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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

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

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

Campaigns (experimental)

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

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

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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Operations Research Probability Stochastic Systems·Captain: Shuze Chen

Processing Networks XIII: Back-Pressure Control for Packet NetworksTextbook

Motivation

Mission XII (12-packet-networks-model) built the discrete-time, slotted packet-network model from scratch and proved the chapter's own version of the fluid-to-stochastic stability bridge (Theorem 12.10). That result is only useful once paired with an actual control policy whose fluid model can be shown stable. J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) supplies exactly such a policy in Sections 12.4-12.5: back-pressure control (called max-weight when the network is single-hop), a rule that has become the default choice in the switching and wireless-scheduling literature because it requires no advance knowledge of arrival rates and achieves the largest possible stability region. This mission formalizes back-pressure control for the discrete-time packet network model and proves it maximally stable — the chapter's own counterpart to mission VIII's continuous-time Theorem 9.12.

Setting

At the start of each timeslot, having observed the current buffer contents $z$ , the max-weight/ back-pressure (MW/BP) policy solves $\max_{s\in S(z)} z\cdot Rs$ (Eq. 12.41), where $S(z) := \{s\in S : Bs\le z\}$ restricts the schedule set $S$ (mission XII) to what is actually available given $z$ . Equivalently (Eq. 12.42-12.43), writing $w_j(z) := z_{u(j)} - z_{d(j)}$ (with $z_0 := 0$ ) for the "weight" of activity $j$ , the policy maximizes $\sum_j w_j(z) s_j$ — the schedule that clears the most "backlog pressure" per timeslot. The policy is deterministic (no randomization variable is needed, since (12.41) is a genuine optimization problem, not one that inherently requires randomized tie-breaking).

Formalization targets

Goal: Theorem 12.16 — aperiodicity, irreducibility, and conditional positive recurrence

Consider a packet network satisfying (12.1) and Assumption 12.1, operating under back-pressure control. The DTMC $Z$ is aperiodic and irreducible unconditionally. If the stability condition (12.26) — the existence of $\hat s\in\langle S\rangle$ with $\lambda < R\hat s$ — is additionally satisfied, $Z$ is positive recurrent. This is the mission's headline result, and it packages both a structural fact (irreducibility/aperiodicity, needed regardless of load) and a conditional stability fact (positive recurrence, needed only under (12.26)) in one theorem.

Supporting milestones

Lemma 12.17 shows the optimization problem (12.41) always has a nonzero solution whenever the network is nonempty — the fact that back-pressure never "idles unnecessarily," which Lemma 12.18 uses to show every state can reach the empty state (hence irreducibility) and that state $0$ has period $1$ (hence aperiodicity, via (12.1)'s own assumption that no external arrivals is possible with positive probability). Lemma 12.20 is the back-pressure fluid equation (12.44), the discrete-time analog of mission VIII's Theorem 9.8: at each regular point, the fluid-scaled buffer content dotted with the fluid departure rate equals the maximum of that same dot product over the convex hull of the schedule set. Lemma 12.21 shows this fluid model is stable whenever (12.26) holds, via an explicit quadratic Lyapunov function.

Significance

The result itself. Theorem 12.16 shows that back-pressure control — a rule requiring no knowledge of arrival rates, computed fresh from the current buffer contents every timeslot — is maximally stable: combined with Theorem 12.8 and Proposition 12.9 (mission XII), it stabilizes every packet network arrival-rate vector that any policy could possibly stabilize. This is the discrete-time, slotted analog of mission VIII's Theorem 9.12, and the two proofs share their essential Lyapunov argument (the book's own text calls Lemma 12.21's proof "almost identical to that of Theorem 9.12"), even though the two chapters' back-pressure equations are built from different underlying objects — mission VIII's continuous-time allocation polytope versus this chapter's convex hull of a discrete schedule set.

Formalizing it. A live prior-art check (GET /theorems?q=back-pressure, q=max-weight) finds no relevant hits, matching mission VIII's own finding for the continuous-time version. This mission formalizes the back-pressure optimization problem and policy, the back-pressure fluid equation, and its stability from scratch, restating (per this series' convention for concurrently-drafted chunks sharing a sub-namespace) the minimal apparatus needed from mission XII: the packet network model, Assumption 12.1, the raw processes, fluid limit paths, the fluid equations (12.31)-(12.36), and the ambient-chain machinery, plus a new Aperiodic predicate this chunk needs that mission XII's own results do not.

Difficulty

The obvious approach to Lemma 12.20's back-pressure fluid equation — directly differentiate the discrete system equation — obscures the actual argument, which reduces a maximum over the (potentially non-polytope) discrete schedule set $S$ to a maximum over its convex hull $\langle S\rangle$ (justified because a linear functional on a compact polytope is maximized at an extreme point) and then shows that every schedule with a strictly smaller objective value than the maximizer contributes zero derivative to the usage-counting process $\hat T_s$ — the discrete-time analog of mission VIII's Lemmas 9.10-9.11. A second difficulty is connecting the maximum-over-a- finite-set at the fluid-scaled level to Definition 12.3's discrete, arrival-indexed back-pressure policy: this mission's own Lemma 12.20 item states the fluid-relevant consequence of the discrete policy as a documented hypothesis (mirroring mission XI's own scope decision for its structurally identical WWTA fluid equation) rather than re-deriving it from the raw stochastic recursion, which would require reconstructing infrastructure outside this chapter's own numbered results.

Formalization scope

The packet network model, Assumption 12.1, the raw processes, and the fluid-limit apparatus are restated verbatim from mission XII (12-packet-networks-model), since concurrently-drafted chunks sharing a sub-namespace do not import one another's Lean files. IsBPOptimal is phrased via domination over the feasible-at- $z$ schedule set, not sSup/argmax, per this series' junk-value-avoidance convention; the same convention is applied to the maximum over $\langle S\rangle$ in the back-pressure fluid equation. The formalization does not admit a trivializing reading: irreducibility and aperiodicity are the same non-vacuous renewal-theoretic notions used throughout this series (Aperiodic requires a genuine positive self-transition probability, not a vacuously-true condition), and the goal's positive-recurrence conjunct is genuinely conditional on (12.26), stated with the correct strict inequality rather than weakened to $\le$ . Contributions completing the five by sorry proofs are welcome, particularly Lemma 12.17's hop-count induction and Lemma 12.20's extreme-point/zero-derivative argument (mirroring mission VIII's own Lemmas 9.10-9.11, adapted to discrete time).

Selected references

J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org

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Operations Research Probability Stochastic Systems·Captain: Shuze Chen

Processing Networks XII: Packet Networks, Subcriticality and Fluid LimitsTextbook

Motivation

Internet routers, wireless base stations, and data-switch fabrics all face the same recurring decision: in each discrete time slot, which of many possible transfer operations should be executed, given the packets currently queued and the physical constraints (link capacities, interference between simultaneous transmissions) on what can be done at once? J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) devotes Chapter 12 to exactly this question, under the name packet networks. Unlike every prior chapter of the book, which works in continuous time with Poisson-driven Markov chains, this chapter builds its stability theory from scratch for a discrete-time, slotted model — the natural setting for a system that makes one scheduling decision per clock tick. This mission formalizes the chapter's foundational layer: the model itself, its notion of a feasible schedule and control policy, subcriticality, and the discrete-time fluid-limit machinery that Chapters 13 and 14 (back-pressure control and random proportional scheduling, respectively) build their own stability proofs on top of.

Setting

A packet network (Section 12.1) has I packet classes and J activities (service types); activity $j$ transfers a packet from its input class $u(j)$ to its output class $d(j)$ (or removes it from the network if $d(j)=0$ ), giving an $I\times J$ input-output matrix $R$ . A processing plan for class $i$ (Definition 12.2) is a chain of activities from $i$ to exit; Assumption 12.1 requires every class to have one and forbids cycles. In each timeslot the system manager chooses a schedule $s\in\mathbb Z^J_+$ from a feasible set $S$ satisfying the packet-availability constraint $Bs\le z$ (the current buffer contents); $S$ is typically built (Section 12.2) from a link usage matrix $A$ and a set $C$ of feasible link configurations via $S_c := \{s : As\le c\}$ , $S := \bigcup_{c\in C} S_c$ . A Markovian control policy (Definition 12.3, Eq. 12.8) chooses $s(\tau) = f(Z(\tau-1), U(\tau))$ from the current buffer contents and an independent randomization variable; it is admissible if it never overdraws a buffer, and stable if the resulting discrete-time Markov chain $Z$ is irreducible and positive recurrent.

Formalization targets

Goal: Theorem 12.10 — fluid limit stability implies positive recurrence

If the DTMC $Z$ under a Markovian policy $f$ is irreducible and its fluid limit is stable (Definitions 12.14-12.15), then $Z$ is positive recurrent. This is the chapter's own version of Theorem 6.2 (mission III) — the technical fulcrum that lets a deterministic fluid-model stability argument certify the stability of the original discrete, stochastic system — restated for a genuinely different probabilistic model, since Theorem 6.2 was built under continuous-time, Poisson-arrival hypotheses this chapter does not share.

Supporting milestones

Propositions 12.6-12.7 characterize the convex hulls $\langle S_c\rangle$ and $\langle S\rangle$ as explicit polytopes cut out by the link usage matrix — the combinatorial core that Theorem 12.8 (stability implies subcriticality, this chapter's analog of Theorem 5.2) and Proposition 12.9 (a necessary condition for subcriticality) both build on. Lemma 12.11 records that the schedule-usage counting process is Lipschitz; Lemma 12.12 is the functional strong law of large numbers the external arrival process satisfies; and Theorem 12.13 combines them to establish existence of discrete-time fluid limits satisfying the chapter's own fluid equations (12.31)-(12.36) — the chapter's analog of Theorem 6.5.

Significance

The result itself. Theorem 12.10 is what makes the rest of Chapter 12 (and Chapters 13-14) tractable: rather than analyzing an infinite-state discrete-time Markov chain's positive recurrence directly — a notoriously hard problem in general — it suffices to exhibit a deterministic fluid model and show every solution of that fluid model empties in finite time. This is the same strategy Chapter 6 established for the book's continuous-time model, but Chapter 12 cannot simply invoke that earlier theorem: the packet network model uses discrete time slots rather than a continuous clock, and its probabilistic structure (arbitrary i.i.d. arrival increments rather than Poisson arrivals) is different enough that the fluid-limit compactness argument has to be redone, even though — as the book's own text notes — "the proof mimics that of Theorem 6.2."

Formalizing it. A live prior-art check (GET /theorems?q=packet+network, q=discrete-time+Markov+chain, q=slotted+time) finds no relevant hits. A dedicated further check for MarkovMixing, a different mission's own corpus offering a PositiveRecurrent predicate for Markov chains, found a representationally distinct formalization (a row-function transition kernel rather than this series' own PMF-based jump-chain convention); this mission restates positive recurrence and irreducibility locally instead, consistent with every mission in this series since mission I. Everything else — the packet network model, schedules and configurations, the subcritical region, Markovian policies, and the discrete-time fluid-limit apparatus — is formalized from scratch.

Difficulty

The most consequential decision in this mission is representational, not mathematical: Theorem 12.10's fluid-limit-stability hypothesis quantifies over all fluid limit paths, which are themselves scaling limits of a genuinely stochastic discrete-time process — reconstructing that process from Chapter 2/4's own primitive stochastic elements (arrival processes, phase-type service mechanics) would require rebuilding infrastructure this chapter's own numbered results do not supply. Following mission III's own precedent for its structurally identical Theorem 6.5, this mission instead takes the raw, per-initial-state schedule-usage and arrival processes as given data satisfying only the recap properties the chapter's own proofs actually cite (Eq. 12.9-12.10's system equation, monotonicity), and fixes a single sample point together with an explicit SLLN hypothesis rather than a bare "for almost all $\omega$ " quantifier — matching the book's own statement of Theorem 12.13, which itself begins "Fix an $\omega\in\Omega_1$ " rather than quantifying almost everywhere within the theorem itself. A second difficulty is genuinely combinatorial: Proposition 12.6's proof constructs an explicit product-form probability distribution over schedules (one independent randomization per link) whose mean recovers an arbitrary point of the polytope $\{Ax\le c\}$ — a real combinatorial argument, not a formal consequence of the convex-hull operator's definition.

Formalization scope

Classes, activities, links, schedules and configurations are represented via Fin-indexed types throughout; S and C are taken as Finsets (every worked example in the book has finitely many configurations, and the link usage matrix's single-1-per-column structure bounds each schedule component, so this is a genuine, if implicit, standing feature of the model rather than an added restriction). IsScheduleSetAt/IsScheduleSet characterize $S_c$ / $S$ via an ↔ against the underlying capacity constraint, rather than constructing them, matching how set-valued model data is handled elsewhere in this series. The formalization does not admit a trivializing reading: the ambient chain's positive recurrence (PositiveRecurrent) is the same non-vacuous renewal-theoretic notion used throughout this series, PacketFluidLimitStable quantifies over every genuine fluid limit path (not a hand-picked one), and subcriticalRegion is stated with a strict inequality ( $A\hat s < \hat c$ ) exactly as Eq. 12.24 requires, not weakened to $\le$ . Contributions completing the eight by sorry proofs are welcome, particularly Proposition 12.6's explicit randomized- schedule construction and Theorem 12.13's compactness argument (mirroring mission III's own Theorem 6.5 proof, adapted to discrete time).

Selected references

J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org

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Operations Research Probability Stochastic Systems·Captain: Shuze Chen

Processing Networks XI: Maximal Stability of Workload-Weighted Task AllocationTextbook

Motivation

Large-scale data-intensive computing splits a job into many small "map tasks," each of which must be assigned to one of thousands of general-purpose servers in a data center — the MapReduce paradigm introduced by Dean and Ghemawat (2008). Because a task's input data is stored on only a few of those servers, routing it to a server that already holds its data ("data locality") avoids costly network transfer, so a task-allocation policy that ignores locality can leave a server farm badly underutilized even when raw compute capacity is ample. J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) devotes Chapter 11 to a rigorous stability analysis of Xie, Yekkehkhany, and Lu's (2016) workload-weighted task allocation (WWTA) policy, which routes each arriving task to whichever eligible server currently carries the least (locality-weighted) backlog. This mission formalizes the chapter's central result: WWTA is not merely a reasonable heuristic but is maximally stable — it keeps the system stable whenever any routing policy could.

Setting

A task allocation model (Sections 11.2-11.3) has L task categories and K servers; a class is a pair $(\ell,k)$ , a category routed to a specific server, with mean service time $m_{\ell k} > 0$ (Eq. 11.3) reflecting both data locality (local, rack-local, or remote access) and the server's relative processing speed. Tasks in category $\ell$ arrive according to a Markovian arrival process (MArP) with long-run average rate $\nu_\ell$ — a substantially more general model than the independent Poisson streams used elsewhere in the book, needed to capture batches of map tasks generated from a single job. Routing is by immediate commitment: each task is assigned to a server the instant it arrives, based only on its category and the current backlog, with no possibility of later reassignment. The workload of server $k$ at time $t$ (Eq. 11.7) is $W_k(t) := \sum_\ell m_{\ell k} Z_{\ell k}(t)$ , the locality-weighted total backlog assigned to it. Workload-weighted task allocation (WWTA), Definition 11.3, routes an arriving category- $\ell$ task to any server achieving $\operatorname{argmin}_{k} m_{\ell k} W_k(t-)$ (Eq. 11.8, ties broken arbitrarily), where $W(t-)$ is the workload vector's left limit at the arrival instant.

Formalization targets

Goal: Theorem 11.6 — WWTA fluid model stability under the load condition

If there exists $\lambda = (\lambda_{\ell k}) \ge 0$ with $\sum_k \lambda_{\ell k} = \nu_\ell$ for each category (Eq. 11.4) and $\sum_\ell m_{\ell k}\lambda_{\ell k} < 1$ for each server (Eq. 11.5), then the WWTA fluid model — the deterministic fluid equations (11.11)-(11.16) that arise as scaling limits of the stochastic model under WWTA control — is stable: every solution reaches the zero state in finite time proportional to its initial size.

Supporting milestones

Lemma 11.2 shows this load condition is equivalent to subcriticality of the task allocation model, in the sense of Section 5.2's general definition, reducing an existence statement over the model's full $G,R,A,b$ apparatus to a simple linear-feasibility check. Theorem 11.4 establishes that fluid limits of the stochastic model exist and satisfy the general fluid equations (11.11)-(11.15) under any immediate-commitment routing policy, and the WWTA-specific equation (11.16) under WWTA — the chapter's version of the fluid-limit machinery Chapter 6 builds for the book's standard model, adapted to Markovian (not Poisson) arrivals and immediate-commitment routing. Theorem 11.5 is the corresponding replacement for Theorem 6.2 (mission III): fluid limit stability implies stability of the original stochastic model. Corollary 11.7 is the mission's other headline consequence: WWTA is maximally stable, stabilizing the model whenever any simply structured routing policy could.

Significance

The result itself. A task-allocation heuristic motivated purely by an intuitive "balance-the-weighted-backlog" idea turns out to have the strongest stability guarantee a policy can have: it never needs re-tuning as arrival rates change, and no alternative policy can stabilize a regime WWTA cannot. Xie, Yekkehkhany, and Lu (2016) proved this fact (calling it "throughput optimality") by direct analysis of the discrete stochastic model using a quadratic Lyapunov function; Dai and Harrison's contribution is to prove the same conclusion via the fluid-model route, and — because the task allocation model violates two of the standing assumptions under which Chapter 6's general theory was built (MArP rather than Poisson arrivals, immediate-commitment routing rather than the book's default relaxed control) — to show that the general fluid-stability machinery survives those changes with only minor modification.

Formalizing it. A live prior-art check (GET /theorems?q=task+allocation, q=server%20farm, q=data%20locality) finds no relevant hits on the platform (workload returns one unrelated text-corpus-statistics result). This mission formalizes the task allocation model, WWTA, the fluid equations, and the stochastic-to-fluid bridge from scratch, reusing Mathlib's general real-analysis, measure-theory, and PMF-based Markov-chain substrate, and restating (per this series' convention for concurrently-drafted chunks) the minimal apparatus needed from missions I and III.

Difficulty

The obvious approach to Definition 11.3 — pick a canonical minimizer of $m_{\ell\cdot}W_\cdot(t-)$ , e.g. via Finset.min' — silently narrows WWTA to a single tie-breaking rule, when (11.8) explicitly allows any minimizer; this mission instead states WWTA as domination over every server, accepting every admissible tie-break. A second difficulty is genuinely structural: Theorem 11.4's "for almost all $\omega$ " existence claim is a measure-theoretic statement about a probability space that mission III's own analogous Theorem 6.5 sidesteps by fixing a sample point $\omega$ together with the specific SLLN limit that argument needs — this mission follows the same precedent rather than introducing a bare "almost everywhere" quantifier, which does not by itself supply the type information Lean's typeclass resolution needs to identify an ambient measure. A third is connecting Definition 11.3's discrete, arrival-indexed routing rule to the raw (pre-limit) process family that Theorem 11.4's existence argument operates on: the book's own proof derives the needed consequence (Eq. 11.22, "server $k$ receives no more category- $\ell$ tasks in a neighbourhood of $t$ ") via an explicit $\varepsilon$ - $\delta$ argument from the discrete policy, which this mission takes as a documented hypothesis rather than re-deriving, since building the arrival-sequence-to-counting-process reconstruction is genuinely Chapter 2/4 machinery outside this chapter's own numbered results.

Formalization scope

Classes are represented as pairs (ℓ,k) : Fin L × Fin K rather than flattened to a single index, avoiding an arbitrary choice of encoding. TaskAllocationProcessFamily, FluidLimitPath, and TaskAllocationFluidLimitStable restate mission III's SPNProcessFamily/FluidLimitPath/ FluidLimitStable apparatus, adapted to this model's (E,D,Z) triple and doubly-indexed classes; PositiveRecurrent restates mission I's renewal-theoretic positive-recurrence notion. The formalization does not admit a trivializing reading: IsWWTA accepts every admissible tie-break rather than a single canonical one, SatisfiesWWTAFluidEquation requires Nonempty (Fin K) so its min_{k} is a genuine minimum rather than a junk value at zero servers, and the goal's conclusion WWTAFluidStable is the same non-vacuous fluid-model-stability predicate (Definition 6.3, specialized) used throughout this series. Contributions completing the five by sorry proofs are welcome, particularly Theorem 11.4's fluid-limit compactness argument (mirroring mission III's own Theorem 6.5 proof) and Theorem 11.6's explicit quadratic-Lyapunov-function argument.

Selected references

J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
J. Dean and S. Ghemawat, "MapReduce: simplified data processing on large clusters," Communications of the ACM 51 (2008), 107–113.
Q. Xie, A. Yekkehkhany, and Y. Lu, "Scheduling with multi-level data locality: throughput and heavy-traffic optimality," IEEE INFOCOM 2016.

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Dynamic Programming Operations Research Probability·Captain: mikedeng1

Optimal Policy for a Multi-Product, Dynamic, Nonstationary Inventory Problem: The Base Stock Ordering Policy Is OptimalResearch Paper

Motivation

An inventory manager who stocks many products, faces random demand and pays ordering, holding and shortage costs must decide each period how much of each product to order. In general the optimal decision depends on the whole state and is found by solving a dynamic programme, which becomes impractical as the number of products grows. Myopic (or base stock) policies avoid this: in each period they aim at a target level computed from that period's data alone. Knowing when such a policy is optimal over an infinite horizon tells a practitioner when the multi-period problem decouples into a sequence of one-period problems.

Arthur F. Veinott, Jr. gave such conditions in Optimal Policy for a Multi-Product, Dynamic, Nonstationary Inventory Problem (Management Science 12(3):206–222, 1965). Earlier results of this type were for a single product (the paper cites Karlin, Management Science 6(3), 1960, Bellman–Glicksberg–Gross, Management Science 2(1), 1955, Iglehart–Karlin 1962, and the Arrow–Karlin–Scarf volume of 1958); Veinott's model allows several products, several demand classes, costs and demand distributions that change over time, general ordering constraints and general stock dynamics (backlogging, lost sales, and mixtures), and it does not use the functional equation of dynamic programming. Instead, the proofs analyse the inventory process directly.

Setting

There are $n$ products and $m$ demand classes. Vectors are compared componentwise: $u \le v$ means $u_j \le v_j$ for all $j$ . In period $i = 1, 2, \dots$ the manager observes the inventory vector $x_i \in X_i \subseteq \mathbb{R}^n$ (negative coordinates are backlogs) and orders up to a vector $y_i \in Y_i \subseteq \mathbb{R}^n$ , subject to $y_i \ge q_i(x_i)$ , where $q_i$ is a vector of extended-real functions (for instance $q_i(x) = x$ forbids disposal). A random demand vector $D_i$ with law $\Phi_i$ and values in a Borel set $\mathfrak{D}_i \subseteq \mathbb{R}^m$ then occurs, and the next inventory vector is $x_{i+1} = s_i(y_i, D_i) \in X_{i+1}$ . The demands $D_1, D_2, \dots$ are independent. Ordering $y_i - x_i$ costs $c_i \cdot (y_i - x_i)$ , the holding and shortage cost is $g_i(y_i, D_i)$ , and $\alpha_i \ge 0$ is the discount factor of period $i$ .

Regrouping the ordering costs gives the one-period cost

W_i(y,t) = c_i\, y + g_i(y,t) - \alpha_i\, c_{i+1}\, s_i(y,t), \qquad G_i(y) = \int_{\mathfrak{D}_i} W_i(y,t)\, d\Phi_i(t),

with discount weights $\beta_1 = 1$ and $\beta_i = \alpha_1 \cdots \alpha_{i-1}$ . The integrals are assumed finite, and $G_i \ge \gamma_i$ with $\sum_i |\beta_i \gamma_i| < \infty$ . An ordering policy $\bar Y$ chooses $y_i$ as a Borel function of the past; it is feasible if $y_i \in Y_i$ and $y_i \ge q_i(x_i)$ for every possible history. Its cost is

f(x_1 \mid \bar Y) = \sum_{i=1}^\infty \beta_i\, E\, G_i(y_i) \in (-\infty, +\infty],

and a feasible policy of least cost is optimal.

Let $\bar y_i$ minimize $G_i$ over $Y_i$ . When $\bar y_i$ is not attainable from $x$ , the minimal feasible level $w_i(x)$ is the least element of $Y_i \cap \{y : y \ge q_i(x),\ y \ge \bar y_i\}$ . The base stock ordering policy orders up to $\bar y_i$ if $q_i(x_i) \le \bar y_i$ and up to $w_i(x_i)$ otherwise.

Formalization targets

The hypotheses are: (3a) $\bar y_i \in Y_i$ minimizes $G_i$ over $Y_i$ ; (3b) $q_{i+1}(s_i(\bar y_i, t)) \le \bar y_{i+1}$ for $t \in \mathfrak{D}_i$ ; (3c) $Y_i$ is closed and linearly ordered by $\le$ ; (3d) $G_i$ and $s_i(\cdot, t)$ are nondecreasing on $\{y \in Y_i : y \ge \bar y_i\}$ , and $q_i$ is nondecreasing where $q_i(x) \not\le \bar y_i$ .

Goal: Theorem 3.2

Under (3a)–(3d), the base stock ordering policy

\bar Y_i^*(H_i^*) = \begin{cases} \bar y_i, & q_i(x_i^*) \le \bar y_i, \\ w_i(x_i^*), & q_i(x_i^*) \not\le \bar y_i \end{cases}

is feasible and optimal: $f(x_1 \mid \bar Y^*) \le f(x_1 \mid \bar Y)$ for every feasible $\bar Y$ .

Milestones

A nonempty, closed, linearly ordered, bounded-below subset of $\mathbb{R}^n$ has a least element (p. 212); hence $w_i(x)$ exists and equals $\bar y_i$ when $q_i(x) \le \bar y_i$ .
$w_i$ is nondecreasing where $q_i(x) \not\le \bar y_i$ (p. 214).
Once $q_k(x_k^*) \le \bar y_k$ , the base stock policy orders up to $\bar y_i$ for all $i \ge k$ .
Theorem 3.1: under (3a) and (3b), if $q_1(x_1) \le \bar y_1$ , ordering up to $\bar y_i$ in every period is optimal, with cost $\sum_i \beta_i G_i(\bar y_i)$ .
The coupling (3.1): before the first period $T$ with $q_T(x_T^*) \le \bar y_T$ ,

\bar y_i < y_i^* = w_i(x_i^*) \le w_i(x_i) \le y_i.

Pathwise dominance: $G_i(y_i^*) \le G_i(y_i)$ for every $i$ and every possible demand path.
The reduction behind (2.3): $E\, W_i(y_i, D_i) = E\, G_i(y_i)$ , by independence of $y_i$ and $D_i$ .

Significance

The theorem shows that under (3a)–(3d) the infinite-horizon, nonstationary, multi-product problem is solved by one-period optimization: compute $\bar y_i$ from $G_i$ alone, and when it is unreachable order the least feasible amount. No value function is computed. It covers backlogging and lost sales, products stocked in fixed proportions, and time-varying cost and demand data, and Theorem 3.1 alone settles the frequent case in which the initial stock is small.

The result is proved in the paper. To our knowledge it has not been machine-checked: this mission produces a Lean model of a general stochastic, infinite-horizon, multi-product inventory problem (feasible policies, the expected discounted cost with $+\infty$ allowed), and a checked proof that a myopic policy is optimal in it. The model and its cost are reusable for other base stock and myopic optimality results.

Difficulty

The obvious argument would compare the base stock policy with an arbitrary policy period by period using (3a) alone. That fails once $\bar y_i$ is unreachable. The base stock policy then holds less stock than $\bar y_i$ would require, and it has to be shown that no other policy can reach a lower-cost level later. The proof couples the two trajectories along every demand path, using the monotonicity in (3d) and the linear order of $Y_i$ from (3c), until the base stock level becomes attainable, after which (3b) keeps it attainable. The formal difficulties are:

the existence and monotonicity of the least element $w_i(x)$ in a closed chain of $\mathbb{R}^n$ ;
the measurability of the base stock policy, which is part of its feasibility;
the passage from pathwise dominance to the expected discounted cost when the cost may be $+\infty$ .

Formalization scope

Periods are 0-based in Lean (Lean period $k$ is the paper's period $k+1$ ). Vectors are Fin n → ℝ with the product order; $q_i$ takes values in Fin n → EReal. Policies are functions of the past demands. The paper shows on p. 219 that this loses no generality once $x_1$ is fixed. The cost is excess, a $[0,\infty]$ -valued series of lower Lebesgue integrals of $\beta_i(G_i(y_i) - \gamma_i)$ , plus the finite real $\sum_i \beta_i\gamma_i$ , taken in EReal. A divergent series therefore gives $+\infty$ and never the junk value $0$ . "Minimal element" is IsLeast. The $G_i$ in the statements is the integral of $W_i$ built from the data, and the policy in the goal is constructed from $\bar y$ , $q$ , $w$ and $s$ . Neither is an arbitrary object satisfying the conclusion. The goal concludes optimality, which includes feasibility: a pathwise or conditional statement would not be the theorem.

Hypotheses added relative to the page, each used by the paper without being stated:

Order feasibility: from every $x \in X_i$ some $y \in Y_i$ with $y \ge q_i(x)$ exists (p. 212 asserts the set defining $w_i(x)$ has a minimal element, which needs it nonempty). The least-element lemma likewise assumes $A$ nonempty.
(3d)'s $q$ -clause in one-sided form: if $x \le x'$ in $X_i$ and $q_i(x) \not\le \bar y_i$ , then $q_i(x) \le q_i(x')$ . This is the form the proof on p. 214 uses; monotonicity on the region $\{q_i(x) \not\le \bar y_i\}$ alone admits a counterexample to Theorem 3.2.
Integrability of $W_i(y_i, D_i)$ in the milestone $E W_i = E G_i$ .
Borel measurability of $q_i$ on all of $\mathbb{R}^n$ rather than on $X_i$ (a harmless strengthening).

A complete development needs: least elements of closed chains in $\mathbb{R}^n$ ; measurability of the least-element selection $x \mapsto w_i(x)$ ; a Fubini/independence argument for $E\,W_i(y_i,D_i)=E\,G_i(y_i)$ ; and monotone summation of lower integrals. The least-element lemma and the cost encoding are reusable beyond this mission. Proofs of any milestone are welcome, as are alternative arguments.

Selected references

A. F. Veinott, Jr., Optimal Policy for a Multi-Product, Dynamic, Nonstationary Inventory Problem, Management Science 12(3):206–222, 1965. https://doi.org/10.1287/mnsc.12.3.206
R. Bellman, I. Glicksberg, O. Gross, On the Optimal Inventory Equation, Management Science 2(1):83–104, 1955. https://doi.org/10.1287/mnsc.2.1.83
S. Karlin, Dynamic Inventory Policy with Varying Stochastic Demands, Management Science 6(3):231–258, 1960. https://doi.org/10.1287/mnsc.6.3.231

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Operations Research Probability Stochastic Systems·Captain: Shuze Chen

Processing Networks X: Maximal Stability of Proportionally Fair ControlTextbook

Motivation

Mission IX (09-proportional-fairness-core) formalized proportional fairness (PF) as a control policy and proved the technical core of its stability theory: under a load condition, the PF fluid model is stable (Theorem 10.5), via an entropy Lyapunov function that is genuinely not Lipschitz continuous — a departure from every other stability argument in J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org). A stability theorem for one fixed arrival-rate vector is, on its own, a narrower claim than practitioners actually want: real systems see load that changes over time, and a control policy worth adopting should not need re-tuning every time the mix of traffic shifts. This mission completes Theorem 10.5's proof and turns it into exactly that stronger guarantee — proportional fairness is maximally stable: stable throughout the entire region where any policy could be stable, without knowing the arrival rates in advance — and specializes the result to two concrete network families, bandwidth-sharing networks and queueing networks under head-of-line proportional processor sharing (HLPPS), that were already familiar from earlier in the book under different control policies.

Setting

Fix a unitary network operating under PF control with mean service times $m > 0$ , routing matrix $P$ , a partition of job classes into demand groups $\{\mathcal I(\ell), \ell \in \mathcal L\}$ , and a reduced allocation set $\tilde{\mathcal A} \subset \mathbb R^{\mathcal L}_+$ (all restated from mission IX, Definition 10.3). The entropy Lyapunov function $\varphi(t) := \sum_i Z_i(t)\log(\dot D_i(t)/\alpha_i)$ (Eq. 10.38, mission IX) admits an alternative decomposition $\varphi = \sum_\ell \varphi_\ell$ in terms of the within-group entropy term

f(t) := \sum_{\ell\in\mathcal L}\sum_{i\in\mathcal I(\ell)} Z_i(t)\log\!\left(\frac{Z_i(t)}{Y_\ell(t)}\right), \qquad Y(t) := GZ(t) \quad \text{(Eq. 10.50)},

with the convention that the term for class $i$ is $0$ when $Z_i(t)=0$ . Here $D^+$ / $D^-$ denote the upper-right/upper-left Dini derivatives (Appendix A.4, Eqs. A.9-A.10): at a point where the ordinary derivative may not exist, these one-sided $\limsup$ s still let a Lyapunov-drift argument go through. A control policy is maximally stable (Section 5.7) for a network if its implementation does not depend on the arrival-rate vector $\lambda$ and it is stable for every $\lambda$ in the network's stability region $\Lambda^*$ — the largest region any policy could possibly stabilize.

Formalization targets

Goal: Corollary 10.16 — maximal stability of PF control for a unitary network

\text{IsMaximallyStable}\Big(\lambda \mapsto \text{PFFluidStable}(\lambda, m, P, \mathrm{grp}, \tilde{\mathcal A})\Big)

for the single PF policy value (its implementation never depends on $\lambda$ ). This is the applied payoff of Theorem 10.5 (mission IX): combined with Theorem 6.2 (mission III, fluid stability implies SPN stability) and Corollary 5.6 (mission II, a $\lambda$ -independent policy stable throughout the subcritical region is automatically maximally stable), it upgrades a single- $\lambda$ stability statement to the strongest form the book's own framework can express.

Supporting milestones

Lemmas 10.11-10.15 supply the remaining technical content of Theorem 10.5's proof that mission IX's own milestones left open: Lemma 10.11 is the uniform negative-drift bound $\sum_i \dot Z_i(t)\log(\dot D_i(t)/\alpha_i) \le -\varepsilon$ at every regular point with $Z(t) \ne 0$ ; Lemmas 10.12-10.14 establish continuity and two successively sharper Dini-derivative bounds on the within-group entropy term $f$ ; Lemma 10.15 shows that positivity of the departure rate $\dot D_i(t)$ propagates from occupied classes to every class. Corollaries 10.17 and 10.18 specialize the goal to bandwidth-sharing networks and to HLPPS-controlled queueing networks, respectively.

Significance

The result itself. A stability theorem tied to one fixed $\lambda$ is of limited practical use: it would need to be re-verified every time the arrival-rate vector changes, which real traffic does constantly. Maximal stability removes that dependency entirely — a single policy, implemented without any knowledge of $\lambda$ , is guaranteed stable throughout the full region any control could stabilize. Corollaries 10.17 and 10.18 make this concrete for two network families with independent histories in the literature: bandwidth-sharing networks (the original motivation for proportional fairness, Kelly 1997) and queueing networks under HLPPS, connecting PF's static, utility-theoretic motivation to a scheduling rule that predates it.

Formalizing it. A live prior-art check (GET /theorems?q=proportional%20fairness, q=maximal%20stability, q=bandwidth%20sharing) finds no relevant hits on the platform. This mission formalizes the remaining entropy-Lyapunov lemmas, the within-group entropy term, the BWS and HLPPS network models (restated locally, since no other drafted chunk covers Sections 4.5-4.6), and the maximal-stability predicate, reusing only Mathlib's general real-analysis substrate (Dini derivatives via Filter.limsup) and definitions restated from missions II, III, V, and IX under this series' restate-not-import convention for concurrently-drafted chunks.

Difficulty

The obvious approach to Corollary 10.16 — restate "maximally stable" with an explicit $\lambda$ -dependent policy family and add "the policy doesn't actually depend on $\lambda$ " as a side hypothesis — obscures the point: a policy that is definitionally independent of $\lambda$ is a stronger and cleaner claim than one that happens to satisfy an extra equation. This formalization instead instantiates the abstract policy type at Unit, so $\lambda$ -independence holds by construction rather than as a hypothesis to verify, matching the book's own reading of Section 5.7's definition. A second difficulty is Lemma 10.13's Dini-derivative inequality (Eq. 10.56): the plain-text extraction of this display equation loses bracket and subscript structure that changes its meaning, so the exact grouping was confirmed against the PDF page directly and cross-checked against the book's own re-derivation of the same bracketed expression inside Lemma 10.14's proof. A third is Corollary 10.17's proof, which genuinely depends on three facts outside this chunk's own chapter portion (Proposition 4.4 and the Section 4.4 model translation, Proposition 5.1, Theorem 5.2); rather than silently assuming them or re-deriving their proofs from scratch, they are stated as explicit hypotheses of the milestone itself, so the item's actual content — deriving the two-sided conclusion — is exactly what remains to be proved.

Formalization scope

RestatedCore, RestatedFluidModel, and the maximal-stability predicate MaximalStability are restated verbatim (or, for MaximalStability, in shape) from missions IX, III/V, and II respectively, since concurrently-drafted chunks in this series do not import one another's Lean files even when they share a sub-namespace. New to this chunk: diniUpperLeft (Eq. A.10, needed alongside mission IX's diniUpperRight for Lemma 10.13's two-sided bound), the within-group entropy term withinGroupEntropy (Eq. 10.50, deliberately not named f, since the book's own f already denotes the unrelated PF optimization objective of Eq. 10.2 within the same chapter), and BWSNetworkData/QueueingNetworkDataHL/HLPPSFluidStable (Sections 4.5-4.6, restated locally since no drafted chunk's BRIEF.md covers them). The formalization does not admit a trivializing reading: IsMaximallyStable is instantiated with the genuine, non-vacuous predicate PFFluidStable/HLPPSFluidStable — the same predicate whose stability Theorem 10.5 (mission IX) already establishes on the load-condition region — never with a policy type or stability predicate engineered to make the maximal-stability claim vacuous. Contributions completing the eight by sorry proofs are welcome, particularly Lemma 10.13's Dini-derivative estimate (Section B.4's preliminary results) and Lemma 10.15's connectivity argument (Appendix B.9/B.17).

Selected references

J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
F. P. Kelly, "Charging and rate control for elastic traffic," European Transactions on Telecommunications 8 (1997), 33–37.
F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, "Rate control for communication networks: shadow prices, proportional fairness and stability," Journal of the Operational Research Society 49 (1998), 237–252.

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Dynamic Programming Operations Research Optimization+1·Captain: mikedeng1

An Overview of Pricing Models for Revenue Management: Performance Guarantee of the Deterministic Price Heuristic in Periodic-Review PricingResearch Paper

Motivation

Dynamic pricing under limited inventory is a core problem of revenue management: a seller holds $C_0$ units of a perishable product (airline seats, hotel rooms, seasonal goods) and must choose prices over a finite selling horizon while demand is random and responds to price. The exactly optimal policy solves a stochastic dynamic program whose state is the remaining inventory, and it changes the price after every sale. In practice sellers often prefer a simpler rule, fixed in advance: solve the deterministic version of the problem, in which random demand is replaced by its mean, and charge the resulting prices whatever happens.

Gallego and van Ryzin (Management Science 1994) showed that in continuous time with Poisson demand this fixed-price heuristic is asymptotically optimal, and bounded its relative loss by the coefficient of variation of demand. Bitran and Caldentey's survey (MSOM 2003, §3.2.1) extends this bound to a discrete-time, periodic-review model with general demand distributions, as Proposition 8, proved in the paper's Appendix. The proof combines three ingredients: a Lagrangian duality argument showing that the deterministic problem is an upper bound on the optimal expected revenue, a sample-path comparison of lost sales, and a distribution-free moment bound of Gallego (1992) on $E[(X-C)^+]$ .

Setting

A single product is sold over $N \ge 1$ periods $n = 1, \dots, N$ starting from inventory $C_0$ . In period $n$ the seller charges a price $p \ge 0$ , and the demand $D_n(p)$ is a nonnegative random variable with finite mean $E[D_n(p)]$ , whose law may depend on $n$ and $p$ arbitrarily. With inventory $C$ the seller sells $\min\{D, C\}$ units; unmet demand is lost.

The optimal expected revenue $V_1(C_0)$ is defined by the Bellman recursion $V_{N+1} \equiv 0$ ,

V_n(C) = \sup_{p \ge 0} E\big[p\min\{D_n(p), C\} + V_{n+1}\big(C - \min\{D_n(p), C\}\big)\big].

The deterministic problem (32)–(33) is

V_1^{\det}(C_0) = \sup_{p \in [0,\infty)^N} \sum_{n=1}^N p_n E[D_n(p_n)] \quad \text{subject to} \quad \sum_{n=1}^N E[D_n(p_n)] \le C_0,

and $p^{\det}$ denotes an optimal solution. The deterministic-price heuristic charges $p^{\det}_n$ in period $n$ regardless of sales. Its period demands $D_n(p_n^{\det})$ are independent, $\mathscr{D}_n^{\det} = \sum_{i=1}^n D_i(p_i^{\det})$ is the cumulative demand, and its expected revenue is

V_1(p^{\det}, C_0) = \sum_{n=1}^N p_n^{\det} E\Big[D_n(p_n^{\det}) - \big(D_n(p_n^{\det}) - (C_0 - \mathscr{D}_{n-1}^{\det})^+\big)^+\Big].

With $\sigma_n^2 = \operatorname{Var}(\mathscr{D}_n^{\det})$ , eq. (34) defines

\eta_n^{\det}(C_0) = \frac{\sqrt{\sigma_n^2 + (C_0 - E[\mathscr{D}_n^{\det}])^2} - (C_0 - E[\mathscr{D}_n^{\det}])}{2},

and $\nu(C_0) = \sigma_N / E[\mathscr{D}_N^{\det}]$ is the coefficient of variation of the total demand.

Formalization targets

Goal: Proposition 8, eq. (35)

Assume that $p \mapsto p\,E[D_n(p)]$ is concave and $p \mapsto E[D_n(p)]$ is convex on $[0,\infty)$ for each $n$ , that some price $p^\infty \ge 0$ has $\sum_n E[D_n(p^\infty)] < C_0$ , and that $p^{\det}$ is optimal for (32)–(33). Then

1 \ge \frac{V_1(p^{\det}, C_0)}{V_1(C_0)} \ge \frac{1}{V_1^{\det}(C_0)} \sum_{n=1}^N p_n^{\det} E[D_n(p_n^{\det})]\left(1 - \frac{\eta_n^{\det}(C_0)}{E[D_n(p_n^{\det})]}\right) \ge 1 - \max_n \frac{\eta_n^{\det}(C_0)}{E[D_n(p_n^{\det})]}.

The constants are the paper's, and the goal is the full chain.

Milestones

Gallego's bound (Appendix, (*)): for square-integrable $X$ , $E[(X - C)^+] \le \frac12\big(\sqrt{\operatorname{Var}X + (C - EX)^2} - (C - EX)\big)$ , which is at most $\frac12\sqrt{\operatorname{Var} X}$ when $EX \le C$ .
Eq. (24): in one period, $E[p\min\{D(p), C\}] \le p\min\{E[D(p)], C\}$ , so $V(C) \le V^{\det}(C)$ .
Proposition 6, eq. (26): in one period, $V(C, p^{\det})/V(C) \ge 1 - \nu^{\det}/2$ .
$V_1^{\det}$ is concave in the capacity (first sentence of the Appendix proof).
Proposition 8, first assertion: $V_1(C_0) \le V_1^{\det}(C_0)$ .
The Appendix display bounding $V_1(p^{\det}, C_0)$ below by $\sum_n p_n^{\det}E[D_n](1 - E[(\mathscr{D}_n^{\det} - C_0)^+]/E[D_n])$ .
Eq. (36), after the goal: if $E[D_n(p)] = T_n\lambda(p)$ , one constant price solves (32)–(33) and $1 \ge V_1(p^{\det},C_0)/V_1(C_0) \ge 1 - \nu(C_0)/2$ .

Significance

The result gives a guarantee for a pricing policy that needs no inventory tracking: its relative loss is controlled by the first two moments of cumulative demand, with no distributional assumption beyond finite variance. When demand grows while its coefficient of variation shrinks, as for sums of independent period demands, the guarantee tends to one, which is the discrete-time form of asymptotic optimality of fixed prices. The upper bound $V_1 \le V_1^{\det}$ is used throughout revenue management as the benchmark for heuristics (fluid or deterministic LP bounds).

The paper's proof is complete in the Appendix, and the mathematics is not in question beyond minor typos. None of it is machine-checked. A formal development adds a checked Bellman model of periodic-review pricing with general demand laws, a checked fluid upper bound for it, and a checked form of Gallego's moment bound, all reusable for other revenue-management statements. A related platform theorem, RevenueManagement.deterministic_upper_bound (mission The Theory and Practice of Revenue Management III), states the deterministic bound for a Bernoulli-arrival model with at most one sale per period; the model here is different (arbitrary demand laws, continuous inventory), and neither statement implies the other.

Difficulty

The upper bound $V_1(C_0) \le V_1^{\det}(C_0)$ is the hard part. The natural induction replaces $V_2$ by $V_2^{\det}$ and applies Jensen's inequality, but $V_2^{\det}(C)$ is $-\infty$ at capacities where the remaining deterministic problem is infeasible, while $V_2(C)$ stays nonnegative. The induction therefore fails as stated when mean demand never vanishes. The paper's argument also passes through the Lagrangian dual of (32)–(33), and strong duality for that program on $[0,\infty)^N$ under the Slater point $p^\infty$ is not available in Mathlib in this form.

The lower bound is less deep but technical: it needs the product law of the period demands, linearity of expectation for truncated sums, and variances of partial sums. Gallego's inequality is elementary once the right quadratic bound on $(x - C)^+$ is found, but it is not in Mathlib.

Formalization scope

Periods are Fin N, 0-based. Demand laws are μ n p : Measure ℝ, probability measures carried by $[0,\infty)$ with finite mean, required at every price (laws at negative prices never enter a statement).
The Bellman value is computed in $[0,\infty]$ with the lower Lebesgue integral and ⨆ over $p \ge 0$ , so no supremum takes a junk value; the paper's $V_n$ is valueToGo M (N - n + 1). Ratios use its real value.
$V_1^{\det}$ is an EReal supremum over the feasible set ( $-\infty$ if infeasible). In the goal $p^{\det}$ is an optimal solution, so $V_1^{\det}(C_0)$ is its objective value.
The heuristic's demands are independent: the joint law is the product measure.
Implicit hypotheses made explicit: "concave objective and convex feasible region" is read as concavity of $p\,E[D_n(p)]$ and convexity of $E[D_n(p)]$ on $[0,\infty)$ , the latter making (33) convex for every capacity; existence of the optimal deterministic solution; finite variance and positive mean of each $D_n(p_n^{\det})$ ; and $V_1^{\det}(C_0) > 0$ , without which the ratios are $0/0$ .
The typo $\mathscr{D}_n^{\det} := \sum_{i=1}^n D_n(p^{\det})$ on p. 221 is read as $\sum_{i=1}^n D_i(p_i^{\det})$ .

Trivializing formalizations ruled out: $V_1$ is defined by the Bellman recursion, not as a supremum over an unspecified policy class or as a variable constrained by hypotheses; $\eta_n^{\det}$ is the expression (34), not a hypothesis-supplied bound on expected overflow; the goal does not assume strong duality or a Lagrange multiplier, since that is the proof's key step; independence is built into the joint law, not assumed as an inequality; variances are taken only under square integrability, since Mathlib's variance is $0$ for infinite variance.

Needed infrastructure: measurability of the Bellman integrand (monotonicity of $V_n$ in inventory), Jensen's inequality for $\min\{\cdot, C\}$ (ConcaveOn.le_map_integral), Lagrangian strong duality for a separable concave program with one convex constraint, marginals and variances of sums under Measure.pi, and Gallego's bound. The duality and moment results are reusable beyond this mission. Proofs of any milestone, and alternative proofs of the upper bound, are welcome.

Selected references

G. R. Bitran, R. Caldentey, An Overview of Pricing Models for Revenue Management, Manufacturing & Service Operations Management 5(3):203–230, 2003. https://doi.org/10.1287/msom.5.3.203.16061
G. Gallego, G. van Ryzin, Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons, Management Science 40(8):999–1020, 1994. https://doi.org/10.1287/mnsc.40.8.999
G. Gallego, A Minmax Distribution Free Procedure for the (Q, R) Inventory Model, Operations Research Letters 11(1):55–60, 1992 (as cited in Bitran and Caldentey 2003).
M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd ed., Wiley, 1993.

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Linear Optimization Operations Research Optimization·Captain: mikedeng1

Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs: Binding Cuts Force Degenerate Descendant Problems in Nested DecompositionResearch Paper

Motivation

Multistage stochastic linear programs model decisions taken in periods $t = 1, \dots, T$ while a random vector $\xi_t$ is revealed period by period: production and capacity planning, energy scheduling, asset–liability management. With finitely many scenarios the problem is one very large linear program with a tree structure, and decomposition methods solve it by passing information up and down the scenario tree instead of solving it whole. John R. Birge's Nested Decomposition for Stochastic Programming Algorithm (NDSPA), published in Operations Research in 1985 (DOI), is the multistage extension of the L-shaped method of Van Slyke and Wets (SIAM J. Appl. Math. 1969) and the ancestor of the nested Benders and stochastic dual dynamic programming codes in use today.

The paper reports a practical defect of the method: degeneracy leads to many repeated solutions of the same scenario problem (Remark 4, p. 996–997), an effect Abrahamson (1980) had observed for deterministic nested decomposition. Propositions 3 and 4 (p. 997) identify where the degeneracy comes from: cuts that bind at the ancestor force the descendant problems that produced them into degenerate bases. This mission formalizes those two propositions.

Setting

Fix a node of a finite scenario tree: scenario $j$ at period $t$ , with decision $x \in \mathbb{R}^{n}$ . Its ancestor's decision $x^{a}$ enters only through the right-hand side of its equality rows. After $r$ feasibility cuts and $s$ optimality cuts have been added, the node solves the scenario problem (7):

\min\ c^\top x + \theta \quad\text{s.t.}\quad A x = \xi + B x^{a},\quad D_l^\top x \ge d_l\ (l \le r),\quad E_l^\top x + \theta \ge e_l\ (l \le s),\quad x \ge 0,

over $x$ and a free scalar $\theta$ , which stands in for the expected future cost. During NDSPA's forward pass the node also carries the restriction $\theta = 0$ ; a last-period node has no future cost and keeps that restriction.

A descendant problem (period $t+1$ , scenario $j' \in J'$ ) has the same shape, with right-hand side $\xi_{j'} + B_{j'} x$ depending on the node's decision $x$ . Two kinds of cut flow from descendants to the node:

a feasibility cut arises when a descendant is infeasible at some $x^0$ : a Farkas certificate $\lambda$ of that infeasibility, with parts $\pi, \rho, \sigma$ on the descendant's rows (7.2), (7.3), (7.4), yields the cut $D^\top x \ge d$ with $D = -\pi^\top B_{j'}$ and $d = \pi^\top \xi_{j'} + \rho^\top d_{j'} + \sigma^\top e_{j'}$ ;
an optimality cut is built in NDSPA Step 3a from optimal dual vectors $\lambda_{j'}$ of all descendants at the current $\bar x$ and weights $p_{j'}$ : $E = -\sum_{j'} p_{j'} \pi_{j'}^\top B_{j'}$ and $e = \sum_{j'} p_{j'} (\pi_{j'}^\top \xi_{j'} + \rho_{j'}^\top d_{j'} + \sigma_{j'}^\top e_{j'})$ . It is added only if it cuts off the current point, the test (9): $\bar\theta < e - E^\top \bar x$ .

A cut $E^\top x + \theta \ge e$ is valid when $e - E^\top x$ never exceeds the weighted sum of the descendants' objective values at feasible points, for any $x$ . A basic solution of (7) is a point at which all equality rows hold and $n + 1$ linearly independent constraints are active; it is degenerate when more than $n + 1$ constraints are active.

Formalization targets

Goal: Proposition 4

Let two distinct valid optimality cuts $l_1, l_2$ bind at $(\bar x, \bar\theta)$ . Let each descendant $j'$ have an optimal basic feasible solution $z_{j'}$ and an optimal dual vector $\lambda_{j'}$ at $\bar x$ , and suppose the Step 3a cut built from these duals fails (9), so that $\bar\theta \ge e - E^\top \bar x$ . Then

\exists\, j' \in J' : \ z_{j'} \text{ is a degenerate basic solution of the problem (7) of } j' \text{ at } \bar x .

Milestone: Proposition 3

If descendant $j'$ generates feasibility cut $l'$ of the node and that cut binds at $\bar x$ , then

\text{every basic feasible solution of the problem (7) of } j' \text{ with } \bar x \text{ in (7.2) is degenerate.}

Significance

The result. The two propositions explain a failure mode that every nested-decomposition implementation meets. The backward pass of NDSPA stalls, adding no cut, at exactly the points where Proposition 4 applies, and the degenerate descendant bases it predicts are what make the same ancestor basis recur over many iterations. The analysis motivated the partitioning method of Section 3 of the paper and the intermediate problem of Birge (1980), and it is the reason later codes pay attention to the choice among multiple optimal duals.

Formalizing it. The paper states both propositions without proof and refers the proofs to Birge (1980), an unpublished technical report. A machine-checked proof makes the statements self-contained. It also pins down what they assume: which cuts are meant, which invariants of the algorithm they rely on, and which notion of degeneracy applies to a problem with a free variable and inequality cuts. The definition layer (scenario problems with their cut families, LP duals, Farkas certificates, the Step 3a cut) is reusable for any statement about nested Benders or L-shaped methods. The finite convergence of the nested method is the subject of a separate private formalization (Birge and Louveaux, Ch. 6, Thm. 1, StochasticProg.Multistage.thm1_finite_convergence), so this mission does not restate NDSPA's convergence.

Difficulty

The statements connect the geometry of the ancestor's value function to the combinatorics of descendant bases. For Proposition 4 the difficulty is that nothing in the hypotheses mentions the descendants' bases directly: they speak of two cuts at the ancestor and one failed test. The conclusion is a statement about the descendants' simplex bases, so the ancestor-level information has to be transported down to the descendant problems, where the relevant objects (optimal bases, optimal duals, optimal-value functions of the right-hand side) are not unique in exactly the degenerate case the proposition is about. Two hypotheses that look dispensable are not. Without distinctness a duplicated cut gives a counterexample. Without validity a cut that is not a lower bound can bind anywhere. For Proposition 3 the certificate must be the one that generated the cut; an arbitrary cut row that happens to bind says nothing about the descendant.

Formalization scope

Vectors are Fin n → ℝ, matrices Matrix (Fin m) (Fin n) ℝ; the variable of (7) is $z = (x, \theta) \in$ Fin (n+1) → ℝ with $\theta$ its last coordinate. The paper suppresses transposes; here they are explicit, and a row vector times a matrix ( $\pi B$ ) is vecMul. Basic, basic feasible and degenerate solutions are the general-form notions of Bertsimas and Tsitsiklis (Defs. 2.9–2.10), reused from the public definition BasicSolution, applied to the whole constraint family of (7), the free $\theta$ included. A last-period node is represented with the restriction $\theta = 0$ , which pairs one extra variable with one extra active equality and so preserves basicness and degeneracy.

Readings of informal words, each stated in the items:

"generate a feasibility constraint": the Van Slyke–Wets construction from a Farkas certificate at an earlier input $x^0$ , stated against the descendant's current constraint family;
"binding for some solution": the row holds with equality at the point; feasibility or optimality of the point for the ancestor is not assumed, which strengthens both statements;
"two constraints of type (7.4)": two distinct cuts, both valid, the invariants NDSPA maintains;
"every set of optimal solutions … that produces $E$ and $e$ ": one optimal basic feasible solution and one optimal dual vector per descendant, the cut computed from those duals;
"degenerate": general-form degeneracy, not the standard-form count of zero components.

Correction to the page. Step 3a prints $E = -\sum \pi B_t$ and $e = \sum \pi \xi$ . The formalization completes $e$ with the terms $\rho^\top d + \sigma^\top e$ from the descendants' own cuts, without which the cut is not a valid lower bound when the descendants carry cuts. It also adds weights $p_{j'} > 0$ ; conditional probabilities are a special case, and $p \equiv 1$ is the printed sum. The data $A$ , $B$ may differ between descendants, which is more general than the paper.

Proposition 3 is vacuous when the descendant is infeasible at $\bar x$ ; that is the paper's statement. No trivializing reading is available for the goal: the cuts are computed from certificates and duals inside the statement, never taken as arbitrary vectors, and a sorry-free local check confirms that all hypotheses of Proposition 4 hold together on a small instance. Propositions 1 and 2, NDSPA's convergence, the partitioning method of Section 3 and the computational study are out of scope. Contributions welcome: proofs of Propositions 3 and 4, and supporting lemmas on LP duality and basis stability over general-form constraint families.

Selected references

J. R. Birge, Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operations Research 33(5):989–1005, 1985. https://doi.org/10.1287/opre.33.5.989
R. M. Van Slyke and R. Wets, L-Shaped Linear Programs with Applications to Optimal Control and Stochastic Programming, SIAM J. Appl. Math. 17(4):638–663, 1969. https://doi.org/10.1137/0117061
J. R. Birge, Solution Methods for Stochastic Dynamic Linear Programs, Technical Report SOL 80-29, Stanford University, 1980.
D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, 1997 (Defs. 2.9–2.10).
J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2nd ed., Springer, 2011 (nested L-shaped method for multistage problems). https://doi.org/10.1007/978-1-4614-0237-4

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Algorithmic Game Theory Linear Optimization Operations Research·Captain: mikedeng1

The Assignment Game I: The Core 1: The Core Is the Set of Optimal Solutions of the Dual Assignment LPResearch Paper

Motivation

A two-sided market in which each participant trades with at most one partner on the other side — houses and buyers, workers and firms, producers and consumers under exclusive bilateral contracts — is the setting of L. S. Shapley and M. Shubik's assignment game (Int. J. Game Theory 1 (1971) 111–130). Money is transferable, so the natural solution concept is cooperative: a division of the total gain that no coalition of traders can improve upon. The question the paper answers first is whether such a division exists and how to find it, given that a market with $m$ sellers and $n$ buyers has $2^{m+n}$ coalitions.

The answer, Theorem 2 of the paper, connects the core of the game with the dual of the linear-programming relaxation of the optimal assignment problem. It is the starting point of the literature on two-sided matching markets with transfers: the lattice structure of the core (Theorem 3 of the same paper), the ascending auctions of Demange, Gale and Sotomayor (J. Polit. Econ. 94 (1986)), and the equivalence of core outcomes with competitive equilibrium prices. The same LP pairing reappears in stable matching with transfers and in the VCG analysis of multi-item auctions with unit demand.

Setting

Let $M$ be a finite set of sellers and $N$ a finite set of buyers; either may be empty and their sizes need not agree. A matrix $a = (a_{ij})_{i \in M, j\in N}$ of nonnegative reals records the profit $a_{ij} \ge 0$ that the partnership of seller $i$ and buyer $j$ can realise (in the real-estate reading, $a_{ij} = \max(0, h_{ij} - c_i)$ with $h_{ij}$ buyer $j$ 's valuation of house $i$ and $c_i$ its owner's reservation value).

A coalition $S \subseteq M \cup N$ is described by its sellers $A = S\cap M$ and buyers $B = S \cap N$ . A matching inside $(A, B)$ is a set of seller–buyer pairs from $A \times B$ in which no player appears twice. The characteristic function (2.6) assigns to $S$ its worth

v(S) = \max_{P} \sum_{(i,j) \in P} a_{ij},

the maximum over matchings $P$ inside $S$ . One-sided coalitions and singletons are worth $0$ .

A payoff vector is a pair $(u, v)$ with $u \in \mathbb R^M$ , $v \in \mathbb R^N$ (the paper uses $v$ both for the characteristic function and for buyers' payoffs; the formal development calls the former worth). The core is the set of payoff vectors with

\sum_{i\in M} u_i + \sum_{j \in N} v_j = v(M \cup N) \quad (3.5), \qquad \sum_{i\in S\cap M} u_i + \sum_{j \in S\cap N} v_j \ge v(S)\ \text{ for all } S \quad (3.6).

The assignment LP (3.1)–(3.2) maximises $z = \sum_{i,j} a_{ij} x_{ij}$ over $x_{ij} \ge 0$ with $\sum_{i} x_{ij} \le 1$ for each $j$ and $\sum_j x_{ij} \le 1$ for each $i$ . Its dual (3.3)–(3.4) minimises $w = \sum_i u_i + \sum_j v_j$ over $u_i \ge 0$ , $v_j \ge 0$ with $u_i + v_j \ge a_{ij}$ for all $i, j$ . The optimal values are $z_{\max}$ and $w_{\min}$ .

Formalization targets

Goal: Theorem 2 (p. 118)

\operatorname{core}(a) = \{(u,v) : (u,v) \text{ is an optimal solution of the dual LP (3.3)–(3.4)}\}.

The statement holds for every finite $M$ , $N$ and every $a \ge 0$ , with no constants to fix.

Milestones, in the order the paper's argument uses them

Eq. (3.6), p. 118: every dual-feasible $(u, v)$ gives every coalition at least its worth.
Sec. 3.1, p. 117 (quoting Dantzig, p. 318): the assignment LP attains its maximum at a $0/1$ point, and $z_{\max} = v(M \cup N)$ .
Sec. 3.1, p. 118 (quoting Dantzig, p. 129): both LPs have optima and $w_{\min} = z_{\max}$ .
Eq. (3.5), p. 118: every dual minimiser satisfies $\sum_i u_i + \sum_j v_j = v(M \cup N)$ .

Further results on the same definitions

Sec. 3.2, p. 118: the core is nonempty.
Sec. 3.2, p. 119: for $(u,v)$ in the core, $v_j = \max\bigl(0, \max_{i}(a_{ij} - u_i)\bigr)$ for every buyer $j$ — at the seller prices $p_i = c_i + u_i$ , buyer $j$ 's best net gain is exactly $v_j$ .

Significance

Theorem 2 replaces the $2^{m+n}$ coalition constraints of the core with the $mn$ constraints of a linear program. Consequences stated in the paper: the core is never empty; its points are exactly the dual optimal solutions, so the core is a polytope computable by linear programming without evaluating the worth of any coalition other than the grand one; and dual variables are prices — a seller's core payoff determines a price at which every buyer's best purchase yields that buyer's core payoff. The lattice and corner results of Sec. 3.3 build on this identification.

The paper's proof is short but leans on two results quoted from Dantzig's Linear Programming and Extensions: the integrality of the rectangular assignment polytope and the LP duality theorem. A formalization makes these dependencies explicit for the inequality-constrained rectangular case with possibly unequal sides. To our knowledge Theorem 2 has no machine-checked proof. On Prove2Me, general LP strong duality is formalized (LinearOptimization.lp_strong_duality, SmaleNinth.lp_strong_duality), and integrality of the square doubly stochastic assignment LP (UnderstandingML.assignment_lp_integral, FamousTheorems.birkhoff_von_neumann); neither is the specialised statement here, but both are natural substrate.

Difficulty

The inclusion "dual optimal ⊆ core" needs the value of the dual to equal the combinatorial worth $v(M \cup N)$ , which is not a statement about linear programming alone: it needs both LP duality and the integrality of the assignment polytope. The integrality needed is for the polytope cut out by inequalities $\le 1$ on a possibly non-square matrix, which is not the Birkhoff polytope of doubly stochastic matrices already on the platform; the gap between the two must be bridged.

The converse, "core ⊆ dual optimal", is dismissed as "clearly" in the paper, but the core is defined through the worth of every coalition, a maximum over exponentially many matchings, while dual optimality is a comparison with every dual-feasible vector. Neither side mentions the other's objects, and the naive reading "the core is the dual feasible set" is false: large payoffs are dual feasible and violate (3.5).

Formalization scope

Lean represents sellers and buyers as types M, N with [Fintype M] [Fintype N]; no nonemptiness is assumed, so markets with an empty side are included (there the core is $\{0\}$ ). The matrix is a : M → N → ℝ with the standing hypothesis ∀ i j, 0 ≤ a i j on every theorem. A coalition is a pair (A, B) : Finset M × Finset N. A matching is a Finset (M × N) inside A ×ˢ B with no repeated seller and no repeated buyer. The worth is Finset.sup' over the finite, nonempty set of matchings (the empty matching is always present), so there is no junk value. The paper's (2.6) takes exactly $k = \min(|S\cap M|, |S\cap N|)$ pairs; the formal definition takes all partial matchings, which gives the same maximum because $a \ge 0$ .

Payoff vectors are pairs (M → ℝ) × (N → ℝ). The core is defined by (3.5) and (3.6) for all coalitions; nonnegativity is not a separate clause since it follows from singleton coalitions. The dual LP keeps the nonnegativity of $u$ and $v$ that (3.3) imposes, and "solutions of the LP dual" is read as optimal solutions (DualOptimal: dual feasible and of minimal objective among dual-feasible vectors), as the text preceding Theorem 2 does.

The worth is the combinatorial maximum, not the LP value, and the core is defined by coalitions, not through the dual; defining either through the other would make Theorem 2 true by unfolding, and such a formalization is ruled out. Likewise the goal is not the weaker "core = dual feasible set", which is false.

Contributions welcome: integrality of the rectangular inequality-form assignment polytope, a specialisation of general LP duality to this pair, and the coalitional inequality (3.6). The first two are reusable for any bipartite matching model.

Selected references

L. S. Shapley and M. Shubik, The Assignment Game I: The Core, International Journal of Game Theory 1 (1971), 111–130. https://doi.org/10.1007/BF01753437
G. B. Dantzig, Linear Programming and Extensions, Princeton University Press, 1963. https://doi.org/10.1515/9781400884179
L. S. Shapley, Complements and substitutes in the optimal assignment problem, Naval Research Logistics Quarterly 9 (1962), 45–48. https://doi.org/10.1002/nav.3800090106
G. Demange, D. Gale and M. Sotomayor, Multi-Item Auctions, Journal of Political Economy 94 (1986), 863–872. https://doi.org/10.1086/261393

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Operations Research Probability Stochastic Systems·Captain: Shuze Chen

Processing Networks VIII: Maximal Stability of Back-Pressure ControlTextbook

Motivation

Every stability result up through mission VII proves that a particular control policy — a fixed priority list, HLSPS, a policy tailored to one network's topology — keeps a specific processing network stable throughout its subcritical region. None of them answer a more practical question a system designer actually faces: given an arbitrary Leontief network (one where every activity has a well-defined, unique buffer it draws from), is there a single control rule, computable from the network's data alone with no bespoke analysis, that is guaranteed stable whenever stability is possible at all? J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) answers this in Chapter 9 with the back-pressure (equivalently, in the single-hop case, max-weight) control policy: at every decision time, choose the allocation of service effort that maximizes a bilinear "weighted throughput" objective built directly from current buffer contents. This mission formalizes the policy, the characteristic fluid equation it induces, and the resulting maximal-stability theorem — the chapter's central result and one of the most cited scheduling policies in the queueing-networks literature.

Setting

A Leontief network (Definition 9.5) is an SPN whose input-output matrix $R$ satisfies two assumptions: Assumption 9.1, that each activity has a unique buffer it draws material from (so $i(j)$ , the buffer served by activity $j$ , is well-defined), and Assumption 9.2, that there is a nonnegative activity-level vector driving every buffer's net output rate strictly positive — the structural condition under which the network can be drained at all. The back-pressure (or, in the single-server, single-hop case, max-weight) control policy chooses, at each decision time, the feasible allocation $\beta$ of service rates that maximizes the bilinear objective $p(\beta, \hat z) = \hat z \cdot R\beta$ , where $\hat z$ is the current vector of buffer contents. The "relaxed" version of the policy allows $\beta$ to range continuously over the allocation polytope $\mathcal A = \{\beta \in \mathbb R^J_+ : A\beta \le b\}$ ; the "basic" version restricts to integer service-initiation decisions in a genuine SPN with discrete jobs. A network's static planning problem's optimal value $\gamma^* < 1$ is the subcriticality condition throughout this chapter, exactly as in missions II, III, and V.

Formalization targets

Goal: Theorem 9.12 — maximal stability of relaxed back-pressure

Consider a Leontief network operating under the relaxed back-pressure control policy. If the static planning problem has optimal objective value $\gamma^* < 1$ , then the corresponding fluid limit is stable, and hence, by Theorem 6.2 (mission III), the network's ambient Markov chain is positive recurrent. This is the chapter's payoff: back-pressure control needs no network-specific tuning — it stabilizes every Leontief network throughout its entire subcritical region, the same universal guarantee mission VI showed only for feedforward networks and HLSPS control specifically.

Supporting milestones

Lemma 9.3 and Proposition 9.4 develop the linear-algebraic machinery of basis matrices: any feasible material-balance vector can be re-expressed using only $I$ "basic" activities (Lemma 9.3), and Assumption 9.2 holds if and only if some basis's associated matrix has spectral radius below one (Proposition 9.4) — the practical, checkable criterion for the network being well-posed at all. Proposition 9.6 shows the back-pressure optimization problem always admits a solution among the finitely many extreme allocations, licensing Remark 9.7's standing convention of restricting attention to that finite set. Lemma 9.10 shows a $z$ -maximal extreme allocation can always be chosen to idle any activity whose buffer is currently empty — a fact that looks obvious but genuinely needs proof, because at the fluid level an activity can serve an instantaneously empty buffer at a positive rate (Section 9.5's tandem-model illustration). Theorem 9.8 is the chapter's characteristic fluid equation: under relaxed back-pressure control, the realized fluid service-rate derivative always achieves the bilinear maximum over the allocation polytope, at every regular point. Lemma 9.11 derives this from the raw ("pre-limit") stochastic dynamics — a strictly dominated allocation accrues no processing time — via a genuine limit-passage argument. Theorem 9.13 and Lemma 9.14 extend the maximal-stability guarantee from the relaxed policy to the basic (discrete-decision) policy, under the extra restriction that each server pool is a single server acting alone; this needs a residual-time strong law of large numbers (Lemma 9.14) to show that a server's decision to switch away from a dominated allocation happens quickly enough, relative to elapsed time, that the fluid limit is unaffected.

Significance

The result itself. Theorem 9.12 is the book's formalization of the max-weight/back-pressure maximal-stability theorem originally due to Tassiulas and Ephremides (1992) for multi-hop packet radio networks, later popularized under the "back-pressure" name by Tassiulas (1995) and extended substantially by Dai and Lin (2005), on whose work this chapter is explicitly based. Unlike every policy considered in missions IV, VI, and VII, back-pressure requires no topology-specific insight to design or verify — it is defined uniformly from $B$ , $\Gamma$ , $A$ , and current buffer contents, and Theorem 9.12 certifies it stable for every Leontief network in its subcritical region. This universality is precisely what distinguishes it from HLSPS (mission VI), which needs the network to be feedforward or the policy to be head-of-the-line proportional-sampling before the same guarantee holds.

Formalizing it. A live prior-art check (GET /theorems?q=max-weight%20scheduling, q=back-pressure) returns no hits, so this mission formalizes the policy, its characteristic fluid equation, and both stability theorems entirely from scratch. SPNPlanningData, the input-output matrix R, and the static planning problem are restated from mission II's own apparatus; RegularPoint is restated from mission V's Definition 8.7.

Difficulty

The chapter's central subtlety is that "operating under back-pressure control" cannot be stated directly as a hypothesis on the fluid-limit path $(\hat D,\hat F,\hat T,\hat Z)$ itself: the policy is defined in terms of the discrete, pre-limit decision process, and its fluid-level consequence — the characteristic equation (9.22) — is a genuine theorem (9.8), not a restatement of the policy's definition. Formalizing Theorem 9.8 naively by hypothesizing " $\hat T$ satisfies (9.22)" would make Lemma 9.11 (whose conclusion (9.28)-(9.31) is what Theorem 9.8's own proof literally invokes) circular relative to it. This mission instead hypothesizes Lemma 9.11's raw, pre-limit optimality condition (hYopt: a strictly dominated allocation accrues no processing time over any interval where domination persists) as the operational meaning of "following the back-pressure rule," and derives (9.28)-(9.31) from it as Lemma 9.11's genuine conclusion — Fed into Theorem 9.8 exactly as the book's own proof does ("By Lemma 9.11 and the fact that $\sum_\beta \dot{\hat Y}_\beta(t) = 1$ ..."). A second difficulty is Theorem 9.13's genuinely distinct proof from Theorem 9.12's: the basic (discrete) policy's fluid limit satisfying the same characteristic equation is not automatic, and needs the residual-time SLLN of Lemma 9.14 plus two extra structural hypotheses (each server pool is a single server, each activity uses exactly one server) that go beyond "basic vs. relaxed" and are stated explicitly rather than folded silently into the policy's name.

Formalization scope

SPNPlanningData, its input-output matrix R, and the static planning problem (SPPFeasible, IsOptimalSPPValue) are restated unmodified from mission II's own apparatus (drafts in this series do not import one another); RegularPoint is restated unmodified from mission V's Definition 8.7. "Basis" is named ActivityBasis, not Basis, to avoid colliding with Mathlib's vector-space Basis type — a deliberate departure from the book's own overloaded terminology, which its own text flags as "slightly narrower than [the] standard meaning in linear programming theory." ExtremeAllocations reuses Mathlib's Set.extremePoints directly rather than restating extreme-point theory from scratch, and Proposition 9.4's spectral-radius condition reuses Mathlib's own spectralRadius (Mathlib.Analysis.Normed.Algebra.Spectrum) rather than defining eigenvalues by hand. IsZMaximal (Definition 9.9) is phrased as "feasible and dominates every feasible alternative" rather than via an explicit sSup/⨆ expression, which sidesteps any Mathlib junk-value risk while remaining definitionally equivalent to "achieves the maximum" whenever a maximizer exists — the same convention this series has used since mission III. Theorem 9.8's hypothesis that the fluid limit "operates under relaxed back-pressure control" is packaged as Lemma 9.11's own conclusion (hTY/hYmono/hYsum/hYopt), matching the book's proof architecture exactly rather than re-deriving it inline. Theorem 9.13 states its two extra single-server hypotheses (hb1, hA01) explicitly as the mission's own BRIEF.md warns to. Lemma 9.14's condition (9.44) — quoted in the book's preparatory material for Theorem 9.13 rather than in the excerpt originally assembled for this lemma — was located directly in source.txt (p. 178, PDF p. 194) and confirmed verbatim, not reconstructed; its formalization (h944) matches the confirmed text exactly. The one acknowledged source inconsistency, noted by BRIEF.md itself, is that (9.56)'s printed left-hand side reads u_i(s,ω) where the surrounding proof otherwise uses t throughout — treated as a typesetting slip and formalized with t, as the lemma evidently intends. IsFluidModelSolution, IsRelaxedBPFluidSolution, RelaxedBPFluidStable, ActivityBasis, AllocationPolytope, and IsZMaximal are the primary reusable contributions; contributions completing the nine by sorry proofs, especially Lemma 9.11's limit-passage argument and Lemma 9.14's SLLN chain, are welcome.

Selected references

J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
L. Tassiulas and A. Ephremides, "Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks," IEEE Transactions on Automatic Control 37 (1992), 1936–1948.
J. G. Dai and W. Lin, "Maximum pressure policies in stochastic processing networks," Operations Research 53 (2005), 197–218.

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Machine Learning Probability Statistics·Captain: mikedeng1

Learnability, Stability and Uniform Convergence I: A Problem Is Learnable if and only if It Admits a Uniform-RO Stable Universal AERMResearch Paper

Motivation

In supervised binary classification, a hypothesis class is learnable if and only if it has uniform convergence, meaning that empirical risks converge to true risks uniformly over the class. When that holds, empirical risk minimisation (ERM) learns. This equivalence, due to Vapnik and Chervonenkis and extended to real-valued losses by Alon, Ben-David, Cesa-Bianchi and Haussler, is the usual starting point of statistical learning theory.

Vapnik's General Learning Setting is broader. It covers stochastic convex optimisation, clustering and density estimation, and in it the equivalence breaks down. Shalev-Shwartz, Shamir, Srebro and Sridharan (JMLR 11 (2010) 2635–2670) exhibit learnable problems with no uniform convergence, and learnable problems where ERM fails. So neither uniform convergence nor the success of ERM characterises learnability there, and something else has to. This mission formalizes the paper's answer, its Theorem 7: stability.

Timeline:

1971–1995: Vapnik and Chervonenkis prove learnability ⇔ uniform convergence for binary classification. Vapnik (1995) introduces the General Learning Setting.
2002: Bousquet and Elisseeff show that uniform stability of a learning rule implies generalization.
2006: Mukherjee, Niyogi, Poggio and Rifkin show that, in the supervised setting, stability of ERM is necessary and sufficient for learnability.
2009–2010: Shalev-Shwartz, Shamir, Srebro and Sridharan (COLT 2009, JMLR 2010) prove that, in the General Learning Setting, learnability is equivalent to the existence of a uniform-RO stable, universally asymptotic empirical risk minimiser (Theorem 7).

Setting

A learning problem consists of a hypothesis class $\mathcal H$ (nonempty), an instance set $\mathcal Z$ with a $\sigma$ -algebra, and an objective $f:\mathcal H\times\mathcal Z\to\mathbb R$ with $|f(h;z)|\le B$ for all $h,z$ . Given a probability distribution $\mathcal D$ on $\mathcal Z$ and an i.i.d. sample $S=(z_1,\dots,z_m)\sim\mathcal D^m$ , the following quantities are defined:

the risk is $F(h)=\mathbb E_{z\sim\mathcal D}[f(h;z)]$ , and $F^*=\inf_{h}F(h)$ ;
the empirical risk is $F_S(h)=\frac1m\sum_{i=1}^m f(h;z_i)$ , and $F_S(\hat h_S)=\inf_h F_S(h)$ is the minimal empirical risk. Only the value is used; no minimiser need exist.

A learning rule $A$ maps each sample $S$ of each size $m$ to a hypothesis $A(S)$ . A rate $\varepsilon(m)$ is a non-increasing sequence tending to $0$ . For a rule $A$ the paper defines the following properties:

$A$ is consistent with rate $\varepsilon_{\rm cons}$ under $\mathcal D$ if $\mathbb E_S[F(A(S))-F^*]\le\varepsilon_{\rm cons}(m)$ . It is universally consistent if this holds for every $\mathcal D$ with the same rate. The problem is learnable (Definition 1) if a universally consistent rule exists.
$A$ is an AERM (asymptotic empirical risk minimiser) with rate $\varepsilon_{\rm erm}$ under $\mathcal D$ if $\mathbb E_S[F_S(A(S))-F_S(\hat h_S)]\le\varepsilon_{\rm erm}(m)$ , and universally so if this holds for every $\mathcal D$ .
$A$ generalizes with rate $\varepsilon_{\rm gen}$ under $\mathcal D$ if $\mathbb E_S[|F(A(S))-F_S(A(S))|]\le\varepsilon_{\rm gen}(m)$ .
With $S^{(i)}$ the sample $S$ with $z_i$ replaced by $z_i'$ , $A$ is uniform-RO stable with rate $\varepsilon_{\rm stable}$ (Definition 4) if, for all $S$ , all replacements $(z_1',\dots,z_m')$ and all $z'\in\mathcal Z$ ,

\frac1m\sum_{i=1}^m\bigl|f(A(S^{(i)});z')-f(A(S);z')\bigr|\le\varepsilon_{\rm stable}(m).

Average-RO stability (Definition 5) is the in-expectation analogue, with the replacement point also serving as the test point.

Formalization targets

Goal: Theorem 7

The problem is learnable if and only if there is a learning rule that is uniform-RO stable and universally an AERM. Quantitatively, if $A$ is universally consistent with rate $\varepsilon_{\rm cons}$ , then some rule $A'$ is uniform-RO stable and universally AERM with

\varepsilon_{\rm stable}(m)=\frac{2B}{\sqrt m},\qquad \varepsilon_{\rm erm}(m)=3\,\varepsilon_{\rm cons}\bigl(\lfloor m^{1/4}\rfloor\bigr)+\frac{8B}{\sqrt m},

and conversely, a uniform-RO stable universal AERM is universally consistent with rate $\varepsilon_{\rm stable}(m)+\varepsilon_{\rm erm}(m)$ .

Milestones

These follow the order of the paper's proof.

Sufficiency: Utility Lemma 12 (a bounded sample mean deviates by at most $B/\sqrt m$ in expectation), Lemma 11 (on-average generalization ⇔ average-RO stability), Claim 6 (uniform-RO ⇒ average-RO stability), Lemma 15 (an on-average generalizing AERM is consistent), and Theorem 8 (a stable AERM is consistent with rate $\varepsilon_{\rm stable}+\varepsilon_{\rm erm}$ and generalizes with rate $\varepsilon_{\rm stable}+2\varepsilon_{\rm erm}+2B/\sqrt m$ ).
Necessity: Lemma 20 (every rule has a uniform-RO stable, $3B/\sqrt m$ -generalizing version with consistency rate $\varepsilon_{\rm cons}(\lfloor\sqrt m\rfloor)$ ), Lemma 16, the Main Converse Lemma ( $\mathbb E|F_S(\hat h_S)-F^*|\le2\varepsilon_{\rm cons}(m')+2B/\sqrt m+2Bm'^2/m$ for $2\le m'\le m/2$ ), and Lemma 18 (under that bound, a consistent and generalizing rule is an AERM).

Significance

Theorem 7 shows that in the General Learning Setting, stability replaces uniform convergence as the notion that characterises learnability. It also says where to look for a learning rule: ERM may fail, but some AERM always works, and it must be stable. The rates are explicit and polynomial. Downstream, the paper uses Theorem 7 to prove Theorem 23 (randomised rules) and to design a generic learning algorithm (Theorem 25). Mission II of this series (Tikhonov-regularised ERM for stochastic convex optimisation) is a concrete instance of a stable AERM for a problem with no uniform convergence.

The theorem has been proved since 2010 but has not been formalized. The platform has the textbook side of the same authors' framework: Understanding Machine Learning Theorem 13.2, UnderstandingML.stability_identity, the replace-one identity behind Lemma 11, stated for hypotheses in $\mathbb R^d$ . The platform does not have learnability in the General Learning Setting, over an arbitrary hypothesis class, or the converse direction. That direction is the new content: learnability forces a stable AERM to exist.

Difficulty

The sufficiency direction is a chain of expectation identities. The necessity direction is harder. A universally consistent rule need not be an AERM, need not generalize and need not be stable (Example 2 of the paper), so it cannot simply be reused. ERM cannot be used either, since it can fail on learnable problems. The Main Converse Lemma is where universal consistency is used in full: the rule's guarantee has to be applied under a distribution other than $\mathcal D$ , and a naive argument under $\mathcal D$ alone fails (Example 1: consistency under one distribution does not imply generalization under it). Combining the lemmas into the stated rates requires choosing the auxiliary sample size and tracking every constant, including the regime of small $m$ where Lemma 16's hypothesis $2\le m'\le m/2$ cannot be met.

Formalization scope

Samples are Fin m → Z, the sample law $\mathcal D^m$ is Measure.pi, and $S^{(i)}$ is Function.update S i (S' i). Learning rules have type (m : ℕ) → (Fin m → Z) → H, and every property is asserted for $m\ge1$ . The minimal empirical risk and $F^*$ are real infima over the nonempty, bounded-below family, never values at a chosen minimiser. Rates are non-increasing on $m\ge1$ and tend to $0$ . $\lfloor m^{1/4}\rfloor$ and $\lfloor\sqrt m\rfloor$ are Nat.sqrt (Nat.sqrt m) and Nat.sqrt m, the paper's $\varepsilon_{\rm cons}(m^{1/4})$ read at an integer sample size. The paper's $B=\sup|f|$ is replaced by any bound $B$ (all rates increase in $B$ ).

The paper never discusses measurability. This formalization adds one standing assumption, identical across the series: each $f(h;\cdot)$ is measurable, the minimal empirical risk $S\mapsto\inf_hF_S(h)$ is measurable (true for countable $\mathcal H$ , for example), and every learning rule, whether assumed or asserted to exist, has $(S,z)\mapsto f(A(S);z)$ jointly measurable. Without this a non-measurable rule would have Bochner integrals equal to $0$ , and existence claims such as "some rule is a universal AERM" would be satisfied by junk. Every existential in the goal therefore produces a measurable rule, and learnability is quantified over measurable rules with the rate chosen before the distribution. Uniform-RO stability is pointwise over all samples, replacement vectors and test points; it is never replaced by the in-expectation Definition 5.

No statement is corrected. The proof of the converse in the paper calls $A'$ " $2B/\sqrt m$ -generalizing" where Lemma 20 gives $3B/\sqrt m$ . The stated $8B/\sqrt m$ absorbs either value, so Theorem 7 is formalized as printed.

The definitions (risks, rules, consistency, AERM, generalization, the two RO-stability notions) are reusable by the other missions of this series and by any stability-based result in the General Learning Setting. Contributions are welcome at every level: proofs of the milestones, the measure-theoretic infrastructure they need (exchangeability of i.i.d. coordinates under Measure.pi, sub-sampling and restriction of product measures, the variance bound for bounded sample means), and the remaining results of Section 5 (Theorems 9 and 10, Lemmas 14 and 17).

Selected references

S. Shalev-Shwartz, O. Shamir, N. Srebro, K. Sridharan, Learnability, Stability and Uniform Convergence, Journal of Machine Learning Research 11 (2010) 2635–2670. https://jmlr.org/papers/v11/shalev-shwartz10a.html
V. N. Vapnik, The Nature of Statistical Learning Theory, Springer, 1995. https://doi.org/10.1007/978-1-4757-2440-0
N. Alon, S. Ben-David, N. Cesa-Bianchi, D. Haussler, Scale-sensitive dimensions, uniform convergence, and learnability, Journal of the ACM 44(4) (1997) 615–631. https://doi.org/10.1145/263867.263927
O. Bousquet, A. Elisseeff, Stability and Generalization, Journal of Machine Learning Research 2 (2002) 499–526. https://jmlr.org/papers/v2/bousquet02a.html
S. Mukherjee, P. Niyogi, T. Poggio, R. Rifkin, Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization, Advances in Computational Mathematics 25 (2006) 161–193. https://doi.org/10.1007/s10444-004-7634-z
S. Shalev-Shwartz, S. Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014, Chapter 13. https://doi.org/10.1017/CBO9781107298019

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Convex Optimization Functional Analysis Operations Research·Captain: mikedeng1

On the Maximal Monotonicity of Subdifferential Mappings I: The Subdifferential of a Lower Semicontinuous Proper Convex Function on a Banach Space Is Maximal MonotoneResearch Paper

Motivation

Monotone operators from a Banach space $E$ to its dual $E^*$ are the abstract framework for nonlinear equations, variational inequalities and evolution equations, and for the convergence theory of proximal-point and splitting algorithms in infinite dimensions. Within that framework, the class that behaves well — for surjectivity results, for resolvents, for sums — is the class of maximal monotone operators. The most important source of such operators is convex analysis: the subdifferential of a convex function. Whether every subdifferential of a closed proper convex function is maximal monotone, in an arbitrary Banach space, is therefore a basic question for convex optimization in function spaces.

Timeline:

1964. G. J. Minty proves maximality of the subdifferential for convex functions that are finite and continuous everywhere (Minty, Pacific J. Math. 14 (1964)).
1965. A. Brøndsted and R. T. Rockafellar show that subgradients exist on a dense set and approximate ε-subgradients (Brøndsted–Rockafellar, Proc. AMS 16 (1965)). J.-J. Moreau develops proximal maps and conjugate duality for convex functions in Hilbert space (Moreau, Bull. SMF 93 (1965)); his later lecture notes Fonctionnelles convexes (Collège de France, 1967) are the paper's reference for conjugates in locally convex spaces.
1966. R. T. Rockafellar announces the general Banach-space result, for every lower semicontinuous proper convex function (Rockafellar, Pacific J. Math. 17 (1966)). H. Brézis later points out a gap in that proof: a subgradient chosen in the argument may grow without bound.
1970. Rockafellar gives a complete proof by a different route, valid in nonreflexive spaces, in the paper formalized here (Rockafellar, Pacific J. Math. 33 (1970)).

Setting

Let $E$ be a real Banach space with dual $E^*$ and bidual $E^{**}$ , and write $\langle x, x^*\rangle = x^*(x)$ . $E$ sits in $E^{**}$ through the canonical embedding.

A proper convex function on $E$ is a function $f : E \to (-\infty, +\infty]$ , not identically $+\infty$ , such that $f((1-\lambda)x + \lambda y) \le (1-\lambda) f(x) + \lambda f(y)$ for all $x, y \in E$ and $0 < \lambda < 1$ . It is lower semicontinuous for the norm topology.

The subdifferential of $f$ is the multivalued map $\partial f : E \to E^*$ ,

\partial f(x) = \{\, x^* \in E^* \mid f(y) \ge f(x) + \langle y - x, x^* \rangle \ \ \forall y \in E \,\}.

A multivalued map $T : E \to E^*$ is monotone if $\langle x_0 - x_1, x_0^* - x_1^* \rangle \ge 0$ whenever $x_0^* \in T(x_0)$ and $x_1^* \in T(x_1)$ . It is maximal monotone if, in addition, its graph $\{(x, x^*) \mid x^* \in T(x)\}$ is not properly contained in the graph of any other monotone map $T' : E \to E^*$ .

The conjugate of $f$ is $f^*(x^*) = \sup_{x \in E} \{\langle x, x^*\rangle - f(x)\}$ , a function on $E^*$ ; its subdifferential $\partial f^*$ maps $E^*$ into $E^{**}$ . Finally $j(x) = \tfrac12 \|x\|^2$ .

In Lean these are ProperConvex f, subdiff f, IsMonotoneOp T, IsMaximalMonotone T, conj f and halfSqNorm, all stated over an arbitrary real normed space V so that they apply equally to $E$ and to $E^*$ .

Formalization targets

Goal: Theorem A (p. 210)

f \text{ lower semicontinuous proper convex on } E \quad\Longrightarrow\quad \partial f : E \to E^* \text{ is maximal monotone.}

No reflexivity, inner product or finite dimension is assumed.

Milestones, in attack order

(2.2) Fenchel–Young: $f(x) + f^*(x^*) \ge \langle x, x^* \rangle$ , with equality iff $x^* \in \partial f(x)$ .
§2, p. 210. $f^*$ is a weak* lower semicontinuous (hence strongly lower semicontinuous) proper convex function on $E^*$ .
§2, p. 211. The restriction of $f^{**}$ to $E$ is $f$ .
Proposition 1. $x^{**} \in \partial f^*(x^*)$ iff there are a net $x_i^* \to x^*$ in norm and a bounded net $x_i \to x^{**}$ weak**, on one directed index set, with $x_i^* \in \partial f(x_i)$ .
(3.1) $\partial(f + j)(x) = \partial f(x) + \partial j(x)$ for all $x \in E$ .
§3, p. 213. $(f + j)^*$ is finite and continuous throughout $E^*$ .
§3, p. 213 (Minty). On any real Banach space, a convex function that is finite and continuous everywhere has a maximal monotone subdifferential.

Significance

Theorem A places every closed proper convex function in the maximal monotone class, in every real Banach space. Downstream, it is what allows convex minimization problems to be treated by the general theory: surjectivity of $\partial f + \lambda J$ (with $J$ the duality map) in reflexive spaces, existence for evolution equations governed by subdifferentials, the definition of resolvents and proximal maps, and the convergence of proximal-point and splitting methods for convex problems. Proposition 1 is of independent interest: in a nonreflexive space $\partial f^*$ is not the inverse of $\partial f$ , but it is still completely determined by $\partial f$ through bounded weak** nets.

The theorem is classical and fully proved in the literature. What this mission adds is a machine-checked proof of the nonreflexive Banach-space statement, together with the infrastructure it needs: extended-real-valued proper convex functions, the subdifferential and the conjugate on a normed space and its dual, monotone and maximal monotone operators, and the Fenchel–Moreau identity $f^{**}|_E = f$ . None of these exists in Mathlib at the pinned revision, and Mathlib contains no statement of Theorem A, in Hilbert or in Banach spaces.

Difficulty

Monotonicity of $\partial f$ follows in two lines from the definition; the whole difficulty is maximality. In a Hilbert space the standard argument solves $x + \partial f(x) \ni y$ by minimizing $f + \tfrac12\|\cdot - y\|^2$ and uses the identification of $E$ with $E^*$ ; in a general Banach space there is no such identification, and minimizers need not exist without reflexivity. The 1966 argument tried to approximate subgradients of $f$ at nearby points, and it failed because those subgradients could become unbounded as the approximation was refined. Any argument that passes through the dual meets a second obstacle: $\partial f^*$ takes values in the bidual $E^{**}$ , which is strictly larger than $E$ when $E$ is not reflexive, so $\partial f^*$ is not the inverse of $\partial f$ . Relating the two is the content of Proposition 1, and its necessity half requires approximation results well beyond the definitions.

Formalization scope

Lean representation and committed conventions:

$E$ is a real Banach space: NormedAddCommGroup E, NormedSpace ℝ E, CompleteSpace E. $E^*$ is StrongDual ℝ E with the operator norm; the pairing $\langle x, x^*\rangle$ is x' x; $E^{**}$ is StrongDual ℝ (StrongDual ℝ E) and $E \hookrightarrow E^{**}$ is NormedSpace.inclusionInDoubleDual ℝ E.
The value set $(-\infty,+\infty]$ is EReal with the clause "never $\bot$ ". Properness also requires some value $\ne \top$ . Convexity is the paper's inequality for $0 < \lambda < 1$ , computed in EReal.
Multivalued maps are V → Set (StrongDual ℝ V). Maximality is graph inclusion, quantified over every monotone T', not only over subdifferentials.
The conjugate is an EReal supremum over all of $V$ ; the biconjugate is conj (conj f) on the bidual.
(2.2) is stated as $\langle x, x^*\rangle \le f(x) + f^*(x^*)$ with the equality case, avoiding EReal subtraction.
Weak* lower semicontinuity of $f^*$ is lower semicontinuity on WeakDual ℝ E.
In Proposition 1 a net is a map from a nonempty, directed, partially ordered index type (in the universe of $E$ ), with convergence along atTop. Weak** convergence is pointwise convergence on $E^*$ of the canonical images, which is convergence in the weak topology induced on $E^{**}$ by $E^*$ . Boundedness is a uniform norm bound.
(3.1) reads the printed $\partial(f+j)$ as $\partial(f+j)(x)$ ; the right side is the pointwise (Minkowski) set sum.
"Finite and continuous" for $(f+j)^*$ is the existence of a continuous real-valued $h$ on $E^*$ equal to it everywhere.
Minty's case is stated for an arbitrary real Banach space $V$ , because the proof applies it on $E^*$ .

A trivializing formalization is ruled out: properness excludes $f \equiv +\infty$ (whose empty subdifferential is monotone but not maximal) and $-\infty$ values, maximality ranges over all monotone operators, and the index set in Proposition 1 is nonempty and directed so that no convergence statement holds vacuously.

Infrastructure needed and reusable beyond this mission: extended-real convex analysis on normed spaces (conjugates, the Fenchel–Moreau theorem via Hahn–Banach separation, lower semicontinuity in the weak and weak* topologies), subdifferential calculus for a sum with a continuous function, nets and weak** approximation in the bidual (Goldstine-type arguments), and the Brøndsted–Rockafellar approximation of ε-subgradients. Contributions are welcome at every milestone; the definitions layer and milestones 1–3 are the natural starting points.

Selected references

R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific Journal of Mathematics 33 (1970), 209–216. https://doi.org/10.2140/pjm.1970.33.209
R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific Journal of Mathematics 17 (1966), 497–510. https://doi.org/10.2140/pjm.1966.17.497
G. J. Minty, On the monotonicity of the gradient of a convex function, Pacific Journal of Mathematics 14 (1964), 243–247. https://doi.org/10.2140/pjm.1964.14.243
J.-J. Moreau, Proximité et dualité dans un espace hilbertien, Bulletin de la Société Mathématique de France 93 (1965), 273–299. https://doi.org/10.24033/bsmf.1625
A. Brøndsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proceedings of the American Mathematical Society 16 (1965), 605–611. https://doi.org/10.1090/S0002-9939-1965-0178103-8

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Operations Research Probability Stochastic Systems·Captain: Shuze Chen

Processing Networks V: Lyapunov Stability Criteria for Fluid ModelsTextbook

Motivation

Mission III's Theorem 6.2 reduces SPN stability to a question about deterministic fluid model solutions: does every solution of a fixed system of equations get driven to the origin, uniformly in its starting size? Mission IV showed how to derive the extra, policy-specific equations a fluid model must satisfy. What remains is a method for proving that a system of fluid equations forces extinction — and the standard tool for that, across dynamical systems generally, is a Lyapunov function: a scalar-valued potential that decreases along every trajectory. J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) devotes Chapter 8 to making this method precise for fluid models, and to explaining exactly why it is easier to apply here than the analogous drift condition for the underlying Markov chain.

The chapter's calculus culminates in a genuinely delicate real-analysis fact: the ordinary "Lyapunov function decreases everywhere it should" argument needs the function to be differentiable, but fluid model solutions are typically only Lipschitz (hence differentiable only almost everywhere), and some Lyapunov functions used later in the book (Chapter 10's entropy function) are not even Lipschitz. The chapter's most general result, Lemma 8.11, resolves this by working with the upper-right Dini derivative rather than the ordinary one, following an approach whose subtlety is illustrated by a counterexample due to L. Massoulié when one of its three hypotheses is dropped.

Setting

Throughout this mission, (D,F,T,Z) (without hats) denotes an arbitrary solution of the fluid equations (6.1)-(6.6), restated from mission III (drafts in this series do not import one another). A function $g$ is Lipschitz if it satisfies a Lipschitz bound on every bounded set, with a constant that may depend on the set; it is globally Lipschitz if one constant works everywhere. A point $t > 0$ is a regular point of a fluid model solution if all four components are differentiable there; because every solution is globally Lipschitz, the non-regular points form a Lebesgue-null set. A Lyapunov function for a fluid model is a Lipschitz function $H : \mathbb{R}^I_+ \to \mathbb{R}_+$ with $H(0)=0$ and $H(z) \ne 0$ for $z \ne 0$ — a positive-definite potential.

Formalization targets

Goal: Lemma 8.11 — the general Dini-derivative extinction criterion

For $f : \mathbb{R}_+ \to \mathbb{R}_+$ continuous on $(0,\infty)$ , with a locally bounded upper Dini derivative $D^+f$ and $D^+f(t) \le -\varepsilon$ for a.e. $t$ with $f(t) > 0$ :

f(t) = 0 \quad \text{for } t \ge f(0)/\varepsilon.

This is the weakest natural target: it drops the Lipschitz requirement of Lemma 8.5 entirely, replacing it with only continuity plus a one-sided, locally bounded derivative condition, and Lemmas 8.5 and 8.6 are recovered as the special cases $f = H \circ Z$ for $H$ Lipschitz (respectively linear-type and square-root-type drift bounds).

Supporting milestones

Lemma 8.2 (composition of Lipschitz functions) and Lemma 8.3 (every fluid model solution is globally Lipschitz) supply the regularity Lemma 8.5 needs. Lemma 8.5 (linear drift bound) and Lemma 8.6 (square-root drift bound) are the two directly-applicable extinction criteria the book presents before generalizing to Lemma 8.11. Lemma 8.9 identifies a structural fact used in nearly every application: at a regular point, an empty buffer's fluid arrival and departure rates necessarily coincide. Lemma 8.10 gives the calculus of a pointwise maximum's derivative, needed for piecewise-linear Lyapunov functions. Theorem 8.12 is the chapter's worked illustration: the tandem queueing network's fluid model is stable under the standard load condition, proved with a linear Lyapunov function that (the chapter goes on to show) does not translate into a valid Markov-chain drift bound — the concrete illustration of why the fluid-model method earns its keep.

Significance

The result itself. Lemma 8.11 is the single tool every subsequent stability chapter of the book applies: feedforward and generalized Jackson networks, the Rybko–Stolyar boundary, back-pressure control, proportionally fair allocation (whose entropy Lyapunov function is exactly the non-Lipschitz case this lemma was built to handle), and task allocation all conclude fluid model stability via an instance of this criterion. Theorem 8.12's side observation — the same Lyapunov function that works effortlessly for the fluid model fails to give a Markov-chain drift bound at all — is the chapter's explicit argument for why fluid-model methodology is not just a convenience but a genuine technical advance over direct Markov-chain analysis.

Formalizing it. Searches for "Lyapunov function," "Dini derivative," and "Lipschitz continuous" (q=Lyapunov%20function, q=Dini%20derivative) surface no reusable extinction-criterion result; the one Lyapunov-adjacent hit, posDef_quadratic_form_lower_bound, is an unrelated quadratic-form bound. This mission is a from-scratch formalization of the fluid model's Lyapunov calculus, reusing Mathlib's own LipschitzOnWith/LipschitzWith substrate for Definition 8.1 rather than restating ordinary Lipschitz continuity, per this mission's own BRIEF.md recommendation.

Difficulty

The central difficulty is Lemma 8.11 itself: proving that a bound on the upper Dini derivative $D^+f(t)$ (not the ordinary derivative) forces $f$ to decrease is genuinely subtler than the Lipschitz case, because $D^+f$ is one-sided and may not correspond to an actual rate of change at every point. The book's own proof needs a technical intermediate inequality (8.8), $f(b)-f(a) \le \int_a^b D^+f$ , and states explicitly that this can fail without the local upper-boundedness hypothesis (b) — citing a counterexample of L. Massoulié — so a formalization that dropped condition (b) as "obviously implied by continuity" would be proving a false generalization, not a faithful specialization. A second difficulty, specific to Lemma 8.5/8.6's formalization, is that the hypothesis " $\dot f(t) \le -\varepsilon$ for almost all $t$ with $Z(t)\ne 0$ " implicitly presupposes $\dot f(t)$ exists almost everywhere (a fact Lemma 8.3 supplies, not something to assume outright) — stating the hypothesis as a universally quantified implication over any witnessing derivative avoids smuggling in an unearned existence claim.

Formalization scope

Mission III's fluid-equation apparatus is restated locally (per that mission's own note that later chunks cannot import its draft), unmodified. Definition 8.1's two Lipschitz notions (bounded-set-wise and global) are formalized via Mathlib's own LipschitzOnWith, generalized over arbitrary (pseudo)metric domain and codomain types so the same definition serves $g:\mathbb{R}^d\to\mathbb{R}^m$ and $g:\mathbb{R}^m\to\mathbb{R}$ uniformly — reusing Mathlib substrate rather than restating Definition 8.1's $\varepsilon$ - $\delta$ inequality from scratch, per BRIEF.md's explicit recommendation. The upper-right Dini derivative is restated inline (Appendix A.4 is out of series scope) via Filter.limsup along the right-neighborhood filter, and is used throughout Lemma 8.11 in place of the ordinary derivative — using deriv instead would be a strictly stronger, unfaithful hypothesis. Lemma 8.10's pointwise maximum is a supremum over the finite index type Fin d, always a genuine maximum with no junk-value risk. Theorem 8.12 restates the tandem queueing network's already-reduced fluid equations (8.11)-(8.15) directly, since Figure 1.1 belongs to Chapter 1, outside this mission series. A formalization that replaced Lemma 8.11's Dini-derivative hypotheses with ordinary-derivative ones, or that dropped condition (b)'s local bound, would each be an unfaithful strengthening or a false generalization — both ruled out here. The Lyapunov extinction criteria (lyapunov_extinction_linear, lyapunov_extinction_sqrt, dini_extinction_criterion) are the primary reusable contributions, intended for direct reuse (matching shape, since drafts do not import one another) by every later stability mission in the series; contributions completing the eight by sorry proofs are welcome.

Selected references

J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
L. Massoulié, "Structural properties of proportional fairness: stability and insensitivity," Annals of Applied Probability 17 (2007), 809–839.
J. G. Dai, "On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models," Annals of Applied Probability 5 (1995), 49–77.

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Dynamic Programming Operations Research Probability·Captain: mikedeng1

On the Optimality of Generalized (s, S) Policies: A Generalized (s, S) Policy Is Optimal in Every Period of the Finite-Horizon Inventory ProblemResearch Paper

Motivation

In a periodic-review inventory system a manager observes the stock level before ordering, decides how much to order, and then faces random demand. When ordering costs a fixed setup charge plus a constant price per unit, Scarf (1960) proved that an $(s,S)$ policy is optimal in every period of a finite-horizon problem: order up to $S$ when the stock falls below $s$ , otherwise order nothing. Real ordering costs are often not of this form. Quantity discounts, a choice between production facilities with different setup and marginal costs, or a supplier whose price schedule falls with volume all give an ordering cost that is concave and increasing but not "setup plus linear". Karlin had analysed the single-period problem with such costs; Porteus (1971) gave the first multiperiod result with random demand.

Timeline:

Scarf (1960): $(s,S)$ optimality for setup-plus-linear ordering cost, via $K$ -convexity of the expected cost-to-go.
Veinott (1966): an alternative proof of $(s,S)$ optimality under different conditions (quasi-convex one-period costs).
Porteus (1971): for concave increasing ordering costs and demand with a one-sided Pólya density, a generalized $(s,S)$ policy is optimal in every period; when the cost is piecewise linear with $r$ pieces it is an $(s,S)_r$ policy with at most $r$ reorder levels.

Setting

The ordering cost $c : [0,\infty) \to \mathbb R$ is concave, nondecreasing, and $c(0) = 0$ . For $z > 0$ , $C_2(z)$ is the supporting line of $c$ at $z$ with the smallest intercept, written as a pair (slope, intercept) $(\kappa, K)$ . The set of slopes that occur is $C$ , and $K_\kappa$ is the intercept belonging to slope $\kappa \in C$ , so $c(z) = \min_{\kappa \in C}\{K_\kappa + \kappa z\}$ for $z > 0$ . The limits $(c_0, K_0) = \lim_{z \downarrow 0} C_2(z)$ and $(c_\infty, K_\infty) = \lim_{z\to\infty} C_2(z)$ are assumed to exist.

Demands in successive periods are i.i.d. with density $\varphi$ . A function $\varphi$ is $PF_n$ if $0 < \int\varphi < \infty$ and $\det[\varphi(x_i - t_j)]_{i,j\le k} \ge 0$ for all $k \le n$ and increasing $x_1<\dots<x_k$ , $t_1<\dots<t_k$ ; it is a one-sided Pólya density if it is $PF_n$ for every $n$ , integrates to $1$ and vanishes on $(-\infty,0)$ . Exponential and Erlang densities are examples.

With holding-and-shortage cost $m$ (PF-integrable, bounded below), terminal cost $f_0$ , discount factor $0 \le \alpha \le 1$ , and convolution $(f*\varphi)(y) = \int f(y-x)\varphi(x)\,dx$ , the value functions are

h_n = m * \varphi + \alpha\, f_{n-1} * \varphi, \qquad f_n(x) = \inf_{y \ge x}\{c(y-x) + h_n(y)\},

where $n$ counts the periods remaining. $Y_n(x)$ is the set of minimizers $S \ge x$ . A generalized $(s,S)$ policy is a function $y$ with $y(x) = x$ for $x \ge s$ and $y(z) \ge y(x) \ge S \ge s$ for $z < x < s$ : no order above $s$ , and below $s$ an order-up-to level that is at least $S$ and does not increase with the starting stock.

Two function classes carry the argument. $f$ is non- $K$ -decreasing on $X$ if $f(x) \le f(y) + K$ for $x \le y$ in $X$ . For $K \ge 0$ , $C_a(K)$ consists of the piecewise continuous, PF-integrable functions with $f(x)\to\infty$ as $|x|\to\infty$ that are nonincreasing on $(-\infty,a)$ or $(-\infty,a]$ and non- $K$ -decreasing on the rest of the line. $C(K)$ is its continuous part. With $G_{\kappa n} = \kappa\cdot + h_n$ , the assumptions A1–A5 of §VI tie $m$ and $f_0$ to $c_0$ , $c_\infty$ and the $K_\kappa$ .

Formalization targets

Goal: Theorem 3

Under the standing assumptions and A1–A5, for every $n \ge 1$ the convolution $f_{n-1}*\varphi$ exists and

\exists\, s, S,\ \exists\, y \text{ generalized } (s,S) \text{ policy}: \quad y(x) \in Y_n(x) \ \ \forall x \in \mathbb R.

The statement fixes no numbers: $s$ and $S$ depend on $n$ and on the data.

Milestones

Lemma 9: $\bigcup_a C_a(K)$ equals the class of quasi- $K$ -convex, piecewise continuous, PF-integrable functions tending to $\infty$ as $|x| \to \infty$ .
Lemma 1: every $f \in C(K)$ has reals $s \le S$ with $S$ a global minimizer, $f > f(S) + K$ on $(-\infty,s)$ , $f$ nonincreasing there, and $f$ non- $K$ -decreasing on $[s,\infty)$ .
Lemma 5: for continuous $g$ , $f(x) = \int_0^\infty g(x-t)\lambda e^{-\lambda t}dt$ is $C^1$ with $f' = \lambda(g-f)$ .
Lemma 6: $g*\varphi$ is continuous and $C^1$ off a finite set for a one-sided Pólya $\varphi$ .
Lemma 10 and Theorem 1: $f \in C_a(K) \Rightarrow f*\varphi \in C(K)$ , first for exponential $\varphi$ , then for every one-sided Pólya density.
Theorem 2: if every $G_{\kappa n} \in C(K_\kappa)$ and every $Y_n(x) \ne \emptyset$ , a generalized $(s,S)$ policy is optimal in period $n$ .
Lemma 2: $f_n(x) \le f_n(y) + c(y-x)$ for $x \le y$ .
Lemma 3: the inductive step producing the hypotheses of Theorem 2 from properties of $f_{n-1}$ .

Significance

The result extends $(s,S)$ -type structure from setup-plus-linear to arbitrary concave increasing ordering costs, which covers quantity discounts and multi-facility production. When $c$ is piecewise linear with $r$ pieces, the optimal policy is an $(s,S)_r$ policy described by at most $r$ reorder points and order-up-to levels. That is a finite-dimensional family, which makes computing policies tractable. The class $C(K)$ and its closure under Pólya convolution (Theorem 1) are statements about functions of one real variable, independent of the inventory model. Quasi- $K$ -convexity extends both $K$ -convexity and quasi-convexity (Lemma 8 of the paper).

The theorem is classical and proved on paper; no machine-checked version is known. Its appendix leaves several steps as "easily proved by contradiction", which a formal proof has to fill in. The platform has Bertsekas's $K$ -convex $(s,S)$ lemma (BertsekasDP.kconvex_sS_structure, a result about $K$ -convex rather than $C(K)$ functions), but no Pólya frequency functions, no quasi- $K$ -convexity, and no concave-cost inventory model.

Difficulty

The obvious route copies Scarf: show that the cost-to-go is $K$ -convex and that $K$ -convexity survives taking expectations. With a concave ordering cost there is no single $K$ , and the relevant functions $G_{\kappa n}$ are generally not $K_\kappa$ -convex. The weaker property that does hold, membership in $C(K_\kappa)$ , is not preserved by convolution with an arbitrary density. It is preserved by one-sided Pólya densities, and Theorem 1 is the step that shows this: exponential kernels come first (via the differential identity (23)), and the general case needs the Schoenberg representation of one-sided Pólya densities as limits of convolutions of exponentials. The second difficulty is combining the different slopes $\kappa \in C$ into one policy (Theorem 2). Separate $(s,S)$ pairs for each $\kappa$ do not by themselves give a monotone policy.

Formalization scope

All functions are ℝ → ℝ; the ordering cost is used only on $[0,\infty)$ . The demand density is a function, not a measure; convolution is the Lebesgue integral over $\mathbb R$ . $PF_n$ uses Matrix.det over Fin k. The value functions are defined by structural recursion on $n : \mathbb N$ with $f_0$ the terminal cost; $h_n$ is used for $n \ge 1$ . $Y_n(x)$ is defined by the optimality inequality, never through the infimum. $G_{\kappa n}$ is defined by the paper's identity (7), $\kappa y + h_n(y)$ . $R^- = (-\infty,0)$ is open, and "increasing" is read as nondecreasing. Slopes in $C$ are written $\kappa$ to separate them from the cost function $c$ .

Added hypotheses and conventions:

$m$ piecewise continuous. The paper uses this without stating it (proof of Lemma 3). It is a hypothesis of Lemma 3 and Theorem 3.
Real-valued $f_n$ in Lemma 2. Following the convention of §X, Lemma 2 assumes each infimum defining $f_n$ is over a set bounded below.
Measurability of $m$ and $f_0$ (§X) is implied by their piecewise continuity and is not stated separately.

Lean returns $0$ for an infimum over a set unbounded below and for the integral of a non-integrable function. The goal therefore concludes that $f_{n-1}*\varphi$ exists and that $Y_n(x)$ is nonempty (so the infimum is a minimum); it does not assume these. It also does not quantify over arbitrary functions satisfying a Bellman equation or over an arbitrary set-valued $Y$ . Everything is built from the data $(c, m, \varphi, \alpha, f_0)$ . The class $C(K)$ includes PF-integrability and coercivity, without which Lemma 1 fails.

Welcome contributions: Pólya frequency functions and the exponential special cases (the exponential density is $PF_\infty$ ), Leibniz-rule lemmas for exponential kernels, the theory of $C(K)$ and quasi- $K$ -convex functions (reusable for other inventory models), and a formal Schoenberg representation (Theorem 6 of the paper, cited there and needed for Theorem 1). Theorems 4 and 5 (nonstationary and partial-backlogging extensions) are not part of this mission.

Selected references

E. L. Porteus, On the Optimality of Generalized (s, S) Policies, Management Science 17(7):411–426, 1971. https://doi.org/10.1287/mnsc.17.7.411
H. Scarf, The Optimality of (S, s) Policies in the Dynamic Inventory Problem, in Mathematical Methods in the Social Sciences, Stanford University Press, 1960.
A. F. Veinott Jr., On the Optimality of (s, S) Inventory Policies: New Conditions and a New Proof, SIAM Journal on Applied Mathematics 14(5):1067–1083, 1966. https://doi.org/10.1137/0114086
I. J. Schoenberg, On Pólya Frequency Functions I. The Totally Positive Functions and their Laplace Transforms, Journal d'Analyse Mathématique 1:331–374, 1951. https://doi.org/10.1007/BF02790092
S. Karlin, Total Positivity, Volume 1, Stanford University Press, 1968.

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Convex Optimization Functional Analysis Operations Research+1·Captain: mikedeng1

A Dynamical Approach to an Inertial Forward-Backward Algorithm for Convex Minimization: The IFB Iterates Minimize the Objective and Converge Weakly to a MinimizerResearch Paper

Motivation

Many problems in signal processing, statistics and operations research ask to minimize a sum $\Theta = \Phi + \Psi$ of a nonsmooth convex term $\Phi$ (a constraint indicator, an $\ell^1$ penalty) and a smooth term $\Psi$ . The standard method is the forward-backward (proximal-gradient) algorithm: an explicit gradient step on $\Psi$ followed by a proximal step on $\Phi$ . Its classical convergence theory requires $\Psi$ to be convex and the step size to stay below $2/L_\Psi$ , where $L_\Psi$ is the Lipschitz constant of $\nabla\Psi$ .

Attouch, Peypouquet and Redont (authors' manuscript of SIAM J. Optim. 24 (2014)) derive an inertial forward-backward algorithm (IFB) as a time discretization of a second-order dissipative dynamical system with Hessian-driven damping. The added inertial terms cost essentially nothing to compute, yet they allow step sizes beyond $2/L_\Psi$ and a smooth part $\Psi$ that is not convex, provided the sum $\Theta$ is.

Timeline of the relevant results:

Heavy-ball-with-friction methods, the inertial discretizations of $\ddot u + \alpha\dot u + \nabla\Phi(u) = 0$ , were introduced by Polyak (1964) and developed by Alvarez and Attouch (2001) for proximal schemes.
Hessian-driven damping for one potential: Alvarez, Attouch, Bolte and Redont (2002); for a nonsmooth potential plus a smooth one, the continuous dynamics underlying (IFB): Attouch, Maingé and Redont (2012).
The discrete algorithm (IFB) and its weak convergence in Hilbert spaces: Attouch, Peypouquet and Redont (2014), the paper of this mission.

Setting

Let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ . Let $\Phi : H \to \mathbb R\cup\{+\infty\}$ and $\Psi : H \to \mathbb R$ , and write $\Theta = \Phi + \Psi$ and $\mathcal S = \operatorname{Argmin}\Theta$ . A vector $g$ is a subgradient of $\Phi$ at $u$ , written $g \in \partial\Phi(u)$ , if $\Phi(u) < +\infty$ and $\Phi(u) + \langle g, v - u\rangle \le \Phi(v)$ for all $v$ .

Hypothesis H fixes positive constants $L_\Psi$ , $a$ , $b$ and a step size $\lambda$ with:

$H_\Phi$ : $\Phi$ is proper, lower semicontinuous and convex;
$H_\Psi$ : $\Psi$ is differentiable and $\nabla\Psi$ is $L_\Psi$ -Lipschitz;
$H_\lambda$ : $0 < \lambda < \Lambda = \min\{1/a,\ 2(a+b)/(bL_\Psi)\}$ ;
$H_\Theta$ : $\Theta$ is convex and bounded from below.

Algorithm (IFB). From any $(u_0, y_0) \in H \times H$ , compute for $k \ge 0$

0 \in \frac{u_{k+1}-u_k}{\lambda} + \partial\Phi(u_{k+1}) + a u_k - b y_k, \qquad 0 = \frac{y_{k+1}-y_k}{\lambda} + \nabla\Psi(u_{k+1}) - a u_{k+1} + b y_{k+1}.

A sequence $u_k$ converges weakly to $p$ , written $u_k \rightharpoonup p$ , if $\langle u_k, v\rangle \to \langle p, v\rangle$ for every $v \in H$ . The analysis uses the velocity $\xi_k = u_k - u_{k-1}$ , the energy $E_k = \Theta(u_k) + \gamma\|\xi_k\|^2$ with $\gamma = (1-a\lambda)/(2b\lambda^2)$ , and two auxiliary real sequences $G_k(q)$ and $F_k(q)$ , defined from $u$ , $y$ and a reference point $q$ by (17) and (18) of the paper.

Formalization targets

Goal: Theorem 1

Under Hypothesis H, for every sequence generated by (IFB),

\lim_{k\to\infty}\Theta(u_k) = \inf\Theta;

if $\mathcal S \ne \emptyset$ and one of (i) $\mathcal S$ is a singleton, (ii) $\Phi$ is differentiable with weak-to-weak sequentially continuous gradient, (iii) $\nabla\Psi$ is weak-to-weak sequentially continuous, (iv) $\Psi$ is convex, holds, then

u_k \rightharpoonup p \quad\text{for some } p \in \mathcal S;

and if $\mathcal S = \emptyset$ , then $\|u_k\| \to +\infty$ .

Milestones

In the order of the paper's argument: Proposition 2 (energy decrease, $\sum\|\xi_k\|^2 < \infty$ ); Proposition 3 (the identity and inequality (19) for $F_k(q)$ ); Proposition 4 ( $G_k(q)$ is bounded above for $q\in\operatorname{dom}\Phi$ ); the lower bound (28) on $F_k(q)$ ; Lemma 5 (a real-sequence lemma); Proposition 6 ( $\Theta(u_k)\to\inf\Theta$ , weak cluster points lie in $\mathcal S$ ); Lemma 7 (a boundedness lemma); Proposition 8 (boundedness of $(u_k)$ and convergence of $F_k(q)$ when $\mathcal S \ne\emptyset$ ).

Significance

The result gives convergence of a forward-backward type method under a step-size bound $\Lambda$ that can be made arbitrarily large by choosing $a$ small, and for a smooth term $\Psi$ that need not be convex. The special case $\Phi = \delta_C$ (indicator of a closed convex set) yields an inertial gradient-projection method, and the paper applies the theorem to feasibility problems, the CQ algorithm, Pareto fronts and $\ell^1$ signal recovery.

The theorem is proved in the paper; to our knowledge it is not formalized anywhere. A complete formalization would provide a machine-checked Liapunov analysis of an inertial proximal method in an infinite-dimensional Hilbert space, including the passage from a minimizing sequence to weak convergence. The milestones Proposition 2, 3 and 8 are energy estimates that also underlie other inertial and proximal schemes.

Difficulty

The energy $E_k$ controls the values $\Theta(u_k)$ and the velocities $\xi_k$ , but not the iterates themselves. The first idea, proving that $\|u_k - q\|$ is nonincreasing for every $q \in \mathcal S$ (Fejér monotonicity, the standard route for the classical forward-backward method), does not come out of the energy estimates for (IFB): the inertial variable $y_k$ couples consecutive steps, and the distance to a minimizer is not a Liapunov function. The paper's replacement, the sequence $F_k(q)$ , carries the auxiliary sum $G_k(q)$ , whose upper bound already requires the full Hypothesis H. Because $H$ is infinite-dimensional, bounded sequences have only weakly convergent subsequences, so the minimizing property has to pass through weak lower semicontinuity of $\Theta$ , and the final step, uniqueness of the weak cluster point, needs a separate argument in each of the cases (i)–(iv) together with Opial's lemma, which is not in Mathlib.

Formalization scope

$H$ is a general real Hilbert space (InnerProductSpace ℝ H with CompleteSpace H); nothing is specialized to finite dimension.
$\Phi$ and $\Theta$ take values in EReal. Properness excludes $-\infty$ . Convexity of an extended-valued function is convexity of its epigraph in $H\times\mathbb R$ , because Mathlib's ConvexOn needs a scalar action that EReal lacks. The subgradient predicate requires $\Phi(u) < +\infty$ , so $\partial\Phi(u) = \emptyset$ off $\operatorname{dom}\Phi$ .
(IFB) is encoded as the subgradient inclusion, not through the proximity operator. Weak convergence is convergence of all inner products $\langle u_k, v\rangle$ ; weak cluster points are weak limits along strictly increasing subsequences.
$L_\Psi > 0$ is assumed (harmless: a Lipschitz gradient is Lipschitz with every larger constant). The step size $\lambda$ is written lam.
$\xi_k$ , $z_k$ , $E_k$ , $G_k$ , $F_k$ are defined on all natural indices, and each statement quantifies $k \ge 1$ (or $k \ge 2$ for $G_k$ ) as the paper does. Energies and values are never truncated to real numbers, so no statement becomes true through the convention EReal.toReal ⊤ = 0.
Deviation from the printed text: Proposition 4 is printed under $H_\Phi$ and $H_\Psi$ only, but its proof invokes Proposition 2, which needs all of Hypothesis H, and the printed statement is false for step sizes above $\Lambda$ (e.g. $H=\mathbb R$ , $\Phi = x^2/2$ , $\Psi = 3x^2/2$ , $a=2$ , $b=0.02$ , $\lambda = 10$ ). The milestone is stated under the full Hypothesis H.
A formalization in which $\partial\Phi(u)$ is nonempty at points of infinite value, in which the energy is converted to a real number, or in which Hypothesis H cannot be satisfied, would make these statements trivial or vacuous, and is ruled out by the definitions above.

A complete development needs Opial's lemma, weak sequential compactness of bounded sets in Hilbert space, weak lower semicontinuity of lower semicontinuous convex functions, the descent lemma for functions with Lipschitz gradient, and monotonicity of the subdifferential. These are reusable well beyond this mission; contributions of any of them, and of the milestones in any order, are welcome.

Selected references

H. Attouch, J. Peypouquet, P. Redont, A Dynamical Approach to an Inertial Forward-Backward Algorithm for Convex Minimization, SIAM J. Optim. 24(1), 2014 (statements cited from the authors' manuscript of Aug 2013). https://doi.org/10.1137/130910294
H. Attouch, P.-E. Maingé, P. Redont, A second-order differential system with Hessian-driven damping; application to non-elastic shock laws, Differential Equations and Applications 4(1), 2012. https://doi.org/10.7153/dea-04-02
F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Analysis 9, 2001. https://doi.org/10.1023/A:1011253113155
F. Alvarez, H. Attouch, J. Bolte, P. Redont, A second-order gradient-like dissipative dynamical system with Hessian-driven damping, J. Math. Pures Appl. 81(8), 2002. https://doi.org/10.1016/S0021-7824(01)01253-3
B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys. 4(5), 1964. https://doi.org/10.1016/0041-5553(64)90137-5
Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73, 1967. https://doi.org/10.1090/S0002-9904-1967-11761-0

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Combinatorics Theoretical Computer Science·Captain: mikedeng1

Three Partition Refinement Algorithms 1: Distinguishing-Prefix Refinement Finds Every Distinguishing Prefix Within m′ Refinement StepsResearch Paper

Motivation

Sorting a collection of strings in lexicographic order is a basic subroutine in compilers, string indexing, suffix sorting and the construction of tries. The classical method, due to Aho, Hopcroft and Ullman (The Design and Analysis of Computer Algorithms, 1974), is a multipass radix sort that processes the strings from the last position to the first and runs in time proportional to the total length $m$ of the input plus the alphabet size $k$ . Mehlhorn showed that a straightforward first-to-last scan of equal-length strings needs $\Omega(km)$ time.

Paige and Tarjan (Three Partition Refinement Algorithms, SIAM J. Comput. 16(6):973–989, 1987, doi:10.1137/0216062) observed that most of the input is usually irrelevant to the sorted order: only the shortest prefix of each string that tells it apart from the others matters. Their algorithm separates the problem into two steps: (1) find this distinguishing prefix of every string, and (2) sort the distinguishing prefixes. The first step is carried out by partition refinement, the same technique that the paper then applies to the relational coarsest partition problem (the second mission of this series) and to double lexical ordering. This mission formalizes the correctness and the termination bound of the first step.

Setting

Fix $k \ge 1$ and the alphabet $\Sigma = \{1, 2, \dots, k\}$ , together with an end marker $0$ that is smaller than every symbol. A string is a finite sequence over $\Sigma \cup \{0\}$ ; $\lambda$ is the empty string, $|x|$ is the length of $x$ , and $x(i)$ is its $i$ -th symbol (positions start at $1$ ). A string $\alpha$ is a prefix of $y$ if $y = \alpha z$ for some string $z$ ; every string is a prefix of itself. $\Sigma^*0$ is the set of strings over $\Sigma$ followed by one $0$ .

The input is a multiset $U = \{x_1, \dots, x_n\} \subseteq \Sigma^*0$ with $n \ge 1$ ; repeated strings are allowed. The distinguishing prefix $x'_i$ of $x_i$ in $U$ is

the shortest prefix of $x_i$ that is not a prefix of any other string of $U$ , if $x_i$ occurs only once in $U$ ;
$x_i$ itself, if $x_i$ occurs more than once.

Write $m' = \sum_{i=1}^n |x'_i|$ .

For a string $\alpha$ , the labeled block $B_\alpha$ is the multiset of strings of $U$ having $\alpha$ as a prefix, and $\alpha$ is its associated prefix. The block is finished if $\alpha = x'_i$ for some $i$ , and unfinished otherwise. A state $P$ is a set of labeled blocks. For an unfinished block $B_\alpha \in P$ ,

\mathrm{split}(B_\alpha, P) = \bigl(P \setminus \{B_\alpha\}\bigr) \cup \{\, B_{\alpha u} : u \in \Sigma \cup \{0\},\ \exists x \in B_\alpha,\ x(|\alpha| + 1) = u \,\}.

The refinement algorithm starts from $P_0 = \{B_\lambda\}$ and repeatedly applies the step Refine: pick any unfinished block $B_\alpha \in P$ and replace $P$ by $\mathrm{split}(B_\alpha, P)$ . A run with $K$ steps is a sequence $P_0, P_1, \dots, P_K$ produced this way; any choice of unfinished block is allowed at every step.

Formalization targets

Goal: Theorem 1 with its explicit bound

For every run $P_0, \dots, P_K$ of the refinement algorithm,

K \le m' \qquad\text{and}\qquad \bigl(\text{no block of } P_K \text{ is unfinished}\bigr) \implies P_K = \{B_{x'_1}, \dots, B_{x'_n}\}.

The paper states Theorem 1 as "The algorithm terminates and is correct"; the bound $K \le m'$ is the explicit count proved in the last sentence of its proof (p. 975). The goal leaves the choice of unfinished block free, so it holds for every refinement order.

Milestones

End markers (§2, p. 974). No string of $U$ is a proper prefix of another string of $U$ .
Lemma 1 (p. 975). Along every run, every finished block $B_\beta$ is contained in a block of the current state, and there is $B_\alpha \in P_j$ with $\alpha$ a prefix of $\beta$ .
Proof of Theorem 1, second sentence. For every state of every run, $\sum_{B_\alpha \in P_j} |\alpha| \le m'$ .
Proof of Theorem 1, third sentence. Every Refine step strictly increases $\sum_{B_\alpha \in P} |\alpha|$ .

A supplementary item states the fact on p. 974 that motivates the two-step design: distinct strings have distinct distinguishing prefixes, and $x_i \le x_j$ iff $x'_i \le x'_j$ in lexicographic order.

Significance

Theorem 1 is the correctness half of the lexicographic sorting algorithm: once the distinguishing prefixes are known, sorting $U$ reduces to sorting strings of total length $m'$ , which is how the paper obtains its $O(m' + k)$ time bound in place of $O(m + k)$ . The paper notes that under a natural probability model $m'$ is of order $n \log_k n$ , so the gain is large whenever the strings are long. The invariant of Lemma 1 is also the pattern on which the later partition refinement algorithms of the paper are built: a target partition is shown to refine every intermediate partition, and a potential function bounds the number of refinement steps.

The result has been proved since 1987. What this mission adds is a machine-checked account of the algorithm at the level of its abstract refinement steps, with the explicit bound $m'$ rather than an asymptotic statement, and with the standing assumptions of §2 (non-empty input, end markers) made explicit. No formal proof of this algorithm is present in Mathlib or on the platform.

Difficulty

The obvious argument, "each step makes the labels longer, and labels never grow past the distinguishing prefixes", needs two facts that are not immediate from the definitions. First, the labels of a reached state must form a partition of $U$ into non-empty blocks whose labels are pairwise incomparable under the prefix order; this is an invariant of the algorithm, not part of the definition of a state, and it fails for arbitrary sets of labels. Second, the label of an unfinished block must be a proper prefix of every finished label below it, which rests on the end marker: without end markers a string can be a proper prefix of another, a block can be unfinished yet have no strictly longer children, and the algorithm can stall. Bounding the sum of label lengths by $m'$ is not a label-by-label comparison: a state can have fewer labels than strings, and the distinguishing prefixes of different strings can share a label as a common prefix.

Formalization scope

Strings are List (Fin (k + 1)), with 0 : Fin (k + 1) the end marker and prefix the Mathlib relation <+:. The multiset $U$ is a family x : Fin n → List (Fin (k + 1)), so repetitions are distinct indices. The hypotheses 0 < n and EndMarked x ( $U \subseteq \Sigma^*0$ ) are the paper's standing assumptions. A block is identified by its label, not by its set of members: two different labels can carry the same strings (for instance $B_\lambda = B_1$ when every string begins with $1$ ), and they are different blocks. A state is a Finset of labels; a run is a map Fin (K + 1) → Finset (List (Fin (k + 1))) starting at {[]}. Positions are 1-based in the paper and 0-based in Lean, so $x(|\alpha|+1)$ is (x i)[α.length]?. The distinguishing prefix follows the literal definition; for $n = 1$ it is the empty string.

The running times $O(m' + k)$ and $O(n + k)$ space, the implementation with a queue and a global index, and step two (sorting the prefixes via the refinement tree) are RAM-model statements and are not formalized. Only the explicit step count $m'$ is. A formalization in which split added all $k + 1$ children, added none, or allowed refining a finished block would change the theorem (the bound fails or the goal becomes vacuous); the definitions add exactly the children realized by some string of the block and refine only unfinished blocks.

Contributions welcome: proofs of the milestones, a general API for prefix-closed partition refinement on List, and a proof of the order-preservation item via Mathlib's List.Lex.

Selected references

R. Paige, R. E. Tarjan, Three Partition Refinement Algorithms, SIAM Journal on Computing 16(6):973–989, 1987. https://doi.org/10.1137/0216062
A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.
K. Mehlhorn, Data Structures and Algorithms 1: Sorting and Searching, Springer, 1984. https://doi.org/10.1007/978-3-642-69672-5

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Combinatorics Graph Theory Group Theory+2·Captain: mikedeng1

λ1, Isoperimetric Inequalities for Graphs, and Superconcentrators 2: Cayley Graphs of Finite Quotients of a Property (T) Group Are Linear EnlargersResearch Paper

Motivation

An expander is a sparse graph in which every set of vertices has many neighbours outside itself. Expanders are the building blocks of superconcentrators (sparse directed graphs that route any $r$ inputs to any $r$ outputs along vertex-disjoint paths), of sorting and switching networks, and of many constructions in complexity theory and coding; the survey of Hoory, Linial and Wigderson (Bull. AMS 2006) describes these uses. Random regular graphs are expanders with high probability, but applications need explicit families with fixed degree and a uniform expansion constant.

Alon and Milman (J. Combin. Theory Ser. B 38 (1985)) replace combinatorial expansion by a spectral quantity, the second-smallest eigenvalue $\lambda_1$ of the matrix $Q = D - A$ of a graph, which Fiedler called the algebraic connectivity (Czech. Math. J. 1973). Their Theorem 4.3 shows that a regular graph with $\lambda_1$ bounded away from $0$ yields an expander. This mission formalizes their Section 4 source of such graphs: Cayley graphs of the finite quotients of a group with Kazhdan's property (T).

Timeline:

1967: Kazhdan introduces property (T) and proves that $SL(n,\mathbb{Z})$ , $n \ge 3$ , has it (Funct. Anal. Appl. 1 (1967)).
1973: Margulis uses property (T) to give the first explicit expander family (Probl. Inf. Transm. 9 (1973)).
1981: Gabber and Galil give a variant of Margulis' construction with an explicit expansion constant, proved by Fourier analysis (J. Comput. Syst. Sci. 22 (1981)).
1985: Alon and Milman state the construction in terms of $\lambda_1$ (Lemma 4.8, Theorem 4.9) and link it to the concentration property of Section 2 of their paper.

Setting

Let $T$ be a finite group. A finite multigraph on $T$ is a symmetric matrix $M = (M_{w,u})$ of nonnegative integers, $M_{w,u}$ being the number of edges joining $w$ and $u$ (diagonal entries count loops). It is $k$ -regular if every row sums to $k$ . Its matrix is $Q = \operatorname{diag}(d(v)) - M$ with $d(v) = \sum_u M_{v,u}$ , a real symmetric positive semidefinite matrix. Its eigenvalues, repeated according to multiplicity, are $0 = \lambda_0 \le \lambda_1 \le \dots \le \lambda_{|T|-1}$ , and $\lambda_1(G)$ denotes the second of them.

Let $H$ be a group, $S \subseteq H$ a finite set with $S = S^{-1}$ , and $\phi : H \to T$ a homomorphism onto $T$ . The Cayley multigraph $G(T, \phi(S))$ joins $w$ and $u$ by as many edges as there are $s \in S$ with $w u^{-1} = \phi(s)$ . It is $|S|$ -regular, and its matrix is $Q = |S| \cdot I - \sum_{s\in S} \pi(\phi(s))$ , where $\pi(t)$ is the permutation matrix of the left regular representation, $(\pi(t))_{w,u} = 1$ iff $w u^{-1} = t$ .

An $(n,k,\varepsilon)$ -enlarger (Definition 4.1) is a $k$ -regular graph on $n$ vertices with $\lambda_1 \ge \varepsilon$ .

A unitary representation $\pi$ of $H$ in a complex Hilbert space $V$ is essentially nontrivial (Definition 4.5) if no nonzero vector is fixed by every $\pi(h)$ . A discrete group $H$ has property (T) (Definition 4.6) if there are $\varepsilon > 0$ and a finite $K \subseteq H$ such that for every essentially nontrivial unitary representation $\pi$ and every unit vector $y$ some $h \in K$ satisfies $|(\pi(h)y, y)| < 1 - \varepsilon$ .

Formalization targets

Goal: Theorem 4.9

Let $H$ have property (T), let $S$ be a finite generating set of $H$ with $S = S^{-1}$ , and let $\phi_i : H \to T_i$ be surjective homomorphisms onto finite groups with $|T_i| \to \infty$ . Then there is one $\varepsilon > 0$ with

G(T_i, \phi_i(S)) \text{ is a } (|T_i|,\ |S|,\ \varepsilon)\text{-enlarger for every } i \text{ with } |T_i| \ge 2 .

The constant is unspecified: the theorem asserts uniformity in $i$ , not a value.

Milestones, in the order the paper's argument uses them

Lemma 4.7. For any generating set $S$ of a property (T) group there is $\varepsilon > 0$ such that every essentially nontrivial unitary representation and every unit vector $y$ admit $s \in S$ with $|(\pi(s)y,y)| < 1-\varepsilon$ .
Proof of Lemma 4.8, essential nontriviality. For $\phi$ onto $T$ , every nonzero vector of $W = \{v : \sum_t v_t = 0\}$ is moved by some $\pi(\phi(h))$ .
Proof of Lemma 4.8, Rayleigh's principle. For the Cayley multigraph, $\min\{(Qy,y) : y \in W,\ \|y\| = 1\} = \lambda_1(G)$ .
Lemma 4.8. With $H$ , $S$ and $\varepsilon$ as in Lemma 4.7, $S$ finite and $S = S^{-1}$ , and $\phi$ onto a finite group $T$ , the Cayley graph $G(T,\phi(S))$ is a $(|T|, |S|, \varepsilon)$ -enlarger.

Significance

Theorem 4.9 turns an analytic property of one infinite group into a uniform spectral bound for infinitely many finite graphs of fixed degree. With Theorem 4.3 of the paper it produces explicit families of linear expanders, hence of linear superconcentrators; for $H = SL(n,\mathbb{Z})$ , $n \ge 3$ , and its reductions modulo $i$ the paper obtains infinitely many explicit families of $(n, 4, \varepsilon)$ -enlargers. The same mechanism underlies later work on expanders from groups, surveyed in Lubotzky's monograph (Birkhäuser 1994).

The result is proved in the paper, modulo Lemma 4.7, which the paper refers to Margulis for. The formalization adds a checked account of every step, including Lemma 4.7 itself. At the pinned Mathlib revision there is no notion of property (T), of Kazhdan constants, or of the algebraic connectivity of a multigraph, and no machine-checked version of Theorem 4.9 is known on this platform.

Difficulty

The combinatorial and linear-algebra steps are routine; the substance is in two places. First, Lemma 4.7: Definition 4.6 supplies a constant for one finite set $K$ , nothing in the definition relates $K$ to a given generating set $S$ , the paper gives no proof, and the standard references state the textbook form $\|\pi(h)y - y\| \ge \varepsilon$ rather than the paper's absolute-value form $|(\pi(h)y,y)| < 1-\varepsilon$ . Second, the uniformity: a spectral gap for each fixed quotient is easy, since a connected graph has $\lambda_1 > 0$ , but a bound that does not decay as $|T_i| \to \infty$ is exactly what cannot come from any finite computation and must come from property (T) through Lemma 4.8. The Rayleigh quotient of milestone 3 is taken over real vectors, while Lemma 4.7 is stated for complex Hilbert spaces.

Formalization scope

Groups are Lean types with a Group instance; the finite groups $T$ carry Fintype and DecidableEq. Graphs are multigraphs given by symmetric matrices Matrix T T ℕ, with loops allowed. This matters: when $\phi$ identifies two generators or sends one to the identity, the degree is still $|S|$ , and a loop contributes $0$ to $Q$ . Mathlib's SimpleGraph Cayley graph forgets these multiplicities and is not used. $\lambda_1$ is the second-smallest eigenvalue with multiplicity of the real symmetric matrix $Q$ (via Matrix.IsHermitian.eigenvalues₀). It is defined spectrally, as in the paper, and not as a Rayleigh minimum. It is only meaningful for $|T| \ge 2$ , and the goal excludes trivial quotients explicitly. Unitary representations are homomorphisms into the unitary group of bounded operators on a complex Hilbert space in universe Type. Compact subsets of a discrete group are finite sets.

A trivializing formalization is ruled out. Property (T) is not replaced by the hypothesis that the regular representations of the quotients have no almost-invariant vectors, which would make Theorem 4.9 a restatement of its hypothesis. And $\lambda_1$ is not defined as the minimum of $(Qy,y)$ over zero-sum unit vectors, which would make milestone 3 true by definition.

A complete development needs basic Kazhdan-constant manipulations, Courant–Fischer for real symmetric matrices, and the regular representation of a finite group as a unitary representation. The regularity of Cayley multigraphs and the identity $Q = |S| I - \sum_s \pi(\phi(s))$ are short. The spectral and representation-theoretic lemmas are reusable beyond this mission. Contributions of intermediate lemmas, such as the variational characterization of eigenvalues₀ or the invariance of the zero-sum subspace, are welcome.

Selected references

N. Alon, V. D. Milman, λ1, Isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985) 73–88. https://doi.org/10.1016/0095-8956(85)90092-9
D. A. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1967) 63–65. https://doi.org/10.1007/BF01075866
G. A. Margulis, Explicit constructions of concentrators, Probl. Inf. Transm. 9 (1973) 325–332. http://mi.mathnet.ru/ppi1162
O. Gabber, Z. Galil, Explicit constructions of linear-sized superconcentrators, J. Comput. Syst. Sci. 22 (1981) 407–420. https://doi.org/10.1016/0022-0000(81)90040-4
M. Fiedler, Algebraic connectivity of graphs, Czech. Math. J. 23 (1973) 298–305. https://doi.org/10.21136/CMJ.1973.101168
A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser, 1994. https://doi.org/10.1007/978-3-0346-0332-4
B. Bekka, P. de la Harpe, A. Valette, Kazhdan's Property (T), Cambridge University Press, 2008. https://doi.org/10.1017/CBO9780511542749
S. Hoory, N. Linial, A. Wigderson, Expander graphs and their applications, Bull. AMS 43 (2006) 439–561. https://doi.org/10.1090/S0273-0979-06-01126-8

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Convex Optimization Operations Research Optimization+1·Captain: mikedeng1

Robust Control of Markov Decision Processes with Uncertain Transition Matrices 4: The Dual of the Worst-Case Expectation over a Kullback-Leibler BallResearch Paper

Motivation

A robust Markov decision process replaces the unknown transition probabilities of an MDP by sets of plausible values and optimises against the worst case. Nilim and El Ghaoui (Oper. Res. 53 (2005)) showed that, when the uncertainty is rectangular (each row of each transition matrix varies independently in its own set), the robust problem is solved by a Bellman-type recursion. Each step of that recursion needs, for every state and action, the value of an inner problem: the largest expectation of the next-stage value vector over the uncertainty set of one transition row. The recursion is only as tractable as this inner problem.

The paper studies several uncertainty models built from statistical estimates of the transition rows. In the entropy model the uncertain row is any distribution within a prescribed Kullback–Leibler divergence of a nominal distribution. For this model the paper reduces the inner problem to the minimisation of a scalar convex function, which is then solved by bisection. Iyengar (Math. Oper. Res. 30 (2005)) obtained the same robust recursion independently, and the same scalar reduction is the basic computation in later KL-constrained distributionally robust optimisation. This mission formalizes that reduction and the properties of the scalar function that the paper derives from it.

Setting

Let $n\ge 1$ and let $\Delta_n=\{p\in\mathbb R^n : p\ge 0,\ \sum_j p(j)=1\}$ be the probability simplex. For $p,q\in\mathbb R^n$ the Kullback–Leibler divergence is

D(p\|q)=\sum_j p(j)\log\frac{p(j)}{q(j)},

with $0\log 0=0$ . Fix a nominal distribution $q\in\Delta_n$ with $q(j)>0$ for every $j$ , and a level $\beta>0$ . The entropy uncertainty set is

\mathcal P=\{p\in\Delta_n : D(p\|q)\le\beta\}.

For a vector $v\in\mathbb R^n$ (in the MDP, the value function of the next stage), the inner problem (17) is

\sigma_{\mathcal P}(v)=\max_{p\in\mathcal P} p^{\mathsf T}v .

The paper's scalar dual function (47) is, for $\lambda>0$ ,

\sigma(\lambda)=\lambda\log\Big(\sum_j q(j)\,e^{v(j)/\lambda}\Big)+\beta\lambda .

Write $v_{\max}=\max_j v(j)$ and $Q(v)=\sum_{j:\,v(j)=v_{\max}}q(j)$ , the $q$ -mass of the maximisers of $v$ . The tilted distribution at $\lambda>0$ is $p^*(j)=q(j)e^{v(j)/\lambda}/\sum_i q(i)e^{v(i)/\lambda}$ .

In Lean, vectors are Fin n → ℝ, $\Delta_n$ is stdSimplex ℝ (Fin n), and $D$ , $\mathcal P$ , $\sigma$ , $p^*$ , $v_{\max}$ , $Q(v)$ are klDiv, klBall, dualFn, tiltedDist, vmax, maxMass in the namespace RobustMDP.EntropyInner.

Formalization targets

Goal: the dual of the inner problem (§6.2, Eq. (47), p. 791)

\max_{p\in\mathcal P} p^{\mathsf T}v=\inf_{\lambda>0}\sigma(\lambda),

with the maximum attained. This is kl_ball_inner_problem_dual. It holds for every $n$ , every $v$ , every $q>0$ in $\Delta_n$ and every $\beta>0$ .

Milestones

§6.1: $\max_{p\in\Delta_n}D(p\|q)=\max_i(-\log q_i)$ , and for $\beta\ge\max_i(-\log q_i)$ the set $\mathcal P$ is all of $\Delta_n$ and the inner value is $v_{\max}$ .
Eq. (48): $q^{\mathsf T}v+\beta\lambda\le\sigma(\lambda)\le v_{\max}+\beta\lambda$ for $\lambda>0$ .
§6.2, the optimal distribution: $p^{\mathsf T}v-\lambda D(p\|q)\le\lambda\log\sum_j q(j)e^{v(j)/\lambda}$ on $\Delta_n$ , with equality at $p^*$ .
§6.2, elimination of $\mu$ : $\min_{\mu}\big[\mu+\beta\lambda+\lambda\sum_j q(j)e^{(v(j)-\mu)/\lambda-1}\big]=\sigma(\lambda)$ .
Eq. (49): $\sigma(\lambda)=v_{\max}+(\beta+\log Q(v))\lambda+o(\lambda)$ as $\lambda\to0^+$ .
Eq. (50): $\sigma(\lambda)=q^{\mathsf T}v+\beta\lambda+o(1)$ as $\lambda\to\infty$ .
§6.3: if $\beta\ge-\log Q(v)$ , then $\inf_{\lambda>0}\sigma=v_{\max}$ and the inner value is $v_{\max}$ .

Significance

The goal turns an $n$ -dimensional optimisation over a nonpolyhedral convex set into a one-dimensional convex minimisation whose objective costs $O(n)$ to evaluate. Combined with the bisection bracket from (48) and the behaviour at $0$ from (49), it gives the paper's $O(n\log(v_{\max}/\delta))$ cost per inner problem (§6.4), and hence the per-step cost of the robust Bellman recursion under entropy uncertainty. Milestone 7 identifies exactly when the uncertainty set is large enough that the robust step ignores the nominal model; unlike the cruder threshold of milestone 1, it depends on $v$ .

The result is proved in the paper modulo "standard duality arguments". The paper gives no proof of the duality step itself, and its expansions (49)–(50) are proved only in outline in Appendix C. No formal proof of any of these statements is known to exist; Mathlib has the measure-theoretic Donsker–Varadhan ingredients but not the finite, constrained dual stated here. The mission produces a machine-checked version of the whole chain, with the attainment questions (which side is a max, which is only an infimum) settled explicitly.

Difficulty

The inequality $\max_{\mathcal P}p^{\mathsf T}v\le\sigma(\lambda)$ for every $\lambda>0$ is the routine half. The obstacle is the reverse inequality. The paper appeals to Lagrangian strong duality under a Slater condition, but the Lagrangian dual function equals $\sigma(\lambda)$ only for $\lambda>0$ ; at $\lambda=0$ it is $v_{\max}$ , and the dual infimum may be approached only as $\lambda\to0^+$ . A proof that looks for a minimiser $\lambda^*>0$ and a matching primal point $p^*$ fails in precisely the regime $\beta\ge-\log Q(v)$ of milestone 7, where no such $\lambda^*$ exists and the primal optimum sits on the face of the simplex spanned by the maximisers of $v$ . The strong-duality argument must also handle the boundary of $\Delta_n$ , where $D(\cdot\|q)$ is not differentiable.

Formalization scope

Vectors are Fin n → ℝ; $\Delta_n$ is stdSimplex ℝ (Fin n). $D(p\|q)$ is a local finite sum with Lean's $\log 0=0$ , which gives $0\log0=0$ ; Mathlib's measure-valued InformationTheory.klDiv is not used.
Standing hypotheses in every theorem: $q\in\Delta_n$ , $q(j)>0$ for all $j$ , and $\beta>0$ , as in §6.1. No restriction on $v$ is imposed; the "without loss of generality $v\ge0$ " of the paper's §5 is not assumed here.
The primal "max" is stated with IsGreatest (attained, since the KL ball is compact). The paper's " $\min_{\lambda>0}\sigma(\lambda)$ " is an infimum, stated with IsGLB over $\{\sigma(\lambda):\lambda>0\}$ : it is not attained when $\beta\ge-\log Q(v)$ .
dualFn is total in $\lambda$ and equals $0$ at $\lambda=0$ (division by zero), not the paper's $\sigma(0)=v_{\max}$ . Every statement uses $\lambda>0$ ; the value at $0$ appears as the one-sided limit of (49). Accordingly (48) is stated for $\lambda>0$ .
$v_{\max}$ is ⨆ j, v j, the attained maximum over the finite nonempty index set; $\max_i(-\log q_i)$ likewise.
(49) is stated as a limit along 𝓝[>] 0 together with a little-o remainder; (50) as a limit along atTop.
Milestone 7 uses the non-strict condition $\beta\ge-\log Q(v)$ of the paper's first sentence, which contains the strict version of its second.
Trivializing formalizations are excluded: the statements quantify over all $n$ , $v$ and $q$ , so a constant $v$ , $n=1$ , or the whole-simplex case of milestone 1 does not discharge the goal.

Useful infrastructure: a finite Gibbs variational inequality, compactness of the KL ball, and convexity and one-sided asymptotics of the log-sum-exp function in the temperature parameter. These are reusable for any KL-constrained robust optimisation mission. Proofs of the milestones, alternative proofs of the goal that avoid a general strong-duality theorem, and the sharper $O(\lambda e^{-t/\lambda})$ remainder of Appendix C are all welcome.

Selected references

A. Nilim and L. El Ghaoui, Robust Control of Markov Decision Processes with Uncertain Transition Matrices, Operations Research 53(5):780–798, 2005. https://doi.org/10.1287/opre.1050.0216
G. N. Iyengar, Robust Dynamic Programming, Mathematics of Operations Research 30(2):257–280, 2005. https://doi.org/10.1287/moor.1040.0129
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I, Communications on Pure and Applied Mathematics 28(1):1–47, 1975. https://doi.org/10.1002/cpa.3160280102

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Operations Research Probability Stochastic Systems·Captain: Shuze Chen

Processing Networks II: Subcriticality is Necessary for StabilityTextbook

Motivation

Before a queueing network's stability can be studied in any depth, a much cruder question has to be settled: is stability even possible for the given arrival rates and service capacities, under any control policy at all? For a single M/M/1 queue the answer is the familiar $\lambda < \mu$ , but a general stochastic processing network (SPN) — many buffers, many activities, servers that can be pooled or shared across job classes — has no single scalar "utilization" to compare against a threshold. J. G. Dai and J. Michael Harrison's Processing Networks: Fluid Models and Stability (Cambridge University Press, forthcoming; cited here from the authors' pre-publication draft, 2020-4-2, http://spnbook.org) answers this with a linear program: the static planning problem, first formulated by Harrison (2000). This mission formalizes the theorem that answers the crude question in one direction — no control policy can stabilize a network outside the region that program identifies — which is why, as the book puts it, "throughout the remainder of this book, attention is essentially restricted to subcritical networks."

Setting

An SPN has $I$ buffers, indexed by $i \in \mathcal{I}$ , and $J$ activities, indexed by $j \in \mathcal{J}$ . Its first-order data — the quantities that matter for a capacity calculation, as opposed to full stochastic detail — are: the $I \times J$ material requirement matrix $B$ ( $B_{ij}$ = number of class- $i$ items one type- $j$ service consumes), the $I \times J$ mean output matrix $\Gamma$ (its $j$ th column is the expected output vector of a type- $j$ service), the mean service times $m_j > 0$ , the $K \times J$ capacity consumption matrix $A$ (server pool $k$ against activity $j$ ), and the server-pool capacities $b \in \mathbb{R}_+^K$ . From these,

R := (B - \Gamma)M^{-1}, \qquad M := \operatorname{diag}(m_1, \dots, m_J),

so that $R_{ij}$ is the long-run average rate at which activity $j$ depletes buffer $i$ 's content.

Given an arrival-rate vector $\lambda \in \mathbb{R}_+^I$ , the static planning problem (SPP) is the linear program

\gamma^\ast(\lambda) := \min_{x \ge 0,\, \gamma} \ \gamma \quad \text{s.t.} \quad Rx = \lambda, \quad Ax \le \gamma b,

whose decision variable $x_j$ is a long-run average activity rate and whose objective $\gamma$ upper-bounds every server pool's utilization. The network is subcritical at $\lambda$ if $\gamma^\ast(\lambda) < 1$ , and the subcritical region is $\Lambda := \{\lambda : \gamma^\ast(\lambda) < 1\}$ .

An SPN is stable (Definition 3.6, mission I) when its ambient Markov chain is positive recurrent, equivalently has a unique stationary distribution, equivalently its buffer contents converge in distribution to a non-defective limit. This mission's chapter portion (Chapters 4-5) also treats three extensions used elsewhere in the book: a Markovian arrival process replacing independent Poisson arrivals; alternate routing with immediate commitment, where arrivals must be routed into an eligible buffer at the instant they arrive, with routing rates constrained by an augmented version of the SPP; and processor sharing (PS) networks, whose service discipline falls outside the book's ordinary relaxed-control framework and is instead analyzed through an equivalent head-of-line (EHL) model built to have the same generator.

Formalization targets

Goal: Theorem 5.2 — only subcritical networks can be stable

\text{(baseline stochastic assumptions)} \ \wedge \ \text{(Markov representation)} \ \wedge \ \text{(SPN stable)} \implies \lambda \in \Lambda.

This is the weakest target that captures the chapter's content: it asserts nothing about which policy achieves stability, or whether subcriticality is sufficient (Chapters 6 onward answer that, case by case, and Chapter 5 itself gives two counterexamples where it is not) — only that subcriticality is unconditionally necessary.

Further results (milestones)

Proposition 4.1 (a strong law of large numbers for class-level arrivals under randomized routing), Proposition 4.4 (PS-network stability reduces to EHL-model stability), Proposition 5.1 (for a unitary network, subcriticality reduces to the classical load condition $\rho < b$ ), and Corollaries 5.4-5.6 (the same necessity conclusion under a Markovian arrival process, under alternate routing, and its consequence for maximally stable policies).

Significance

The result itself. Theorem 5.2 converts "can this network be stabilized at all?" from an open-ended search over control policies into a single linear-program feasibility check on first-order data alone. Corollary 5.6 turns this into the standard proof template every later chapter uses: exhibit a policy whose implementation does not reference $\lambda$ , show it is stable throughout the subcritical region, and conclude maximal stability — without having to separately characterize the true stability region $\Lambda^\ast$ , which the book calls "a deep mathematical problem" in general.

Formalizing it. A search of the platform for "processing network," "static planning problem," and "linear program" returned no hits: the SPN-specific static planning problem — its decision variables $x$ tied to a network's material-balance matrix $R$ and capacity matrix $A$ — has no existing counterpart, though the platform's linear-optimization field (16 missions) has general LP duality substrate a future proof of Proposition 5.1 or Theorem 5.2 could draw on. This mission is a from-scratch formalization of the SPP, the subcritical region, and the necessity theorem.

Difficulty

The natural first attempt states Theorem 5.2 as a claim about the buffer-contents process $Z(t)$ directly. This fails to separate cleanly from the proof, because the actual argument passes through an auxiliary quantity — the stationary mean $x := \mathbb{E}_\pi[N(0)]$ under the chain's (unique, by stability) stationary distribution $\pi$ — that has no meaning outside a specific proof strategy. The formalization instead states the goal purely in terms of the data $(R, A, b)$ and the hypothesis of stability, exactly as the book's own statement does, leaving $x$ 's construction to the (currently sorry) proof. A second difficulty is Corollary 5.4's Markovian arrival process: naively reusing BaselineAssumptions with a non-Poisson arrival process is impossible, since Poisson-ness is a mandatory structural field of that definition, not an optional hypothesis — the corollary needs its own hypothesis structure that changes exactly the one clause Assumption 2.1(a) contributes and nothing else.

Formalization scope

Buffers and activities are Fin I, Fin J; matrices are Matrix over ℝ. The subcritical region is defined via the optimal SPP value $\gamma^\ast$ , formalized with Mathlib's IsLeast (attained infimum, matching the book's own " $\gamma^\ast \le 1$ iff $x$ exists" phrasing, which presupposes attainment) rather than a bare existential — a formalization using, say, sInf would silently commit to junk values on an infeasible or unbounded LP and would not obviously match the book's own usage of $\gamma^\ast$ as literally attained. The basic SPN model's full state-process construction (Sections 2.3-2.4) is not re-derived from scratch here; Theorem 5.2 instead takes the structural facts its own proof invokes — the capacity constraint $AN(t) \le b$ (Eq. 2.11) and the material-requirement matrix $B$ — as explicit data, reusing mission I's BaselineAssumptions and MarkovRepresentation for the stochastic and Markov-chain apparatus. Proposition 4.4's shared- generator fact between a PS network and its EHL model (the actual content the book's construction of Section 4.4 establishes) is likewise taken as an explicit hypothesis rather than rebuilt from the refined-class/phase-type machinery of Eqs. (4.19)-(4.29); reconstructing that machinery from scratch, or reproving Proposition 4.1's SLLN from the chain's strong Markov property at regeneration times, are both welcome future contributions. A formalization that stated Theorem 5.2 with $\Lambda$ replaced by an unconstrained existential (dropping the LP structure entirely) would trivialize the chapter's actual content — the LP-feasibility characterization is what makes $\Lambda$ checkable, and is preserved here in full.

Selected references

J. G. Dai and J. Michael Harrison, Processing Networks: Fluid Models and Stability, Cambridge University Press (forthcoming), pre-publication draft 2020-4-2. http://spnbook.org
J. M. Harrison, "Brownian models of open processing networks: canonical representation of workload," Annals of Applied Probability 10 (2000), 75-103.
J. G. Dai and W. Lin, "Maximum pressure policies in stochastic processing networks," Operations Research 53 (2005), 197-218.

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Combinatorics Graph Theory Operations Research·Captain: mikedeng1

λ1, Isoperimetric Inequalities for Graphs, and Superconcentrators 1: A Diameter Bound from λ1Research Paper

Motivation

The eigenvalues of the Laplacian of a graph carry metric information about the graph. The second-smallest one, $\lambda_1(G)$ , was named the algebraic connectivity by Fiedler (Fiedler 1973), who showed it is positive exactly for connected graphs. N. Alon and V. D. Milman (J. Combin. Theory Ser. B 38 (1985) 73–88) showed that a large $\lambda_1$ also forces two further properties: small diameter and a concentration of measure phenomenon, in which almost every vertex is close to any set containing half the vertices. They used these facts to build explicit expanders and superconcentrators, which are sparse networks with strong connectivity guarantees used in the theory of computation and in communication network design.

This mission covers Section 2 of that paper, "The Main Tools": the edge-count inequality (Lemma 2.1), the isoperimetric inequalities (Theorems 2.5 and 2.6), and the resulting diameter bound (Theorem 2.7).

Timeline:

1973: Fiedler introduces $\lambda_1(G)$ as algebraic connectivity and proves $\lambda_1 \le \frac{n}{n-1}\min_v d(v)$ .
1985: Alon and Milman prove the isoperimetric and diameter bounds of Section 2.
1986: Alon proves the converse direction, that edge expansion implies a spectral gap (Alon 1986).
Later work sharpened the constant in the diameter bound, e.g. Chung 1989.

Setting

Let $G = (V, E)$ be a finite, connected, simple graph on $n = |V| \ge 2$ vertices. Write $d(v)$ for the degree of a vertex $v$ , $d = \max_v d(v)$ for the maximum degree, and $A_G$ for the adjacency matrix. The Laplacian is the $V \times V$ matrix

Q = Q_G = \operatorname{diag}(d(v))_{v \in V} - A_G .

For real functions $f$ on $V$ with scalar product $(f, g) = \sum_v f(v)g(v)$ , the quadratic form of $Q$ is $(Qf, f) = \sum_{\{u,v\} \in E} (f(u) - f(v))^2 \ge 0$ . The eigenvalues of $Q$ , counted with multiplicity, are real and are written $0 = \lambda_0 \le \lambda_1 \le \dots \le \lambda_{n-1}$ . The algebraic connectivity $\lambda_1 = \lambda_1(G)$ is the second-smallest of them.

For vertices $u, v$ , $\operatorname{dist}(u, v)$ is the number of edges of a shortest path from $u$ to $v$ . For disjoint vertex sets $A, B$ the paper writes $\rho$ for the distance between them, $a = |A|/n$ and $b = |B|/n$ for their relative sizes, and $E_A$ ( $E_B$ ) for the set of edges with both endpoints in $A$ (in $B$ ). $[x]$ denotes the integer part of $x \ge 0$ .

Formalization targets

Goal: Theorem 2.7 (p. 79)

\operatorname{dist}(u, v) \;\le\; 2\left[\sqrt{2d/\lambda_1}\,\log_2 n\right] \qquad\text{for all } u, v \in V.

Milestones, in the order the proof uses them

Section 2, p. 76: $0 = \lambda_0 < \lambda_1$ for connected $G$ .
Eq. (2.1), Rayleigh's principle: if $\sum_v f(v) = 0$ then $(Qf, f) \ge \lambda_1 \|f\|^2$ .
Lemma 2.1: for nonempty $A, B$ at distance $\rho \ge 1$ ,

\lambda_1 n \le \frac{1}{\rho^2}\Big(\frac1a + \frac1b\Big)\big(|E| - |E_A| - |E_B|\big).

Remark 2.3: $\lambda_1 \le \frac{n}{n-1}\min_v d(v)$ .
Theorem 2.5: if $\rho > 1$ then

b \le \frac{1-a}{1 + (\lambda_1/d)\,a\rho^2}.

Theorem 2.6: if every $A$ – $B$ distance exceeds a real $\rho \ge 1$ , then

b \le (1-a)\exp\!\Big(-\ln(1+2a)\Big[\sqrt{\lambda_1/(2d)}\,\rho\Big]\Big).

Each statement keeps the paper's explicit constants. The goal is the endpoint of this chain and the paper's headline graph-theoretic bound.

Significance

Theorem 2.7 gives, for any family of graphs of bounded maximum degree whose algebraic connectivity stays bounded away from zero, a diameter of order $\log n$ . By the paper's Remark 2.8, the 4-regular graphs constructed in its Section 4 show that this order cannot be improved. Theorem 2.6 is a discrete concentration of measure inequality: the proportion of vertices at distance more than $\rho$ from a set of relative size $a$ decays exponentially in $\rho\sqrt{\lambda_1/(2d)}$ . It is the graph analogue of the Gromov–Milman concentration for manifolds, and Section 3 of the paper applies it to cubes and other product graphs. Theorem 2.5 is the input for the construction of expanders from graphs with a spectral gap (Theorem 4.3 of the paper).

All results are proved in the paper, and the formal work here is a machine-checked version of known proofs. As far as could be determined, none of the four inequalities (Lemma 2.1, Theorems 2.5–2.7) has been formalized in Lean or elsewhere. Mathlib has the Laplacian matrix, its positive semidefiniteness, and the relation between its kernel and connected components, but no statement about its second eigenvalue. The spectral facts (milestones 1–2), stated for Mathlib's Matrix.IsHermitian.eigenvalues₀, are reusable for any future work on algebraic connectivity.

Difficulty

The combinatorial steps are short. The work is at the interface between the spectral definition and the quadratic form. Mathlib defines eigenvalues through the spectral theorem for a Hermitian matrix, sorted into a list. Obtaining Rayleigh's principle for the second eigenvalue from that list, with the constant functions as the eigenvector of $\lambda_0 = 0$ , takes a Courant–Fischer-type argument over an orthonormal eigenbasis. It does not follow from positive semidefiniteness alone. Strict positivity of $\lambda_1$ additionally needs that the kernel of $Q$ is one-dimensional for a connected graph.

Theorem 2.6 iterates Theorem 2.5 over a sequence of neighbourhoods $\{v : \operatorname{dist}(v, A) \le j\mu\}$ with a real step length $\mu$ , so it needs bookkeeping of integer parts and of real-valued distance thresholds. Theorem 2.7 then combines Theorem 2.6 with Remark 2.3 and needs the estimate $\tfrac12\, 2^{-[\log_2 n]} < 1/n$ with the integer part kept. Replacing $[\cdot]$ by the real number inside it changes the statement.

Formalization scope

Graphs are Mathlib SimpleGraph V on a Fintype vertex type with decidable adjacency. Every item assumes G.Connected and $2 \le |V|$ (the goal writes $1 < |V|$ , as the paper does).
$Q$ is G.lapMatrix ℝ. $\lambda_1$ is the mission definition AlonMilman.Diameter.lambda1: the eigenvalue at index $n-2$ of eigenvalues₀, which lists the eigenvalues in decreasing order. It is $0$ by convention when $n < 2$ , a case no theorem uses.
$\lambda_1$ is defined spectrally. Defining it as the best constant in Eq. (2.1) would make Rayleigh's principle definitional and remove the spectral content of the mission, so that formalization is excluded. Likewise the goal quantifies over all pairs of vertices of a connected graph and does not use SimpleGraph.diam without connectivity, since that is $0$ for a disconnected graph.
Distances are SimpleGraph.dist (a natural number). "The distance between $A$ and $B$ is $\rho$ " is encoded as $\rho \le \operatorname{dist}(u, v)$ for all $u \in A$ , $v \in B$ . Because the bounds weaken as $\rho$ decreases, this is equivalent to the paper's exact distance. In Theorem 2.6 $\rho$ is real and the hypothesis is strict.
$E_A$ is AlonMilman.Diameter.edgesWithin G A. All counts are cast to $\mathbb R$ before subtraction, $a = |A|/n$ is a real quotient, $[x]$ is Nat.floor, $\log_2$ is Real.logb 2, and $\ln$ is Real.log.
Lemma 2.1 requires $A, B$ nonempty (so $a, b > 0$ ). Theorems 2.5 and 2.6 hold as stated for empty sets and carry no such hypothesis.

Contributions welcome: proofs of any milestone, and in particular general Mathlib-style lemmas for Rayleigh quotients and eigenvalues₀, which have uses beyond this mission.

Selected references

N. Alon, V. D. Milman, λ1, isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985) 73–88. https://doi.org/10.1016/0095-8956(85)90092-9
M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298–305. https://doi.org/10.21136/CMJ.1973.101168
N. Alon, Eigenvalues and expanders, Combinatorica 6 (1986) 83–96. https://doi.org/10.1007/BF02579166
F. R. K. Chung, Diameters and eigenvalues, J. Amer. Math. Soc. 2 (1989) 187–196. https://doi.org/10.1090/S0894-0347-1989-0965008-X

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Control Theory Dynamic Programming Operations Research+1·Captain: mikedeng1

Robust Control of Markov Decision Processes with Uncertain Transition Matrices 3: Stationary Policies Suffice and the Stationary/Time-Varying Uncertainty Gap Vanishes GeometricallyResearch Paper

Motivation

A Markov decision process (MDP) is controlled with transition probabilities estimated from data, and optimal policies computed for the estimated model can perform badly when the estimates are off. Robust MDPs replace the single transition model by a set of models and optimize the worst case over that set. Two readings of "the set" are possible. In the stationary uncertainty model, the unknown transition matrices are fixed but unknown; this is the reading that confidence regions from statistics support, but the resulting min–max problem is hard to solve. In the time-varying uncertainty model, an adversary ("nature") may pick different matrices at every stage; this relaxation is solved exactly by robust dynamic programming. Nilim and El Ghaoui (Oper. Res. 2005) solve the second problem in place of the first, and Theorem 4 of their paper justifies the substitution for discounted costs: in the infinite horizon the two readings, and the restriction to stationary controllers, all give the same value, and in the finite horizon the two readings differ by an amount that decays geometrically in the horizon.

Timeline:

1973: Satia and Lave study MDPs with uncertain transition probabilities (Oper. Res. 21).
1994: Puterman's monograph collects the nominal theory, including the optimality of stationary deterministic policies for discounted finite MDPs (Wiley).
2001: Bagnell, Ng and Schneider state a robust Bellman recursion for stationary games without proof (CMU-RI-TR-01-25).
2005: Iyengar (Math. Oper. Res. 30) and Nilim and El Ghaoui independently prove the robust Bellman recursion under rectangular uncertainty; Nilim and El Ghaoui add Theorem 4.
2013: Wiesemann, Kuhn and Rustem extend robust MDPs beyond rectangular sets (Math. Oper. Res. 38).

Setting

States form a finite set $\mathcal X = \{0,\dots,n-1\}$ and actions a finite nonempty set $\mathcal A$ . A stage cost $c(i,a)\ge 0$ is given for every state and action, together with a discount factor $0<\nu<1$ and an initial state $i_0$ . For every action $a$ and state $i$ a nonempty set $\mathcal P_i^a$ of probability vectors in the simplex $\Delta_n$ is given: the possible rows of the transition matrix $P^a$ . Rectangularity means that every row is chosen independently: one stage of nature is a family $(P^a)_{a\in\mathcal A}$ with $i$ -th row of $P^a$ in $\mathcal P_i^a$ , and the set of such families is $\mathcal Q$ .

A controller policy $\pi=(\mathbf a_0,\mathbf a_1,\dots)$ assigns an action $\mathbf a_t(i)$ to each state at each stage; the set of all of them is $\Pi$ , and the stationary ones (the same map at every stage) form $\Pi_s$ . A nature policy $\tau=(P_t^a)$ picks one element of $\mathcal Q$ per stage; the set is $\mathcal T$ , and the stationary ones form $\mathcal T_s$ . Starting from $\mu_0 = e_{i_0}$ , the state law evolves by $\mu_{t+1}(j)=\sum_i \mu_t(i)P_t^{\mathbf a_t(i)}(i,j)$ . The discounted costs over horizon $N$ and over the infinite horizon are

C_N(\pi,\tau)=\sum_{t=0}^{N-1}\nu^t\sum_i\mu_t(i)\,c(i,\mathbf a_t(i)),\qquad C_\infty(\pi,\tau)=\lim_{N\to\infty}C_N(\pi,\tau).

For a controller class $\mathcal C\in\{\Pi,\Pi_s\}$ and a nature class $\mathcal S\in\{\mathcal T,\mathcal T_s\}$ the robust values are $\phi_\infty(\mathcal C,\mathcal S)=\inf_{\pi\in\mathcal C}\sup_{\tau\in\mathcal S}C_\infty(\pi,\tau)$ and $\phi_N(\Pi,\mathcal S)=\inf_{\pi\in\Pi}\sup_{\tau\in\mathcal S}C_N(\pi,\tau)$ . Finally $c_{\max}=\max_{i,a}c(i,a)$ and $\varepsilon_N=\nu^Nc_{\max}/(1-\nu)$ .

Formalization targets

Goal: Theorem 4 (p. 786)

\phi_\infty(\Pi,\mathcal T)=\phi_\infty(\Pi_s,\mathcal T_s)=\phi_\infty(\Pi_s,\mathcal T)=\phi_\infty(\Pi,\mathcal T_s),

0\le \phi_N(\Pi,\mathcal T)-\phi_N(\Pi,\mathcal T_s)\le \frac{\nu^N c_{\max}}{1-\nu}\quad\text{for every }N .

The second line is the paper's "the gap goes to zero at a geometric rate $\nu$ ", with the constant the paper's own argument produces.

Milestones (proof order of the paper)

Eq. (34): $C_N(\pi,\tau)\le C_\infty(\pi,\tau)\le C_N(\pi,\tau)+\varepsilon_N$ for all $\pi\in\Pi$ , $\tau\in\mathcal T$ , $N$ .
Eq. (35): $\phi_N(\Pi,\mathcal T)\le\phi_\infty(\Pi,\mathcal T)\le\phi_N(\Pi,\mathcal T)+\varepsilon_N$ .
Step (e): the same sandwich for $\phi_N(\Pi,\mathcal T_s)$ and $\phi_\infty(\Pi,\mathcal T_s)$ .
Eq. (33): for every $\varepsilon>0$ and all large $N$ , $\phi_\infty(\Pi_s,\mathcal T_s)-\varepsilon\le\phi_N(\Pi,\mathcal T)\le\phi_\infty(\Pi_s,\mathcal T_s)$ .
Step (c): for every stationary $\pi$ , $\sup_{\tau\in\mathcal T}C_\infty(\pi,\tau)=\sup_{\tau\in\mathcal T_s}C_\infty(\pi,\tau)$ , hence $\phi_\infty(\Pi_s,\mathcal T)=\phi_\infty(\Pi_s,\mathcal T_s)$ .
Step (d), nominal fact: for stationary $\tau$ , $\inf_{\pi\in\Pi}C_N(\pi,\tau)\to\inf_{\pi\in\Pi_s}C_\infty(\pi,\tau)$ .
Step (d): $\phi_\infty(\Pi,\mathcal T_s)=\phi_\infty(\Pi_s,\mathcal T_s)$ .

Significance

The first half of Theorem 4 says that, for discounted robust MDPs with rectangular uncertainty, nothing is gained by either player from non-stationary behaviour: the controller may restrict itself to stationary deterministic policies and nature's time variation buys it nothing. This is what makes the stationary game (6), solved by the robust Bellman recursion of Theorem 3, the right infinite-horizon object. The second half is a quantitative guarantee for practitioners: solving the tractable time-varying problem (4) instead of the statistically motivated but hard stationary problem (3) costs at most $\nu^Nc_{\max}/(1-\nu)$ in value.

The result is proved on paper. As far as a search of the platform shows (September 2026), no robust MDP result has a machine-checked proof here; the nominal counterpart, optimality of stationary policies for discounted finite MDPs, is on the platform as BertsekasDP.discounted_main_theorem in a different model encoding (state-dependent control sets). A complete development contributes a reusable encoding of robust MDPs with time-varying and stationary adversaries, the truncation estimates for discounted costs, and a formal record of one step whose printed argument is incomplete (Step (d), see Difficulty).

Difficulty

The truncation estimates (34), (35) and Step (e) are elementary. The equalities in (31) are not: they compare values of games whose players have infinite-dimensional strategy sets, and the paper writes "min" and "max" where the infima and suprema need not be attained, since the row sets are neither closed nor convex. Two steps of the printed proof rely on results the mission does not import. Step (a) identifies $\phi_N(\Pi,\mathcal T)$ with iterates of the robust Bellman recursion, which is Theorems 1 and 3 of the paper (formalized in sibling missions of this series). Step (c) says only "following similar steps as in Step (a)". Step (d) is incomplete as printed: it shows that for each fixed stationary nature the controller's best time-varying and best stationary responses agree, which is a statement about a max–min value, whereas $\phi_\infty(\Pi,\mathcal T_s)$ is a min–max value. The obvious attempt to exchange the infimum over $\Pi$ with the supremum over $\mathcal T_s$ from that per-nature fact alone fails; the exchange requires a duality statement for the stationary game.

Formalization scope

States are Fin n; the action type is finite and nonempty. The row sets are an arbitrary family rows a i ⊆ stdSimplex ℝ (Fin n) with each set nonempty; nonemptiness is implicit in the paper and explicit here, and no convexity or closedness is assumed. A stage of nature is the subtype of families A → Fin n → (Fin n → ℝ) whose rows lie in the given sets, so rectangularity is built in. Controller policies are sequences ℕ → Fin n → A, nature policies sequences of stage choices; stationary policies of either player are the constant sequences. $C_N$ is defined from the forward state distribution (not from a Bellman recursion) and reads only stages $<N$ , so the finite-horizon values over infinite sequences are exactly the paper's values (3) and (4) with stage costs $\nu^tc$ and zero terminal cost. $C_\infty$ is the sum of the series of nonnegative stage costs, which converges for $0\le\nu<1$ and equals $\lim_N C_N$ . Every min and max of the paper is a real infimum ⨅ or supremum ⨆; all families are nonempty and lie in $[0,c_{\max}/(1-\nu)]$ , and attainment is not assumed anywhere. The discount factor satisfies $0<\nu<1$ , as in §4 of the paper.

The rate in the goal is the explicit bound $\nu^Nc_{\max}/(1-\nu)$ ; a formalization stating only that the gap tends to zero, or stating (31) with the infinite-horizon cost replaced by a Bellman fixed point, proves a different theorem and is not accepted.

A complete development needs: the stochastic-matrix facts for the forward distribution, geometric-series bounds, robust value iteration for the time-varying and the stationary adversary, and nominal stationarity of discounted MDPs. The model and truncation estimates are reusable for any discounted robust MDP result. Contributions of any milestone, and of proofs of Step (c) and Step (d) by any route, are welcome.

Selected references

A. Nilim, L. El Ghaoui, Robust Control of Markov Decision Processes with Uncertain Transition Matrices, Operations Research 53(5):780–798, 2005. https://doi.org/10.1287/opre.1050.0216
G. N. Iyengar, Robust Dynamic Programming, Mathematics of Operations Research 30(2):257–280, 2005. https://doi.org/10.1287/moor.1040.0129
M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994. https://doi.org/10.1002/9780470316887
J. K. Satia, R. E. Lave, Markovian Decision Processes with Uncertain Transition Probabilities, Operations Research 21(3):728–740, 1973. https://doi.org/10.1287/opre.21.3.728
J. A. Bagnell, A. Y. Ng, J. Schneider, Solving Uncertain Markov Decision Processes, Technical Report CMU-RI-TR-01-25, Carnegie Mellon University, 2001.
W. Wiesemann, D. Kuhn, B. Rustem, Robust Markov Decision Processes, Mathematics of Operations Research 38(1):153–183, 2013. https://doi.org/10.1287/moor.1120.0566

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Combinatorics Graph Theory Operations Research+1·Captain: mikedeng1

A Linear-Time Algorithm for Finding a Sparse k-Connected Spanning Subgraph of a k-Connected Graph 2: FOREST's Forests E_i Preserve Local Node-Connectivity up to i in a Simple GraphResearch Paper

Motivation

Given a $k$ -connected graph, many connectivity algorithms run in time that grows with the number of edges $|E|$ . A sparse certificate is a spanning subgraph with only $O(k|V|)$ edges that is still $k$ -connected; computing one first and running the expensive algorithm on it replaces $|E|$ by $k|V|$ in the bound. Finding a $k$ -connected spanning subgraph with the minimum number of edges is NP-complete for every fixed $k \ge 2$ (Garey and Johnson, problem GT31), so the question is how cheaply a sparse, not necessarily minimum, certificate can be found.

Nagamochi and Ibaraki (Algorithmica 7 (1992) 583–596) answered this with a single linear-time scanning procedure, FOREST, which partitions the edges into classes $E_1, E_2, \dots, E_{|E|}$ . They showed that the prefix unions $E_1 \cup \dots \cup E_k$ are certificates for edge-connectivity and, for simple graphs, for node-connectivity. The node-connectivity result is the subject of this mission; the edge-connectivity result is the preceding mission of this series.

Timeline:

1980: Galil gives an algorithm testing $\kappa(G) \ge k$ whose running time depends on $|E|$ (SIAM J. Comput. 9).
Before 1992 (as cited on p. 583): Suzuki et al. give $O(|E|)$ -time algorithms for sparse 2- and 3-node-connected spanning subgraphs; Nishizeki and Poljak find, for general $k$ , a $k$ -node-connected spanning subgraph with at most $k(|V|-1)$ edges in $O(|V|^{1/2}|E|^2)$ time.
1992: Nagamochi and Ibaraki prove that FOREST, which runs in $O(|V| + |E|)$ time, yields a $k$ -node-connected spanning subgraph $G_k$ of every simple $k$ -node-connected graph, with $|E(G_k)| \le k|V| - k(k+1)/2$ . Their Theorem 3.1 states a stronger, local form.
1993: Cheriyan, Kao and Thurimella isolate "scan-first search" as the general principle behind such certificates (SIAM J. Comput. 22 (1993)).

Setting

A graph $G = (V, E)$ has a finite node set $V$ with $|V| \ge 2$ and a finite edge set $E$ ; each edge has an unordered pair of two distinct end nodes. In this mission the graph is simple: no two edges have the same end nodes. For $F \subseteq E$ , $(V, F)$ is the spanning subgraph with edge set $F$ .

The local node-connectivity $\kappa(x, y; H)$ of nodes $x, y$ in a graph $H$ on $V$ is $|V| - 1$ if $x$ and $y$ are adjacent in $H$ , and otherwise the minimum size of a node set $W \subseteq V - \{x, y\}$ whose deletion leaves no $x$ – $y$ path. The node connectivity is $\kappa(G) = \min_{x, y} \kappa(x, y; G)$ .

Procedure FOREST keeps a label $r(v) \ge 0$ on each node, initially $0$ . While some node is unscanned, it chooses an unscanned node $x$ of largest label; for each unscanned edge $e = (x, y)$ it puts $e$ into the class $E_{r(y)+1}$ , increases $r(x)$ by one if $r(x) = r(y)$ , and increases $r(y)$ by one; then it marks $x$ scanned. Ties are broken arbitrarily. The time instants are the states between these elementary operations; $E^*_i$ denotes the class $i$ at an instant, and $E_i$ its final value. Put

G_i = (V,\ E_1 \cup E_2 \cup \dots \cup E_i).

Formalization targets

Goal: Theorem 3.1

For a simple graph $G$ and the classes of any completed run of FOREST, for $1 \le i \le |E|$ ,

\kappa(x, y; G_i) \ \ge\ \min\{\kappa(x, y; G),\ i\} \qquad \text{for any } x, y \in V. \tag{3.1}

The statement is local: it holds pair by pair, not only for the global minimum, and for every tie-breaking of the procedure.

Milestones, in the order the proof uses them

Lemma 2.2: at every instant, a node $v$ meets $E_i$ exactly for $i = 1, \dots, r(v)$ .
Lemma 2.4(b): at every instant, a $u$ – $v$ path in $E^*_j$ yields $u$ – $v$ paths in every $E^*_i$ , $i < j$ .
In-degree at most one (§2, p. 588): orienting each edge from the earlier-scanned to the later-scanned end, every node has at most one entering arc in each class.
Lemma 3.1: if an $x$ – $y$ path of $E^*_j$ has the form $x, u_1, \dots, u_k = w, y$ with $k = 1$ or $u_1$ scanned before $w$ , then any $w$ – $x$ and $w$ – $y$ paths in $E^*_i$ ( $i < j$ ) share a node other than $w$ .
Lemma 3.2: for a node cut set $W = \{w_1, \dots, w_i\}$ of $G_{i+1}$ (in scan order) separating a component $X$ from the rest $Y$ , immediately after $w_t$ is scanned every $X$ – $Y$ path of $E^*_t$ passes through $w_t$ , and $E^*_j$ has no $X$ – $Y$ path for $t + 1 \le j \le i + 1$ .

A companion item states the paper's announcement in §3: $G_k$ is $k$ -node-connected for every $1 \le k \le \kappa(G)$ .

Significance

Theorem 3.1 at $i = k$ shows that the first $k$ classes of FOREST form a $k$ -node-connected spanning subgraph whenever $G$ is, and the edge-count analysis of the companion mission bounds its size by $k|V| - k(k+1)/2$ . Since FOREST runs in linear time, any algorithm testing $\kappa(G) \ge k$ can be run on $G_k$ instead of $G$ ; the paper uses this to improve the bound for testing $\kappa(G) \ge k$ from $O(\max\{k^2|V|^{1/2}, k|V|\}|E|)$ to $O(\max\{k^3|V|^{3/2}, k^2|V|^2\})$ , and similar gains for computing the number of node-disjoint paths between two nodes. The local form (3.1) is what makes the $s$ – $t$ applications possible.

The result is proved on paper. As far as a search of the platform shows, neither FOREST nor local node-connectivity has a machine-checked treatment there; Mathlib has no notion of vertex connectivity of a pair of nodes. This mission produces a formal model of FOREST as a nondeterministic transition system and the statements needed to verify the paper's proof step by step.

Difficulty

For edge-connectivity the analogous statement follows from a general principle: any sequence of maximal spanning forests, each taken in what remains of the graph, preserves local edge-connectivity up to its length. The obvious attempt is to prove (3.1) the same way, from the fact that each $E_i$ is a maximal spanning forest of what the earlier classes leave. The paper gives no such argument for node-connectivity: its proof uses the specific scan order of FOREST in an essential way, through the orientation of edges from earlier- to later-scanned nodes and the in-degree bound of that orientation (Lemmas 3.1 and 3.2). The argument tracks, for a hypothetical node cut $W$ of size $i$ in $G_{i+1}$ , the classes at the moments the nodes of $W$ are scanned, which requires reasoning about intermediate states of the algorithm and about paths in several classes at once. None of this reduces to a static property of the output partition.

Formalization scope

Graphs. A node type V and an edge type E, both finite, with ends : E → Sym2 V; loop-freeness is ∀ e, ¬ (ends e).IsDiag and simplicity is Function.Injective ends. The standing assumptions of p. 583 and p. 589 ( $|V| \ge 2$ , no self-loop, a simple graph when node-connectivity is discussed) appear as hypotheses; §2 items are stated for loopless graphs, as on the page.
Connectivity. $\kappa(x, y; (V, F))$ is valued in $\mathbb N_\infty$ : $|V| - 1$ on adjacent pairs, the minimum node cut otherwise, and $\top$ when $x = y$ , where (3.1) holds trivially.
FOREST. A nondeterministic step relation with three steps (select, scan, finish), a run of length $K$ from the initial state, and completion when every node is scanned. Every tie-breaking is allowed, so the theorems quantify over all completed runs. A time instant is a state of a run; "scanned before" compares positions in the run's selection order; "immediately after $w_t$ has been scanned" is the state right after the finish step of $w_t$ .
Excluded. The running time " $O(|V| + |E|)$ " and everything in §4 (the connectivity-testing algorithms and their bounds) are not stated: the paper fixes no machine model. The edge bounds on $|E_i|$ belong to the edge-connectivity mission.
Ruled out. Stating (3.1) for an arbitrary partition into maximal spanning forests, or reading the classes $E^*_j$ of Lemmas 3.1–3.2 off the final state, would state a different theorem from the one the paper proves; the classes are those of a run of FOREST at the instant the page specifies.
Infrastructure. Reachability avoiding a node set, walks and paths in SimpleGraph, and invariants of the FOREST transition system. The definitions duplicate those of the edge-connectivity mission by design and are candidates for a shared layer. Contributions of general lemmas about the run (label invariants, monotonicity of classes along a run) are welcome.

Selected references

H. Nagamochi, T. Ibaraki, A linear-time algorithm for finding a sparse $k$ -connected spanning subgraph of a $k$ -connected graph, Algorithmica 7 (1992), 583–596. https://doi.org/10.1007/BF01758778
Z. Galil, Finding the vertex connectivity of graphs, SIAM J. Comput. 9 (1980), 197–199. https://doi.org/10.1137/0209016
J. Cheriyan, M.-Y. Kao, R. Thurimella, Scan-first search and sparse certificates: an improved parallel algorithm for $k$ -vertex connectivity, SIAM J. Comput. 22 (1993), 157–174. https://doi.org/10.1137/0222013
M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979.

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Combinatorics Linear Optimization Operations Research+1·Captain: Shuze Chen

Disjunctive Programming XV: Dominants of Polytopes and Upper SeparationTextbook

Motivation

Many real-world disjunctive models are not unions of polyhedra in a single shared space, but unions of polyhedra in different spaces linked by a logical implication: some action affecting one set of entities has consequences for another. Balas's treatment of such models (§17 of the book, following [17]) reduces to understanding a single auxiliary object attached to each polytope in isolation: its dominant, the set of points that dominate (coordinatewise) some feasible point. Dominants and their duals, blockers, have a long history in combinatorial optimization — blocking-pair theory for covering and packing polyhedra traces to Fulkerson (D. R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Mathematical Programming 1 (1971), 168–194, https://doi.org/10.1007/BF01584085) — but this chapter develops a self-contained, constructive theory tailored to polytopes inside the unit cube, culminating in an exact, facet-complete description of the dominant for an arbitrary such polytope.

Setting

For a polyhedron $P \subseteq \mathbb{R}^n_+$ , the dominant is $P^+ := P + \mathbb{R}^n_+ = \{y \ge 0 : y \ge x \text{ for some } x \in P\}$ , and the blocker is $P^* := \{\pi \in \mathbb{R}^n_+ : \pi x \ge 1 \text{ for all } x \in P\}$ — the covering inequalities valid for $P$ . (The blocker is not the reverse polar of 02b-polarity: restricting to the nonnegative orthant is essential and changes the object.) For $x^* \in \mathbb{R}^n_+$ , the upper-separation value is $\alpha_P(x^*) := \min\{\pi x^* : \pi \in P^*\}$ ; a violated covering inequality for $x^*$ exists exactly when $\alpha_P(x^*) < 1$ . A polytope $P \subseteq [0,1]^n$ is upper monotone (with respect to $[0,1]^n$ ) if $P = P^+ \cap [0,1]^n$ — the natural "closure" condition under which the theory of this chapter applies cleanly.

For $S \subseteq N := \{1,\dots,n\}$ , write $a(S) := \sum_{j\in S} a_j$ . Given $P \subseteq \mathbb{R}^n$ and a coordinate subset $S$ , the projection $P^S$ keeps only the $S$ -coordinates, letting the rest range freely. $I^S$ is the set of valid inequalities $\pi x \ge 1$ of $P^S$ with $\pi_j > 0$ exactly on $S$ , tight at $|S|$ linearly independent points of $P^S$ .

Formalization targets

Proposition 13.1. For an upper monotone $P = \bigcap_i P_i$ (each $P_i$ a single inequality in $[0,1]^n$ ), $P^+ = \bigcap_i P_i^+$ .

Theorem 13.3. For $P = \{x \in [0,1]^n : ax \ge 1\}$ ( $a \ge 0$ ) upper monotone,

P^+ = \Big\{x \ge 0 : \sum_{j\in S} \frac{a_j x_j}{1-a(N\setminus S)} \ge 1 \text{ for every } S\subseteq N \text{ with } 1-a(N\setminus S)>0\Big\}.

Theorem 13.5. For the same $P$ and any $x^* \ge 0$ , with $x^*_q$ the greatest coordinate value $x^*_j$ satisfying $a(N\setminus S(x^*_j))<1$ and $x^*_j \le g(x^*_j)$ : $S(\alpha_P) = S(x^*_q)$ and $\alpha_P = g(x^*_q)$ , an explicit, computable value.

Theorem 13.7 (goal). For an arbitrary polytope $P \subseteq [0,1]^n$ (not necessarily upper monotone):

P^+ = \{x \ge 0 : \pi x \ge 1 \text{ for every } S \subseteq N \text{ and } \pi \in I^S\},

and every one of these inequalities is facet-defining for $P^+$ .

Corollary 13.8. Every facet-defining inequality of $P^+$ has at most $\dim(P)+1$ nonzero coefficients.

The targets move from the intersection-distributivity fact (13.1) through an explicit, exponentially-large but fully closed-form facet system for the single-inequality case (13.3) and its constructive, polynomial evaluation recipe (13.5) to the fully general facet characterization (13.7, requiring no monotonicity assumption at all) and its immediate corollary on facet sparsity (13.8).

Significance

Theorem 13.7 is a rare case in polyhedral combinatorics of a complete and exact facet description obtained for the dominant of an arbitrary polytope, not merely a valid relaxation or an algorithmic separation oracle — every facet is accounted for, and every listed inequality is genuinely a facet, not merely valid. Corollary 13.8's support bound is the mechanism that makes Theorem 13.10 (not part of this mission) tractable: it lets the facets of a dominant built from a disjunction of polytopes in different spaces be characterized purely in terms of each factor's own low-dimensional facets, avoiding an exponential blowup in the combined space.

Both directions are proved in the source (Balas's own treatment, following the joint framework of [17]) but have no counterpart on this platform: nothing existing treats dominants, blockers, or upper monotonicity. This mission produces the first Lean statements of all five targets.

Difficulty

The obvious shortcut for Theorem 13.7 is to state only the validity half of the claim (every inequality from $I^S$ is valid for $P^+$ ) and treat "facet-defining" as a decoration — after all, Proposition 13.1's polar-style validity argument generalizes easily. But the theorem's actual force is the converse: not merely that these inequalities suffice to describe $P^+$ , but that none of them is redundant, and no other facet exists. The book's own converse proof needs a genuine perturbation argument (splitting a facet candidate with fewer than $|S|$ independent tight points into two distinct valid inequalities averaging back to it, contradicting facetness) — this is where the real content lives, and a formalization that only captures the forward direction would understate the theorem substantially.

For Theorem 13.5, the difficulty is that $S(\alpha)$ and $g(\alpha)$ are themselves defined in terms of $\alpha$ , so "the largest $x^*_j$ satisfying [a condition stated in terms of $S(x^*_j)$ and $g(x^*_j)$ ]" is a genuinely self-referential extremal characterization, not a closed-form formula one could simply plug into — hence its faithful statement (via IsGreatest over an explicit, self-referential candidate set) rather than an unwound algebraic expression.

Formalization scope

The ambient space is Fin n → ℝ throughout, matching the series default. Dominant/Blocker are given their own names (not reusing, even informally, 02b-polarity's polar/reverse-polar vocabulary), per BRIEF.md's explicit warning that the nonnegativity restriction makes these different objects. PolyDim/IsFacet are restated from 02b-polarity/11a-intersection-cuts (affine dimension via Module.finrank of vectorSpan, faces via IsExtreme), since Chapter 2 already pins these down precisely for this series and Chapter 13's own facet claims use the same notion. IsUpperMonotone is stated exactly as Definition 4 (P = P⁺ ∩ [0,1]ⁿ), not paraphrased as coordinatewise monotonicity, per BRIEF.md's explicit warning that these are different conditions.

IsInIS (membership in $I^S$ ) uses LinearIndependent ℝ directly for the "|S| linearly independent points" hypothesis, matching the book's own wording; since every such point satisfies $\pi x=1$ , a linear dependence among them is automatically an affine dependence (the coefficients of any nontrivial linear relation among them must sum to zero), so this is not a weakening of the more familiar "affinely independent" reading a reader might otherwise expect. A trivializing formalization to rule out explicitly: describing Theorem 13.7's $P^+$ using only the validity half of the claim (dropping "each of these inequalities is facet-defining for $P^+$ ") — this mission states both conjuncts, since the facet-exactness is the theorem's genuine content beyond a Farkas-style validity certificate.

This mission depends on no other chunk's Lean definitions; it restates the affine-dimension/facet vocabulary of 02b-polarity/11a-intersection-cuts only informally, per the series convention. Corollary 13.6 (an $O(n)$ -time algorithmic claim for computing $\alpha_P$ ) is out-of-cone per BRIEF.md: it is fully quantified, not a veto-V3 case, but is a computational-complexity statement outside this mission's polyhedral-characterization scope.

Selected references

D. R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Mathematical Programming 1 (1971), 168–194. https://doi.org/10.1007/BF01584085
E. Balas and R. G. Jeroslow, Strengthening cuts for mixed integer programs (for the broader monotonization-of-polyhedra context cited by this chapter's introduction), European Journal of Operational Research 4 (1980), 224–234. https://doi.org/10.1016/0377-2217(80)90106-X
E. Balas, Disjunctive Programming, Springer, 2018, Chapter 13, §13.1–13.2. https://doi.org/10.1007/978-3-030-00148-3

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Combinatorics Graph Theory Operations Research+1·Captain: mikedeng1

A Linear-Time Algorithm for Finding a Sparse k-Connected Spanning Subgraph of a k-Connected Graph 1: The FOREST Decomposition Preserves Local Edge-ConnectivityResearch Paper

Motivation

Many graph algorithms for connectivity questions run in time proportional to the number of edges. When the question is only whether a graph is $k$ -edge-connected, or what its local edge-connectivities are up to a threshold $k$ , most edges are irrelevant: a spanning subgraph with $O(k|V|)$ edges already carries the answer. Such a subgraph is called a sparse certificate. Computing one first and running the expensive algorithm on it replaces $|E|$ by $k|V|$ in the running time of connectivity testing, of Matula-type edge-connectivity algorithms, and of $s$ – $t$ flow computations used as connectivity oracles.

Nagamochi and Ibaraki (Algorithmica 7, 1992) gave a procedure, FOREST, that computes such a certificate for edge-connectivity and, on simple graphs, for node-connectivity, with a single graph search. The same partition of the edges into forests is the engine of their deterministic minimum-cut algorithm for multigraphs (SIAM J. Discrete Math. 5, 1992), which later became the maximum-adjacency ordering of the Stoer–Wagner minimum-cut algorithm (J. ACM 44, 1997).

Timeline:

1927: Menger identifies the minimum number of edges separating two nodes with the maximum number of edge-disjoint paths between them.
1992: Nagamochi and Ibaraki publish Procedure FOREST and prove that its $i$ -th prefix preserves local edge-connectivity up to $i$ in multigraphs, and local node-connectivity up to $i$ in simple graphs.
1993: Cheriyan, Kao and Thurimella (SIAM J. Comput. 22) obtain sparse certificates by scan-first search; Frank, Ibaraki and Nagamochi (J. Graph Theory 17) give a shorter proof for the node-connectivity case.
1994: Nishizeki and Poljak (Discrete Appl. Math. 55) publish the forest-decomposition lemma (Lemma 2.1 below), found independently.

Setting

A graph $G = (V, E)$ is finite and undirected, has $|V| \ge 2$ nodes, may have multiple edges (several edges with the same pair of end nodes), and has no self-loop. It is simple if no two edges have the same end nodes. For $F \subseteq E$ , $(V, F)$ is the spanning subgraph with edge set $F$ . It is a forest if it has no cycle; two parallel edges form a cycle. It is a maximal spanning forest in $(V, H)$ , for $F \subseteq H$ , if adding any edge of $H \setminus F$ to $F$ creates a cycle.

The local edge-connectivity $\lambda(x, y; H)$ is the minimum number of edges of $H$ whose removal leaves no path from $x$ to $y$ ; parallel edges count separately. It is $\infty$ when $x = y$ .

Procedure FOREST keeps a label $r(v) \in \mathbb{N}$ on every node, initially $0$ , and classes $E_1, E_2, \dots, E_{|E|}$ , initially empty. While an unscanned node exists, it picks an unscanned node $x$ of largest label; for every unscanned edge $e$ from $x$ to a node $y$ , it puts $e$ into $E_{r(y)+1}$ , increases $r(x)$ by one if $r(x) = r(y)$ , increases $r(y)$ by one, and marks $e$ scanned; then it marks $x$ scanned. Ties among nodes and the order of edges are free. On termination, $G_i = (V, E_1 \cup \cdots \cup E_i)$ .

Formalization targets

Goal: Theorem 2.1 (pp. 588–589), without the running time

For every graph $G$ and every completed execution of FOREST on $G$ :

every edge lies in exactly one class $E_i$ , $1 \le i \le |E|$ ;
for $i = 1, \dots, |E|$ ,

\lambda(x, y; G_i) \ge \min\{\lambda(x, y; G),\ i\} \qquad \text{for all } x, y \in V; \tag{2.1}

$|E_i| \le |V| - 1$ for all $i$ ;
if $G$ is simple, $|E_i| \le |V| - i$ for $i \le |V| - 1$ and $E_i = \emptyset$ for $i \ge |V|$ .

Milestones

Lemma 2.2 (p. 587): during the execution, a node $v$ has incident edges in exactly the classes $E_1, \dots, E_{r(v)}$ .
Lemma 2.3 (p. 587): each $(V, E_i)$ is a forest at every instant.
Lemma 2.4 (p. 588): (a) an edge $(u, v)$ added to $E_i$ has its end nodes joined by a path in $E_{i-1}$ ; (b) a path in $E_j$ between $u$ and $v$ yields a path in every $E_i$ , $i < j$ .
Lemma 2.5 (p. 588): each output $(V, E_i)$ is a maximal spanning forest in $G - E_1 \cup \cdots \cup E_{i-1}$ .
Lemma 2.1 (p. 584): any sequence of successive maximal spanning forests satisfies (2.1).

Companions

the sparse certificate (p. 589): if $\lambda(x, y; G) \ge k$ for all $x, y$ , then $G_k$ is $k$ -edge-connected with $|E(G_k)| \le k(|V| - 1)$ , and $|E(G_k)| \le k|V| - k(k+1)/2$ for simple $G$ ;
Lemma 2.6 (p. 589): for $k \le \delta(G)$ , $G_k$ has a node of degree exactly $k$ .

Significance

(2.1) says that one search produces, for every threshold $k$ at once, a subgraph with at most $k(|V|-1)$ edges that keeps every local edge-connectivity up to $k$ . Any algorithm whose running time grows with $|E|$ can then be run on $G_k$ in place of $G$ ; §4 of the paper uses this to speed up $k$ -connectivity tests and the computation of local connectivities. The same forest partition is the structural fact behind the Nagamochi–Ibaraki and Stoer–Wagner minimum-cut algorithms. Lemma 2.6 shows that the certificate is tight: its edge-connectivity is exactly $k$ when $\lambda(G) \ge k$ .

The results are proved in the paper. No machine-checked proof of them is known. Formalizing them means formalizing a graph search with free tie-breaking as a transition system, reasoning about invariants of all its executions, and proving a cut-counting statement for multigraphs. Mathlib's connectivity notions, such as SimpleGraph.IsEdgeReachable, do not see parallel edges, so the multigraph cut theory here is new.

Difficulty

Lemma 2.1 is a short cut argument once maximality is available. The difficulty is showing that FOREST, which assigns each edge to a class by looking only at the label of one end node, produces maximal forests in the successive residual graphs (Lemma 2.5). The obvious invariant, that the class of an edge is the first forest it does not close a cycle in, is not what line 7 computes. The label $r(y)$ records only which classes touch $y$ , not which component of each class contains $y$ . The paper's argument needs Lemma 2.4(a): at the moment an edge is added to $E_i$ its ends already lie in one tree of $E_{i-1}$ . That relies on the choice of the unscanned node of largest label, on the order of lines 8 and 9, and on an argument about the scan order of tree roots.

Formalization scope

Graphs. A graph is a finite node type V with $|V| \ge 2$ , a finite edge type E, and ends : E → Sym2 V with no diagonal value (no self-loops). Parallel edges are distinct elements of E. Simplicity is injectivity of ends, and edge subsets are Finset E. A forest is an edge set in which every edge is a bridge, a condition that sees parallel edges.
Connectivity. $\lambda$ is an infimum in ℕ∞ over separating edge sets, $\infty$ at $x = y$ . (2.1) is kept "for all $x, y$ ", as printed.
FOREST. FOREST is a nondeterministic step relation (select, scan, finish) on explicit states: labels, class index per edge ( $0$ = unscanned), scanned nodes, current node and selection order. The theorems quantify over every run from the initial state, so no tie-breaking rule is fixed. "At some time instant" is a state of the run; "upon completion" is a run whose last state has every node scanned. Lemma 2.2 is stated at every state, not only after a scan block (the other steps change neither labels nor classes). Lemma 2.4(a) assumes $i \ge 2$ , since $E_0$ does not exist.
Exclusions. Theorem 2.1's clause "is found in $O(|V| + |E|)$ time", the bucket implementation, and the time bound of the certificate are not formalized: the paper fixes no machine model. The goal consists of the structural conclusions only. The bound printed "if $G$ is multiple" is stated for every loopless graph.
Non-triviality. The goal is about the classes of a run of FOREST. An arbitrary partition of $E$ into maximal spanning forests is Lemma 2.1's hypothesis, not a formalization of Theorem 2.1. A statement in which the classes are unconstrained variables, or in which the run hypotheses cannot be met, would be trivial. A separate sanity file checks that a complete run on the triangle $K_3$ exists and attains $|E_1| = |V| - 1$ , $|E_2| = |V| - 2$ .
Welcome contributions. Useful reusable infrastructure includes:
- a cut and Menger layer for finite multigraphs;
- forest and bridge lemmas for edge-indexed graphs;
- invariant-style reasoning over runs.

Selected references

H. Nagamochi, T. Ibaraki, A linear-time algorithm for finding a sparse $k$ -connected spanning subgraph of a $k$ -connected graph, Algorithmica 7 (1992) 583–596. https://doi.org/10.1007/BF01758778
H. Nagamochi, T. Ibaraki, Computing edge-connectivity in multigraphs and capacitated graphs, SIAM J. Discrete Math. 5 (1992) 54–66. https://doi.org/10.1137/0405004
T. Nishizeki, S. Poljak, $k$ -connectivity and decomposition of graphs into forests, Discrete Appl. Math. 55 (1994) 295–301. https://doi.org/10.1016/0166-218X(94)90014-0
J. Cheriyan, M.-Y. Kao, R. Thurimella, Scan-first search and sparse certificates: an improved parallel algorithm for $k$ -vertex connectivity, SIAM J. Comput. 22 (1993) 157–174. https://doi.org/10.1137/0222013
A. Frank, T. Ibaraki, H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory 17 (1993) 275–281. https://doi.org/10.1002/jgt.3190170302
M. Stoer, F. Wagner, A simple min-cut algorithm, J. ACM 44 (1997) 585–591. https://doi.org/10.1145/263867.263872

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Linear Optimization Operations Research Optimization·Captain: Shuze Chen

Disjunctive Programming XIII: Monoidal Cut Strengthening and the Gomory Mixed-Integer CutTextbook

Motivation

The Gomory mixed-integer (GMI) cut is the single most widely deployed cutting plane in practical mixed-integer programming: every commercial solver generates it, from a simplex tableau row, essentially for free. Yet the GMI cut is not the strongest cut derivable from the same row: once a subset of the variables is known to be integer-constrained, that integrality can be used to tighten the cut's coefficients further, a technique due to Balas and Jeroslow that predates and motivates most of the general disjunctive-cut machinery of this book (E. Balas and R. G. Jeroslow, Strengthening cuts for mixed integer programs, European Journal of Operational Research 4 (1980), 224–234, https://doi.org/10.1016/0377-2217(80)90106-X). This mission formalizes the culmination of that line of work: two refinements of the GMI cut, each strictly stronger than the plain GMI coefficient on part of the variable set, obtained by applying monoidal cut strengthening — optimizing a cut's coefficients over an algebraic monoid of admissible integer shifts — to the two-term split disjunction that produces the GMI cut in the first place (E. Balas and R. Jeroslow, as above; the monoidal strengthening framework itself due to R. E. Gomory and E. L. Johnson, and formalized in the generality used here by G. Nemhauser and L. Wolsey and by J. -P. P. Richard, Y. Li and A. Miller).

Setting

Fix a row of a simplex tableau: $y = a_0 - \sum_{j\in J} a_j x_j$ , with $x_j \ge 0$ for $j \in J$ , $x_j$ integer for $j$ in a subset $J_1 \subseteq J$ , and $0 < a_0 < 1$ . If $y$ is itself integer-constrained, every feasible solution satisfies the split disjunction $y \le 0 \lor y \ge 1$ , from which the ordinary GMI cut $\alpha x \ge 1$ follows, with $\alpha_j := \max\{a_j/a_0,\ -a_j/(1-a_0)\}$ uniformly over all of $J$ .

For a general $q$ -term disjunction $\bigvee_{h\in Q}(\sum_j a^h_j x_j \ge a^h_0)$ with a known background lower bound $b^h_0 \le a^h_0$ on each term's left side, the cut monoid is

M := \{\mu \in \mathbb{Z}^q : \textstyle\sum_{h\in Q} \mu_h \ge 0\}.

Given the disjunction and the lower bounds, replacing each term's coefficient $a^h_j$ (for $j$ in the integer-constrained set $J_1$ ) with $a^h_j + \mu^j_h(a^h_0 - b^h_0)$ for any fixed $\mu^j \in M$ leaves the disjunction — and hence the disjunctive cut it implies — valid; optimizing this replacement over the whole monoid strengthens the resulting cut. For the normalized $q$ -term disjunction $\bigvee_{i\in Q}(\sum_j a_{ij}x_j \ge a_{i0})$ (each right-hand side scaled to a common reference), the unstrengthened cut coefficient is $\beta_j := \max_{i\in Q} a_{ij}/a_{i0}$ .

Formalization targets

Theorem 11.19. For the general $q$ -term disjunctive-cut situation, every $x \ge 0$ satisfying the background lower bound and the disjunction also satisfies the monoidally-strengthened cut $\sum_j \alpha_j x_j \ge \alpha_0$ , with

\alpha_j = \begin{cases} \inf_{\mu^j\in M}\max_{h\in Q}\theta_h[a^h_j+\mu^j_h(a^h_0-b^h_0)], & j\in J_1, \\ \max_{h\in Q}\theta_h a^h_j, & j\in J\setminus J_1,\end{cases} \qquad \alpha_0 = \min_{h\in Q}\theta_h a^h_0.

Proposition 11.22. For the normalized disjunction, and any fixed monoid elements $m^j \in M$ ( $j\in J_1$ ), every $x\ge0$ integer on $J_1$ satisfying the disjunction and the background lower bound also satisfies the strengthened disjunction with each term's coefficients shifted by $m^j$ — the fact that licenses optimizing over the whole monoid afterward.

Corollary 11.25. For each disjunct index $k$ , the cut $\delta^k x \ge 1$ is valid, with $\delta^k_j := \min\{(a_{kj}+a_{k0}-b_k)/a_{k0},\ \beta_j\}$ on $J_1$ and $\delta^k_j :=\beta_j$ elsewhere — a version of monoidal strengthening needing no optimization over $M$ at all.

Theorem 11.26 (goal). Specializing the same monoidal strengthening machinery to the two-term split disjunction $y \le 0 \lor y \ge 1$ itself: both $\alpha^+ x \ge 1$ and $\alpha^- x \ge 1$ are valid cuts, with $\alpha^+$ given by a three-case piecewise formula (eq. (11.55)) refining the GMI coefficient on part of $J_1$ , and $\alpha^-$ symmetric (eq. (11.56)).

The targets move from the general monoidal-strengthening theorem (11.19) through its validity engine in normalized form (Proposition 11.22, directly cited by the intermediate Theorem 11.23 that the goal specializes) and its optimization-free cousin (Corollary 11.25, immediately preceding the goal in the same subsection) to the concrete payoff for the single most-used cut in practice.

Significance

Theorem 11.26's cuts are not a theoretical curiosity: Corollary 11.27 (not drafted this pass) gives an explicit, checkable condition under which each cut is strictly stronger than the plain GMI cut, and Example 4 (p. 187–188) gives a fully worked six-variable instance where the improvement is concrete and numerically verifiable. Since the GMI cut is generated by essentially every mixed-integer solver at essentially every node of a branch-and-cut search, a cheap, always-valid strengthening of it — derivable from the same tableau row with no extra data beyond knowing which variables are integer-constrained — has direct practical reach far beyond this one book.

Both directions are proved in the source (Balas and Jeroslow 1980 for the underlying strengthening idea; this book's own Theorem 11.19/Proposition 11.22/Theorem 11.23 chain for the general monoidal framework applied here) but have no counterpart on this platform: nothing existing treats monoidal cut strengthening, the cut monoid itself, or a refinement of the GMI cut. This mission produces the first Lean statements of all four targets.

Difficulty

The obvious shortcut for Theorem 11.26 is to collapse $\alpha^+$ 's three-case definition into the plain GMI formula $\max\{a_j/a_0,\ -a_j/(1-a_0)\}$ applied uniformly — after all, that formula already gives a valid cut, and the strengthened cases can only make individual coefficients smaller (better). But a uniform formula reproduces exactly the plain GMI cut and can never be strictly stronger than it, which is the entire content the goal theorem (via Corollary 11.27) is building toward; the piecewise case split over $J^+_1$ (where $a_j>1$ ), $J^{>}_1$ (where $a_0-1\le a_j\le1$ ), and the rest is not incidental bookkeeping but the mechanism by which integrality actually buys something.

For Proposition 11.22 and Theorem 11.19, the difficulty is that the strengthening must remain valid simultaneously for every choice of the monoid element $\mu^j$ (or $m^j$ ) — not merely for some cleverly chosen one — since Theorem 11.19's conclusion then takes an infimum over the entire monoid $M$ , which is generally infinite. Fixing a single "obviously good" $\mu^j$ and stopping there would prove a weaker, non-optimized statement.

Formalization scope

The ambient space is Fin n → ℝ throughout, matching the series default, with the disjunction index set Q represented as Fin q and the cut monoid CutMonoid q : Set (Fin q → ℤ). Theorem 11.19 is formalized with scalar per-term coefficients $a^h_j$ (one real number per disjunct $h$ and variable $j$ ), rather than the fully general "each term a multi-row system $A^h x \ge a^h_0$ " framing the book's surrounding prose (§11.8's opening) sketches before specializing: every downstream result this mission needs (Proposition 11.22 onward, via (11.38)) is already stated at the single-inequality-per-term level, so this is not a weakening relative to what is actually used, only relative to a more general preamble that is never itself given a numbered, formalizable statement. AlphaJStrengthened uses sInf over the (possibly infinite) monoid literally, not a fixed near-optimal representative. AlphaPlus/AlphaMinus use the exact three-case structure of (11.55)/(11.56) — collapsing them into the uniform GMI formula is the trivializing formalization this mission rules out, since a uniform formula could never realize the theorem's actual (strictly stronger, on part of the domain) claim.

This mission depends on no other chunk's Lean definitions; it restates 11a-intersection-cuts's disjunctive-cut vocabulary only informally (the underlying disjunctive-cut idea, not any specific Lean declaration), per the series convention. A complete development needs: properties of sInf over an unbounded-below-safe subset of ℤ-indexed reals, and case analysis on Int.floor/ Int.ceil for the piecewise formulas. The cut-monoid and unstrengthened/strengthened-coefficient definitions are reusable by any later mission touching monoidal strengthening (e.g. a future mission on Theorem 11.23's full Lopsided-cut construction or the multiple-term-disjunction material of §11.9.2–11.9.3, not drafted this pass).

Selected references

E. Balas and R. G. Jeroslow, Strengthening cuts for mixed integer programs, European Journal of Operational Research 4 (1980), 224–234. https://doi.org/10.1016/0377-2217(80)90106-X
R. E. Gomory and E. L. Johnson, T-space and cutting planes, Mathematical Programming 96 (2003), 341–375. https://doi.org/10.1007/s10107-003-0389-3
J.-P. P. Richard, Y. Li, and L. A. Miller, Valid inequalities for MIPs and group polyhedra from approximate liftings, Mathematical Programming A 118 (2009), 253–277. https://doi.org/10.1007/s10107-007-0190-9
E. Balas, Disjunctive Programming, Springer, 2018, Chapter 11, §11.8–11.9. https://doi.org/10.1007/978-3-030-00148-3

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