is not needed. Later work removed it: Roosta-Khorasani and Ascher (2015) proved a rank-free bound for the Rademacher case, and Cortinovis and Kressner (2022) extended sample bounds to indefinite matrices. Sample bounds of this type underlie variance-reduced estimators such as Hutch++ (Meyer et al. 2021).

The theorem is proved in the literature. To the platform's knowledge it has no machine-checked proof. Formalizing it produces a checked Rademacher concentration inequality for averages of squared linear forms (Achlioptas' lemma, which is also a core lemma of database-friendly Johnson–Lindenstrauss projections), and a checked spectral reduction from a quadratic-form estimator to its eigen-directions. Neither is currently in Mathlib.

Difficulty

The estimator is not an average of independent copies of a bounded variable with a small range: a single sample $z^TAz$ can deviate from $\mathrm{trace}(A)$ by an amount comparable to $\|A\|_F\sqrt n$ . Chebyshev's inequality with the variance of Lemma 2.1 only gives a polynomial dependence on $1/\delta$ . Hoeffding's inequality applied to $z^TAz$ directly gives a dependence on $n$ and on the size of the entries of $A$ . Unlike the Gaussian case, the estimator cannot be written as a weighted sum of independent chi-squared variables, because rotating a Rademacher vector does not give another Rademacher vector. The central difficulty is Lemma 7.2: a tail bound for $(\alpha^Tz)^2$ that holds uniformly over every unit direction $\alpha$ , including directions in which $\alpha^Tz$ is far from Gaussian (for $\alpha = e_1$ it is a constant).

Formalization scope

Matrices are Matrix (Fin n) (Fin n) ℝ, and positive semi-definiteness is Matrix.PosSemidef. The Rademacher law is $\tfrac12(\delta_1 + \delta_{-1})$ on ℝ. The sample space is Fin M → Fin n → ℝ with the product of $Mn$ copies of this law (hutchinsonSampleMeasure), so the law of the estimator is constructed, not assumed. $H_M(\omega) = (M:\mathbb{R})^{-1}\sum_i \omega_i\cdot(A\omega_i)$ . Probabilities are Measure.real, and the $(\epsilon,\delta)$ -approximator is Definition 4.1 verbatim. $\mathrm{rank}(A)$ is Matrix.rank. In Lemma 7.2 the i.i.d. copies $Q_i$ are the functions $(\alpha\cdot z_i)^2$ on this product space.

Corrections of printed statements, each recorded in the item's Formalization Note:

Theorem 7.1 is printed without a range on $\epsilon$ . Its proof needs $\frac{M}{2}(\frac{\epsilon^2}{2} - \frac{\epsilon^3}{3}) \ge \frac{M\epsilon^2}{6}$ , which holds exactly when $\epsilon \le 1/2$ . Without a range the printed statement is false: take $A = \frac1n\mathbf 1\mathbf 1^T$ , $n = 10^4$ , $\epsilon = 10$ and $\delta = 10^{-4}$ . The condition admits $M = 1$ , while $\Pr(H_1 > 11) \approx 9\cdot10^{-4} > \delta$ . The goal and milestone 2 are therefore stated for $0 < \epsilon \le 1/2$ .
Lemma 2.1 is printed for an arbitrary $n\times n$ matrix. The variance formula fails for non-symmetric $A$ : for $A = \begin{pmatrix}0&1\\0&0\end{pmatrix}$ the variance is $1$ , not $2$ . Symmetry is assumed for the variance part only.
Proof of Theorem 7.1. The proof writes $\Lambda = UAU^T$ together with $y_i = U^Tz_i$ . These fit together only for $A = U\Lambda U^T$ , which is the convention of milestone 3.

For $A = 0$ the threshold involves $\ln 0$ . Lean's Real.log 0 = 0 turns the condition into $M \ge 0$ , which agrees with the paper's reading $\ln 0 = -\infty$ . The conclusion then holds because $H_M = \mathrm{trace}(A) = 0$ , so no extra hypothesis is added. A formalization that took the law of the samples as a hypothesis could make that hypothesis unsatisfiable and the theorem vacuous; the constructed product space rules this out.

A complete development needs:

the product Rademacher measure and moment generating functions of Rademacher sums;
a Chernoff bound for averages of i.i.d. bounded variables on a product space;
the spectral theorem for real symmetric matrices, with rank equal to the number of nonzero eigenvalues;
a finite union bound.

The Rademacher concentration results are reusable beyond this mission. Proofs of any milestone are welcome, as are alternative proofs of Lemma 7.2.

Selected references

H. Avron and S. Toledo, Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix, Journal of the ACM 58(2), Article 8, 2011. https://doi.org/10.1145/1944345.1944349
M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines, Communications in Statistics – Simulation and Computation 18(3), 1059–1076, 1989. https://doi.org/10.1080/03610918908812806
D. Achlioptas, Database-friendly random projections, Proceedings of PODS 2001, 274–281. https://doi.org/10.1145/375551.375608
F. Roosta-Khorasani and U. Ascher, Improved bounds on sample size for implicit matrix trace estimators, Foundations of Computational Mathematics 15, 1187–1212, 2015. https://doi.org/10.1007/s10208-014-9220-1
A. Cortinovis and D. Kressner, On randomized trace estimates for indefinite matrices with an application to determinants, Foundations of Computational Mathematics 22, 875–903, 2022. https://doi.org/10.1007/s10208-021-09525-9
R. A. Meyer, C. Musco, C. Musco and D. P. Woodruff, Hutch++: Optimal stochastic trace estimation, SOSA 2021, 142–155. https://doi.org/10.1137/1.9781611976496.16

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Dynamic Programming Operations Research Optimization+1·Captain: Shuze Chen

Markov Decision Processes I: Markov Decision Models and the Bellman EquationTextbook

Motivation

A Markov Decision Model (MDM) formalizes sequential decision-making under uncertainty: a controller observes the current state of a system, chooses an action, receives a reward, and the system moves to a new (random) state whose law depends on the current state and action. This framework underlies dynamic programming across operations research, economics, and engineering — inventory control, sequential portfolio choice, queueing control, and reinforcement learning are all instances of it. The finite-horizon theory developed here is the foundation on which every later chapter of Bäuerle and Rieder's Markov Decision Processes with Applications to Finance (Springer, 2011) builds, including the infinite-horizon, partially observed, and optimal-stopping variants treated later in the book.

The classical treatment of dynamic programming for finite state and action spaces goes back to Bellman (1957) and is standard textbook material (see e.g. Puterman, Markov Decision Processes, 1994). The generalization to Borel state and action spaces — needed as soon as a state variable is continuous, as in almost every financial application — requires genuine measure-theoretic care: suprema over an infinite (even uncountable) action set need not be attained, and the resulting value function need not be measurable. Bertsekas and Shreve's Stochastic Optimal Control: The Discrete Time Case (1978) is the classical reference for this general theory; Bäuerle and Rieder's treatment isolates the exact abstract hypothesis — the Structure Assumption (SAN) below — under which the finite-horizon theory goes through cleanly, separating the recursive (Bellman) machinery from the case-by-case verification of when it applies.

Setting

A Markov Decision Model with planning horizon $N \in \mathbb{N}$ consists of a state space $E$ and action space $A$ (measurable spaces), and, for each stage $n = 0,\dots,N-1$ : a measurable set $D_n \subseteq E \times A$ of admissible state-action pairs (containing the graph of some measurable selection $E \to A$ ); a stochastic transition kernel $Q_n(\cdot \mid x,a)$ giving the law of the next state; a measurable one-stage reward $r_n : D_n \to \mathbb{R}$ ; and a terminal reward $g_N : E \to \mathbb{R}$ .

A decision rule at time $n$ is a measurable $f_n : E \to A$ with $f_n(x) \in D_n(x) := \{a : (x,a) \in D_n\}$ for every $x$ ; an $N$ -stage policy $\pi = (f_0,\dots,f_{N-1})$ is a sequence of such rules. Given $\pi$ and an initial state $x$ at time $n$ , the process evolves as a (non-stationary) Markov chain, and the value of $\pi$ is the expected total reward

V_n^\pi(x) := \mathbb{E}^\pi_{n,x}\!\left[\sum_{k=n}^{N-1} r_k\bigl(X_k, f_k(X_k)\bigr) + g_N(X_N)\right],

with value function $V_n(x) := \sup_\pi V_n^\pi(x)$ , the best attainable expected reward. A policy is optimal if $V_0^\pi = V_0$ . Write $\mathrm{IM}(E)$ for the measurable functions $E \to [-\infty,\infty)$ (never $+\infty$ ), and define, for $v \in \mathrm{IM}(E)$ , the one-step operators $L_n v(x,a) := r_n(x,a) + \int v(x')\, Q_n(dx' \mid x,a)$ , $T_n^f v(x) := L_n v(x, f(x))$ , and $T_n v(x) := \sup_{a \in D_n(x)} L_n v(x,a)$ . A decision rule $f$ is a maximizer of $v$ at time $n$ if $T_n^f v = T_n v$ .

Formalization targets

Goal: Theorem 2.3.8 (the Structure Theorem)

Under the Structure Assumption (SAN) — the existence of sets $\mathrm{IM}_n \subseteq \mathrm{IM}(E)$ , $\Delta_n \subseteq F_n$ with $g_N \in \mathrm{IM}_N$ , $T_n$ mapping $\mathrm{IM}_{n+1}$ into $\mathrm{IM}_n$ , and every $v \in \mathrm{IM}_{n+1}$ admitting a maximizer in $\Delta_n$ — the value function satisfies the Bellman equation $V_n = T_n V_{n+1}$ (with $V_n \in \mathrm{IM}_n$ ), and every sequence of maximizers of $V_1,\dots,V_N$ defines an optimal policy. This is the weakest level at which the theorem holds: it names an abstract structural hypothesis rather than a specific sufficient condition (e.g. compactness plus semicontinuity, treated in a later mission), so any future refinement of sufficient conditions for (SAN) leaves this statement untouched.

Significance

Reducing an $N$ -stage optimization over an infinite-dimensional policy space to $N$ one-stage optimizations — literally the content of the Bellman equation — is what makes dynamic programming computationally and theoretically tractable at all. For finite state and action spaces this reduction is elementary (a supremum over a finite set is always attained); the content of the Structure Theorem is doing this correctly when $E$ , $A$ are general Borel spaces, where existence of the supremum and of a measurable maximizing selection are not automatic and must be assumed abstractly.

Formalizing this theorem produces machine-checked statements of the finite-horizon Bellman equation and verification theorem in the generality actually used throughout the book's finance applications (wealth is real-valued, portfolios are vector-valued — never finite sets). The statement, proof, and every hypothesis are original to this textbook chapter; no formalized version of this general-Borel-space theory exists on the platform. The closest prior art, finite state-and-action-space Bellman equations and verification theorems (e.g. discounted infinite-horizon and stochastic-shortest-path theorems for finite MDPs), is a strictly weaker special case in which the Structure Assumption's existence-of-maximizer clause is automatic; this mission's goal is not restated as a reference to that prior art; the generalization is exactly the mission's content.

Difficulty

The obvious first attempt — prove $V_n = T_n V_{n+1}$ directly from the definitions of $V_n$ and $V_n^\pi$ — runs into two separate obstructions that (SAN) is built to bypass simultaneously. First, $\sup_\pi V_n^\pi(x)$ and $\sup_{a \in D_n(x)} L_n V_{n+1}(x,a)$ are a priori different suprema (over policies versus over actions), and showing they agree requires that the pointwise supremum over decision rules $f \in F_n$ of $L_n V_{n+1}(x, f(x))$ equals the supremum over bare actions $a \in D_n(x)$ — which needs a measurable selection achieving (or approaching) the action-wise optimum, not just its existence pointwise. Second, $V_{n+1}$ itself must be shown measurable — an a priori supremum of measurable functions over an uncountable index set (all policies) need not be measurable — before the integral $\int V_{n+1}\, dQ_n$ even makes sense. (SAN)'s three clauses are exactly what supplies both a well-behaved measurability class $\mathrm{IM}_n$ closed under $T_n$ and a measurable maximizing selection at every stage, letting a backward induction on $n$ establish both facts together.

Formalization scope

State and action spaces are arbitrary measurable spaces (MeasurableSpace E, MeasurableSpace A type classes), not restricted to Borel subsets of Polish spaces, since none of this mission's statements use topology. Time is indexed by ℕ rather than Fin N, with n < N as an explicit side condition throughout (documented in MODERATION_NOTES.md); this changes no content but avoids Fin-cast noise in the several backward recursions the chapter's operators require. Extended-real values (EReal) are used throughout for value functions, restricted by hypothesis to never equal $+\infty$ , matching $\mathrm{IM}(E) := \{v : E \to [-\infty,\infty)\}$ exactly; a formalization using plain ℝ-valued value functions would be a strictly stronger — and unfaithful — claim, since it silently assumes no policy can drive the expected reward to $-\infty$ .

The mission does not construct the canonical path measure on the full trajectory space via the Ionescu–Tulcea theorem; the value of a policy is instead built as an explicit backward accumulator recursion over the model's one-step kernels, which computes the same quantity by the tower property of conditional expectation. Definitions 2.1.1, 2.1.5, 2.2.2, 2.3.1, and 2.3.6 — the Markov Decision Model, (Markov and history-dependent) policies, and the operators — are restated from scratch in this mission's own namespace, since drafts in this series cannot import one another; later missions in the same series restate the same vocabulary independently. A trivializing formalization of the goal would state $V_n$ as an unspecified object merely postulated to satisfy the Bellman equation (Theorem 2.3.7's weaker claim) rather than as the supremum-over-policies value function fixed before the theorem — this mission states the latter.

Selected references

N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, Universitext, Springer, 2011. DOI: 10.1007/978-3-642-18324-9.
D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, 1978.
K. Hinderer, Foundations of Non-stationary Dynamical Programming with Discrete Time Parameter, Lecture Notes in Operations Research and Mathematical Systems 33, Springer, 1970.
M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994.

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Linear algebra Numerical Analysis Probability+1·Captain: mikedeng1

Randomized Algorithms for Estimating the Trace of an Implicit Symmetric Positive Semi-Definite Matrix II: Exact Rank of a Projection Matrix from the Gaussian Trace EstimatorResearch Paper

Motivation

Many matrices in scientific computing are available only implicitly: one can multiply a vector by $A$ , for instance by solving a linear system or applying a matrix function, but the entries of $A$ are never formed. Quantities such as the trace must then be estimated from a small number of matrix–vector products. Randomized trace estimators do exactly this: draw random vectors $z$ , compute the quadratic forms $z^TAz$ , and average.

Avron and Toledo (J. ACM 58(2), 2011) gave the first sample bounds of the form " $M$ samples suffice for relative error $\epsilon$ with probability $1-\delta$ " for the standard estimators. Among their results is a case in which the estimate is not merely approximate but exact: when $A$ is a projection matrix, its trace equals its rank, an integer, and rounding a Gaussian trace estimate recovers that integer with high probability. Computing the rank of a projection arises, for example, when charge densities are computed in electronic structure calculations without diagonalization (Bekas, Kokiopoulou and Saad 2007).

Setting

Let $n \ge 1$ and $A \in \mathbb{R}^{n\times n}$ . A projection matrix here is an orthogonal projection: $A$ is symmetric and $A^2 = A$ . Equivalently, $A$ is symmetric and every eigenvalue of $A$ is $0$ or $1$ ; its rank $\mathrm{rank}(A)$ is the number of eigenvalues equal to $1$ .

Fix a number of samples $M \ge 1$ . Let $z_1,\ldots,z_M \in \mathbb{R}^n$ be random vectors whose $Mn$ entries are independent standard normal random variables. The Gaussian trace estimator (Definition 3.1 of the paper) is

G_M = \frac{1}{M}\sum_{i=1}^{M} z_i^T A z_i .

Its expectation is $\mathrm{trace}(A)$ . For $x \in \mathbb{R}$ , $\mathrm{round}(x)$ denotes the nearest integer to $x$ . For $k \ge 1$ , a random variable $X$ has the $\chi^2$ distribution with $k$ degrees of freedom, $X \sim \chi^2(k)$ , if it has the law of $g_1^2 + \cdots + g_k^2$ for independent standard normal $g_1, \ldots, g_k$ .

Formalization targets

Goal: Lemma 5.3 (p. 8:9)

For a projection matrix $A$ , a failure probability $\delta > 0$ , and every integer $M \ge 1$ with $M \ge 24\,\mathrm{rank}(A)\ln(2/\delta)$ ,

\Pr\bigl(\mathrm{round}(G_M) \ne \mathrm{rank}(A)\bigr) \le \delta .

The number of samples depends on the rank and on $\delta$ only; there is no accuracy parameter, and none on $n$ .

Milestones, in the order the proof uses them

Law of $MG_M$ . $MG_M \sim \chi^2(M\,\mathrm{rank}(A))$ .
$\chi^2$ tail bound (cited by the paper from Li, Hastie and Church 2007). For $X \sim \chi^2(k)$ , $k\ge1$ and $0 < \epsilon \le \tfrac12$ ,

\Pr(|X - k| \ge \epsilon k) \le 2\exp(-k\epsilon^2/6).

Tail of $G_M$ . For $\mathrm{rank}(A) \ge 1$ and $0 < \epsilon \le \tfrac12$ ,

\Pr(|G_M - \mathrm{rank}(A)| \ge \mathrm{rank}(A)\epsilon) \le 2\exp(-M\,\mathrm{rank}(A)\epsilon^2/6).

Eq. (2). If moreover $M \ge 6\,\mathrm{rank}(A)^{-1}\epsilon^{-2}\ln(2/\delta)$ , then $\Pr(|G_M - \mathrm{rank}(A)| \ge \mathrm{rank}(A)\epsilon) \le \delta$ .
Trace of a projection. $\mathrm{trace}(A) = \mathrm{rank}(A)$ .

Significance

The lemma turns a randomized estimator into an exact algorithm with a controlled failure probability: the rank of an implicitly given projection is obtained from $O(\mathrm{rank}(A)\log(1/\delta))$ matrix–vector products, independently of the dimension $n$ . It is also an instance where the paper's general relative-error bound for the Gaussian estimator (Theorem 5.2, whose sample count grows like $\epsilon^{-2}$ ) is improved by exploiting the spectrum of $A$ : an absolute error below $\tfrac12$ is a relative error $\epsilon = 1/(2\,\mathrm{rank}(A))$ , for which the general bound would require a number of samples quadratic in the rank, whereas Lemma 5.3 needs only a linear number.

The result is proved in the paper. The mission produces a machine-checked version of it, together with two pieces of reusable substrate: the exact $\chi^2$ law of a Gaussian quadratic form in an orthogonal projection, and a two-sided $\chi^2$ tail bound with explicit constant $1/6$ on the range $0<\epsilon\le 1/2$ . To the best of available knowledge, none of these statements is formalized in Mathlib; a platform mission on the Johnson–Lindenstrauss lemma states a $\chi^2$ concentration bound with a different exponent, $(\epsilon^2-\epsilon^3)/4$ , on the open range $0<\epsilon<1/2$ , which does not cover the value $\epsilon = 1/2$ needed here when $\mathrm{rank}(A)=1$ .

Difficulty

The deterministic part is short; the probabilistic part is not. The step " $y = Uz$ has independent standard normal entries because $U$ is orthogonal" is the rotation invariance of the standard Gaussian measure on $\mathbb{R}^n$ , and it must be combined with the independence of the $M$ samples to identify the law of a sum of $M\,\mathrm{rank}(A)$ squares; this is a statement about product measures and pushforwards, not about moments. The $\chi^2$ tail bound is quoted by the paper without proof. Its constant $1/6$ is not the constant of the usual textbook $\chi^2$ estimates, and the bound is false outside a restricted range of $\epsilon$ (see below), so a generic sub-exponential concentration inequality with unspecified constants does not deliver it. Finally, the rounding step requires matching Mathlib's round with the event $|G_M - \mathrm{rank}(A)| < \tfrac12$ .

Formalization scope

All declarations live in the namespace TraceEstimation.ProjectionRank. Matrices are Matrix (Fin n) (Fin n) ℝ. The sample space of $G_M$ is Fin M → Fin n → ℝ with the product measure Measure.pi (fun _ => Measure.pi (fun _ => gaussianReal 0 1)), so the law of the estimator is constructed, not assumed. Probabilities are Measure.real of events. "Projection matrix" is A.IsHermitian ∧ A * A = A (orthogonal projection); a non-symmetric idempotent also has eigenvalues $0$ and $1$ , but the paper's proof diagonalizes $A$ by a unitary matrix, which requires symmetry. round is Mathlib's round : ℝ → ℤ; the rank is Matrix.rank. The $\chi^2$ distribution is not defined: its role is played by the pushforward of a product of standard normals under $g \mapsto \sum_l g_l^2$ .

Corrections of the printed text, both in the $\chi^2$ tail bound (milestone 2):

The paper prints $\Pr(|X - k| \le \epsilon k) \le 2\exp(-k\epsilon^2/6)$ . The inner $\le$ is a misprint for $\ge$ ; the next display applies the bound with $\ge$ .
The paper gives no range for $\epsilon$ . The bound fails for $\epsilon = 1$ and large $k$ , since $\Pr(X \ge 2k)$ decays like $e^{-k(1-\ln 2)/2}$ , slower than $e^{-k/6}$ . The mission states it for $0<\epsilon\le 1/2$ , and milestones 3 and 4 inherit that range; the proof uses only $\epsilon = 1/(2\,\mathrm{rank}(A)) \le 1/2$ .

The goal keeps the paper's hypotheses: $\delta > 0$ with no upper bound, and $\mathrm{rank}(A) = 0$ allowed (both are true cases). A statement about the event $|G_M - \mathrm{rank}(A)| < \tfrac12$ , or Eq. (2) alone, is not the goal; the goal is about $\mathrm{round}(G_M)$ . The sample measure is a probability measure, so the bound $\le \delta$ is not vacuous.

Contributions welcome beyond the milestones: rotation invariance of the standard Gaussian vector under orthogonal matrices, the moment generating function of $\chi^2(k)$ , and the spectral fact $\mathrm{trace}(A) = \mathrm{rank}(A)$ for idempotents, each reusable well outside this mission.

Selected references

H. Avron and S. Toledo, Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix, Journal of the ACM 58(2), Article 8, 2011. https://doi.org/10.1145/1944345.1944349
P. Li, T. Hastie and K. Church, Nonlinear estimators and tail bounds for dimension reduction in $l_1$ using Cauchy random projections, in Learning Theory (COLT 2007), Lecture Notes in Computer Science 4539, Springer, 514–529, 2007 (the version the paper cites); journal version in Journal of Machine Learning Research 8, 2497–2532, 2007, https://jmlr.org/papers/v8/li07b.html
C. Bekas, E. Kokiopoulou and Y. Saad, An estimator for the diagonal of a matrix, Applied Numerical Mathematics 57(11–12), 1214–1229, 2007. https://doi.org/10.1016/j.apnum.2007.01.003
M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines, Communications in Statistics – Simulation and Computation 19(2), 433–450, 1990. https://doi.org/10.1080/03610919008812866

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Algorithmic Game Theory Operations Research·Captain: mikedeng1

Monotonic Solutions of Cooperative Games 1: For Five or More Players No Core Allocation Rule Is Coalitionally MonotonicResearch Paper

Motivation

Cooperative games model the division of a jointly produced surplus, or a jointly incurred cost, among the participants of an enterprise: towns sharing a water supply system, divisions of a firm sharing overhead, members of a consortium sharing the savings of joint investment. In such applications allocations are rarely fixed once and for all. Costs and revenues are re-estimated, projects are re-scoped, and the allocation is recomputed. A natural requirement on any allocation method is then monotonicity: when the underlying data change in a player's favour, that player's share should not fall.

H. P. Young, Monotonic Solutions of Cooperative Games (Int. J. Game Theory 14, 1985), studies several forms of this principle. The weakest, aggregate monotonicity, asks that no player lose when only the value of the grand coalition increases; Megiddo (1974) had already shown that the nucleolus violates it. The paper's first theorem concerns a stronger and more natural form, coalitional monotonicity, which was first proposed by Shubik (1962) in the context of cost allocation within a firm, and shows that it cannot be reconciled with the core, the most widely used stability requirement. This mission formalizes that impossibility theorem.

Setting

Fix $n$ players, $N = \{1, 2, \dots, n\}$ . A cooperative game on $N$ is a function $v$ assigning a real number $v(S)$ to every coalition $S \subseteq N$ , with $v(\emptyset) = 0$ . No superadditivity or other structural assumption is imposed.

An allocation procedure is a map $\varphi$ that assigns to every game $v$ on $N$ an allocation $\varphi(v) = (\varphi_1(v), \dots, \varphi_n(v)) \in \mathbb{R}^N$ satisfying efficiency:

\sum_{i \in N} \varphi_i(v) = v(N).

The core of $v$ is the set of allocations that no coalition can improve upon on its own:

C(v) = \Big\{x \in \mathbb{R}^N : \sum_{i \in S} x_i \ge v(S) \text{ for all } S \subseteq N,\ \sum_{i \in N} x_i = v(N)\Big\}.

It may be empty. An allocation procedure is a core allocation rule if $\varphi(v) \in C(v)$ for every game $v$ whose core is nonempty. The nucleolus is the standard example.

An allocation procedure is coalitionally monotonic (Eq. (4) of the paper) if raising the value of a single coalition, with all other values unchanged, never decreases the allocation of any member of that coalition: for all games $v, w$ and every coalition $T \subseteq N$ ,

v(T) \ge w(T),\ \ v(S) = w(S)\ \ \forall S \ne T \ \Longrightarrow\ \varphi_i(v) \ge \varphi_i(w)\ \ \forall i \in T.

The paper notes that this is equivalent to its one-player form (5): if every coalition containing player $i$ weakly gains in value and every coalition not containing $i$ is unchanged, then $\varphi_i$ does not decrease.

In the Lean development these objects are Game n, IsAllocationProcedure, IsCoreRule and IsCoalitionallyMonotonic in the namespace MonotonicSolutions.CoreRules, and the core is the published definition Supermodularity.Cooperative.Core.

Formalization targets

Goal: Theorem 1

For every $n \ge 5$ ,

\neg\,\exists\,\varphi \ :\ \varphi \text{ is an allocation procedure on } N=\{1,\dots,n\},\ \varphi \text{ is a core allocation rule, and } \varphi \text{ is coalitionally monotonic}.

Milestones

Eqs. (4)–(5). Coalitional monotonicity is equivalent to the one-player condition (5).
The core of the game $w$ . For the explicit five-player game $w$ of the proof, built from the coalitions $S_1 = \{3,5\}$ , $S_2=\{1,2,3\}$ , $S_3=\{1,3,4\}$ , $S_4=\{2,4,5\}$ , $S_5=\{1,2,4,5\}$ with values $3,3,9,9,9$ and $w(N) = 11$ , the core is the single point $(0,1,2,7,1)$ .
The core of the game $v$ . For the game $v$ equal to $w$ except $v(S_5) = v(N) = 12$ , the core is the single point $(3,0,0,6,3)$ .
The case $|N| = 5$ . The goal with $n = 5$ .

An additional statement records the paper's remark that the egalitarian rule $\varphi_i(v) = v(N)/|N|$ satisfies (5), so that the monotonicity axiom on its own is satisfiable.

Significance

The theorem shows that no refinement of the core, however cleverly designed, can be coalitionally monotonic once there are five or more players. For the nucleolus this strengthens Megiddo's earlier observation, and the paper uses it to motivate the search for monotonic solution concepts outside the core, which leads to its second theorem: the Shapley value is the unique symmetric allocation procedure satisfying an even stronger monotonicity property (the subject of the companion mission). In cost-allocation practice the result says that an allocation method that is always stable against coalitional secession must sometimes penalise a division for making its own operations cheaper.

The result has been proved since 1985, and its proof is a short explicit counterexample. What the formalization adds is a machine-checked version of the counterexample (two explicit games whose cores are single points), a verified equivalence between the coalition-wise and player-wise forms of monotonicity, and the extension from five to arbitrarily many players, which the paper dismisses as an "obvious extension". To the best of the curators' knowledge none of these statements is formalized in any proof assistant.

Difficulty

The individual steps are elementary, but each requires care. Determining the core of each counterexample game exactly means showing both that the given point satisfies all $2^5$ coalition constraints and that no other point does. The equivalence of (4) and (5) is stated in the paper as following from "successive application" of (4), and every game involved must vanish on the empty coalition. The two counterexample games differ on two coalitions, $S_5$ and $N$ , so a single application of (4) does not suffice. Finally, the paper passes from five players to $n \ge 5$ players with the words "by obvious extension"; a formal proof has to make that step explicit, for arbitrary $n$ and for rules defined on all games with $n$ players, not only on those derived from five-player games.

Formalization scope

Players are Fin n; the paper's player $k$ is the Lean index $k-1$ . Thus the core points $(0,1,2,7,1)$ and $(3,0,0,6,3)$ appear as ![0,1,2,7,1] and ![3,0,0,6,3], and the players "2 and 4" whose allocations decrease are indices 1 and 3.
A game is the subtype Game n := {v : Finset (Fin n) → ℝ // v ∅ = 0}. Monotonicity axioms therefore quantify only over normalised games, as in the paper.
Allocations are real vectors Fin n → ℝ. Efficiency is part of the goal's hypotheses, matching the paper's definition of an allocation procedure.
The core is Supermodularity.Cooperative.Core Finset.univ v.1, the published core definition from Supermodularity and Complementarity, which coincides with the set on p. 68 of the paper.
IsCoreRule requires $\varphi(v) \in C(v)$ only when $C(v)$ is nonempty. The unconditional requirement would be unsatisfiable, because some games have an empty core, and would make the goal trivially true; that reading is ruled out.
IsCoalitionallyMonotonic is Eq. (4) with $T$ ranging over all coalitions, including $N$ . Restricting to $T \ne N$ , or restricting the games to superadditive ones, would change the theorem.
The counterexample games are defined in YoungGames: for $S \ne N$ , the value is the largest listed $w(S_k)$ over the $S_k \subseteq S$ , and $0$ if there is none.

Contributions welcome: proofs of the four milestones and the goal; a general lemma that padding a game with null players preserves the core up to zeros, which is reusable for other small-counterexample results in cooperative game theory; and decision procedures for linear feasibility over the coalitions of small games.

Selected references

H. P. Young, Monotonic Solutions of Cooperative Games, International Journal of Game Theory 14 (1985), 65–72. https://doi.org/10.1007/BF01769885
N. Megiddo, On the Nonmonotonicity of the Bargaining Set, the Kernel and the Nucleolus of a Game, SIAM Journal on Applied Mathematics 27 (1974), 355–358. https://doi.org/10.1137/0127026
M. Shubik, Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing, Management Science 8 (1962), 325–343. https://doi.org/10.1287/mnsc.8.3.325
D. B. Gillies, Solutions to General Non-Zero-Sum Games, in Contributions to the Theory of Games IV, Annals of Mathematics Studies 40 (1959), 47–85 (Princeton University Press). Cited as in Young (1985); no DOI checked.

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