Convex Optimization

Motivation

Chapter 9 is where discrete convex analysis meets classical network flow theory: the minimum cost flow problem's three hallmark properties — an optimality criterion by potentials, an optimality criterion by negative cycles, and integrality of optimal solutions — are shown to survive, in a precise and increasingly general form, first for arbitrary polyhedral convex costs (MCFP3), then for the M-convex submodular flow problem (MSFP2/MSFP3), the chapter's own combinatorial generalization of the classical problem. This mission places the potential criterion (Theorem 9.4) and its cascade of six corollaries and generalizations, the block of results this book's own text uses to carry every other result in the chapter.

Setting

A digraph G = (V,A) with tail/head maps ∂⁺,∂⁻ : A → V. A flow ξ : A → R has boundary ∂ξ(v) = Σ{ξ(a) : ∂⁺a=v} − Σ{ξ(a) : ∂⁻a=v}. A potential p : V → R has coboundary δp(a) = p(∂⁺a) − p(∂⁻a). The minimum cost flow problem MCFP3 minimizes Γ₃(ξ) = Σₐ fₐ(ξ(a)) + f(∂ξ) over flows, for polyhedral convex arc costs fₐ : R → R∪{+∞} and boundary cost f : Rⱽ → R∪{+∞}; MCFP0 is its linear-cost, fixed-supply special case. The M-convex submodular flow problem MSFP3 is MCFP3 with f additionally M-convex; MSFP2 is its linear-arc-cost special case.

Formalization targets

Goal: The potential criterion for MCFP3 (Theorem 9.4)

For a feasible flow ξ, ξ is optimal for MCFP3 iff there is a potential p with ξ(a) a minimizer of the reduced arc cost fₐ[δp(a)] for every arc and ∂ξ a minimizer of the reduced boundary cost f[−p]; and any such optimal potential characterizes optimality of every feasible flow. This is the hub result of the whole chunk: the book states Theorem 9.14 is "immediate" from it, and every other placed result either specializes it directly or builds on that specialization.

Supporting structural targets

Theorem 9.5 reformulates MCFP0's optimality as the absence of a negative cycle in an auxiliary network; Theorem 9.6 gives MCFP0's primal and dual integrality, the latter identifying the optimal-potential set as an L-convex polyhedron. Theorem 9.14 specializes the goal to MSFP3; Theorem 9.15 upgrades this to a full polyhedral and integrality structure theorem for MSFP3's optimal-flow-boundary and optimal-potential sets (M2-convex and L-convex polyhedra respectively); Theorem 9.16 is the integer-flow analogue, with the boundary set now literally M2-convex and the integer-optimal-potential set literally L-convex. Theorems 9.18 and 9.20 give the negative-cycle reformulation for MSFP2, real and integer flows respectively, generalizing Theorem 9.5 by admitting a third class of auxiliary arcs governed by the M-convex boundary cost's directional derivative (or its discrete difference, in the integer case).

Significance

This is the chapter's demonstration that M-convexity is not merely an abstract combinatorial axiom but the exact structural hypothesis under which classical network-flow duality survives intact: every one of the four "nice properties" the book opens the chapter with (potentials, negative cycles, integrality, efficient algorithms) is preserved verbatim in the M-convex generalization, and this mission's eight results are the proof of that claim for the first three. The chunk's own internal dependency structure — one foundational theorem (9.4) from which every other placed result descends by specialization or direct generalization — is itself characteristic of how this book organizes its combinatorial machinery around a single convex- analytic core.

None of these results are open — they are Murota's own account of network flow duality under M-convexity (sections 9.1, 9.4, and 9.5). What this mission contributes is a faithful, machine-checked formal statement of each, extending the platform's coverage of chapter 9 begun in mission 12-network-flows (which covered §9.1.1-9.1.2 and §9.3, the feasibility and max-flow min-cut results, deliberately leaving this block for later apparatus); no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The eight results span real- and integer-flow versions of two nested problem hierarchies (MCFP0 ⊂ MCFP3, MSFP2 ⊂ MSFP3) and two distinct optimality certificates (potentials, negative cycles), which this mission handles by building one shared apparatus — FeasibleFlowMCFP3, Gamma3, OptimalFlowMCFP3, IsOptimalPotential — that MCFP0 and MSFP3 both instantiate (MCFP0 literally as the linear-cost/singleton-boundary special case of Eq. (9.11)), and one shared generic cycle/negative-cycle apparatus (IsCycle, CycleLength, HasNegativeCycle) instantiated three times with different auxiliary-arc types (A⊕A for MCFP0, A⊕A⊕(V×V) for MSFP2's extra Cξ arcs governed by the boundary cost's directional derivative). "Primal integral" and "dual integral" polyhedral convex functions (the book's own C[Z|R→R]/C[R→R|Z] notation, used in Theorem 9.15) needed a modeling decision, since the book's own definition of these classes lies outside this chunk's page range; see Formalization scope.

Formalization scope

Ground-set vertices V and arcs A are Fintype with DecidableEq. All base M-/L-convexity vocabulary is redeclared from prior missions in this series. "Primal integral" (C[Z|R→R], M[Z|R→R]) is formalized as integer effective domain (IsDomainIntegerArc/IsDomainIntegerR); "dual integral" (C[R→R|Z], M[R→R|Z]) is formalized as the existence of an integer subgradient at every domain point (IsDualIntegralArc/IsDualIntegralR) — a standard equivalent characterization for polyhedral convex functions, and a deliberate modeling choice recorded in MODERATION_NOTES.md rather than a literal transcription of the book's own (out-of-range) definition of these two notation classes. All eight numbered results found in this chunk's page range are placed in full, with no partial-coverage scope reduction. Contributions completing any of the eight sorrys are welcome; the goal and Theorem 9.15 carry the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• R. T. Rockafellar, Network Flows and Monotropic Optimization, Wiley, 1984 [178] (the classical potential/Fenchel-duality framework this mission's Theorem 9.4 adapts).
• K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313-371 [140] (the Lagrange duality and negative-cycle theory of section 9.5 this mission's Theorems 9.18 and 9.20 draw from).

Motivation

Mission 29-ch08b-conjugacyduality opened chapter 8's account of M2-convex functions — sums of M-convex functions — proving their domains and minimizers are M2-convex and that they are integrally convex. This mission completes that program and builds its exact mirror for L2-convex functions (integer infimal convolutions of L-convex functions), the class that appears on the opposite side of Edmonds's intersection theorem's min-max relation from M2-convexity. It proves optimality and proximity theorems for both classes, shows their subdifferentials add (a discrete analogue of the classical subdifferential sum rule), derives how the Legendre-Fenchel transform interacts with the sum/infimal-convolution operation, and — the technically hardest result in the whole cluster — establishes that L♮₂-convex functions are integrally convex, by a genuinely different and more intricate argument than the M2-side analogue required.

Setting

Fix a finite ground set $V$. A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is L2-convex if $g = g_1 \square g_2$, the integer infimal convolution $g_1\square g_2(p) = \inf\{g_1(p_1)+g_2(p_2) : p_1+p_2=p\}$, of two L-convex functions $g_1, g_2$; L$^\natural_2$-convex if the summands are L$^\natural$-convex. An M2-convex function is a sum $f_1+f_2$ of two M-convex functions (mission 29-ch08b-conjugacyduality). The integer subdifferential $\partial_{\mathbb Z} f(x)$ and real subdifferential $\partial_{\mathbb R} f(x)$ generalize the subgradient set to integer- and real-valued perturbation directions respectively.

Formalization targets

Goal: L$^\natural_2$-convex functions are integrally convex (Theorem 8.42)

Every L$^\natural_2$-convex function is integrally convex, and in particular every L$^\natural_2$-convex set is integrally convex. The book's own proof is the most intricate argument in this cluster: given $p$ in the Minkowski sum $D_1+D_2$ of two L-convex sets, it constructs an explicit representation of $p$ as a convex combination of finitely many integer points of $D_1+D_2$, all lying in $p$'s own integral neighborhood, via the sorted fractional-part values of a chosen decomposition $p=p_1+p_2$ — a genuinely different technique from the M2-side analogue (Theorem 8.31), whose proof is a two-line consequence of convex extensibility.

Supporting structural targets

Eleven further results build the M2-/L2-convex theory in parallel. Theorems 8.33-8.34 give the M2-optimality criterion (a nonnegative-sum condition over cyclic exchange families) and its scaling-based proximity theorem; Theorem 8.35 shows subdifferentials of a sum of M$^\natural$- convex functions add, and that subdifferentials of M2-/M$^\natural_2$-convex functions are L2-/L$^\natural_2$-convex; Theorem 8.36 computes the conjugate of a sum as the infimal convolution of conjugates, with biconjugacy recovering the original sum. Propositions 8.39-8.41 transfer L-(natural-)convexity from summands to the domain and minimizer set of an L2-convex function, and give the precise attainment condition under which a linearly-perturbed infimal convolution's minimizer set splits additively. Theorems 8.43-8.44 give the L2-optimality and L2-proximity theorems, the exact L-side mirrors of Theorems 8.33-8.34; Theorem 8.45 mirrors Theorem 8.35 for subdifferentials of an infimal convolution; and Theorem 8.46 (found by direct reading, immediately following 8.45 and explicitly named by the book as 8.36's counterpart) shows biconjugacy for L$^\natural$-convex infimal convolutions.

Significance

The M2-/L2-convex function classes are where discrete convex analysis's abstract machinery meets concrete combinatorial optimization: Edmonds's matroid intersection theorem and its generalizations are literally statements about M2-convex minimization, with the L2-convex side supplying the dual bound. Theorem 8.35's subdifferential additivity is the discrete analogue of the classical Moreau-Rockafellar sum rule, and its proof (via the M-convex intersection theorem, already a milestone of mission 10-conjugacy-i) shows the sum rule holding without the constraint-qualification technicalities the continuous theory needs — a case where the discrete theory is cleaner than its continuous ancestor. Theorem 8.42's harder, dedicated proof technique is itself informative: it demonstrates that L2-convexity's combinatorial structure is not a routine transcription of the M2-convex case, foreshadowing the book's broader theme that M- and L-convexity, while conjugate, are not interchangeable in how their proofs actually work.

None of these results are open — they are Murota's account of the sum/infimal-convolution closure properties of M-convex and L-convex functions, continuing chapter 8's duality program into its most combinatorially concrete corner. What this mission contributes is a faithful, machine-checked formal statement of each, including one result (Theorem 8.46) the platform's own automated extractor missed, extending the shared Lean vocabulary (InfConv, L2Convex, M2ConvexSet) mission 29-ch08b-conjugacyduality began; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to adapt the M2-side integral-convexity proof (a direct appeal to convex extensibility) verbatim; the book's own proof shows this does not work, requiring instead a from-scratch construction: decompose $p=p_1+p_2$, take fractional parts $a_1 = p_1-\lfloor p_1\rfloor$ and $a_2=\lceil p_2\rceil-p_2$, sort their combined distinct values, build threshold sets exactly as in the Lovász-extension construction, and verify each resulting integer point $q_i = \lfloor p_1\rfloor+\chi_{U_{1i}}+\lceil p_2\rceil-\chi_{U_{2i}}$ both lies in $D_1+D_2$ (via L-convex-set closure properties, Theorem 5.10) and in $p$'s integral neighborhood (a case split on whether $p(v)$ is itself an integer) — a genuinely multi-stage combinatorial argument with no single-inequality shortcut, unlike almost every other result in this mission.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; M2-/L2-convex functions are (V→ℤ)→WithTop ℝ. All twelve numbered results found in this chunk's page range are placed, with no partial-coverage scope reduction needed — every clause of every result, including all three parts of Theorems 8.35 and 8.45 and the full cyclic-exchange condition of Theorems 8.33-8.34, is stated in full. One numbered result nominally in this chunk's page range, Theorem 8.32, is not re-placed here: it was already found and placed as a milestone in mission 29-ch08b-conjugacyduality, whose own page range overlaps this chunk's by one page (PDF245) — see HARD.md. "g1□g2 > −∞" hypotheses are omitted rather than translated, since WithTop ℝ has no −∞ element to violate. This mission's base vocabulary is redeclared verbatim from mission 29-ch08b-conjugacyduality rather than imported, since sibling drafts in this series cannot yet reference one another; ConvexConjugate is redeclared from mission 10-conjugacy-i. Contributions completing any of the twelve sorrys are welcome; the goal and Theorem 8.35 carry the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• K. Murota and A. Shioura, "Extreme points of a generalized polymatroid," Discrete Applied Mathematics, 152 (2005), pp. 268-278 [153] (the L2-convex integral-convexity proof this mission's goal is drawn from).
• K. Murota and A. Tamura, "Application of M-convex submodular flow problem to mathematical economics," Japan Journal of Industrial and Applied Mathematics, 20 (2003), pp. 257-277 [162] (the M2-proximity theorem, Theorem 8.34).

Motivation

Chapter 8 is where discrete convex analysis explains why it needed two separate notions — M-convexity (exchangeability) and L-convexity (submodularity) — rather than one. The answer is conjugacy: under the classical Legendre-Fenchel transform, the two classes turn out to be exactly dual to each other, the discrete analogue of the fact that convex analysis's transform is self-dual within a single class of convex functions. Mission 10-conjugacy-i proved the integer-lattice version of this fact (Theorem 8.12) but explicitly deferred the polyhedral version — Theorem 8.4, the chapter's own headline "Conjugacy theorem" — noting it needed a real-variable M-/L-convex-function layer the series had not yet built. That layer now exists, built across missions 23-24-ch06*-mconvexfunctions and 26-27-ch07*-lconvexfunctions. This mission proves Theorem 8.4 and its companions: the polar-cone correspondence it induces, its nonpolyhedral generalization, the separation and Fenchel-duality theorems for M♮-/L♮-convex functions, and the basic theory of M2-convex functions (sums of M-convex functions), which the Edmonds intersection theorem's own combinatorics is built from.

Setting

Fix a finite ground set $V$. For $f : \mathbb R^V \to \mathbb R \cup \{+\infty\}$, the Legendre-Fenchel transform is $f^\bullet(p) = \sup_x [\langle p,x\rangle - f(x)]$. A polyhedral convex function $f$ is M-convex ($f \in M[\mathbb R \to \mathbb R]$) if it satisfies (M-EXC[R]); $g$ is L-convex ($g \in L[\mathbb R \to \mathbb R]$) if it satisfies (SBF[R]) and (TRF[R]). A concave function $h$ is always represented via $h_2 = -h$, an ordinary convex function, so every "$f \ge h$" hypothesis is restated as "$f + h_2 \ge 0$" — an equivalent formulation avoiding any need to represent $-\infty$ in the codomain. A polyhedral cone's polar is $C^\circ = \{y : \langle y,x\rangle \le 0\ \forall x \in C\}$. A function is M2-convex if it is the sum of two M-convex functions.

Formalization targets

Goal: the conjugacy theorem (Theorem 8.4)

The classes of polyhedral M-convex functions and polyhedral L-convex functions are in one-to-one correspondence under the Legendre-Fenchel transform: $f \in M \Rightarrow f^\bullet \in L$, $g \in L \Rightarrow g^\bullet \in M$, and the transform is an involution ($f^{\bullet\bullet}=f$, $g^{\bullet\bullet}=g$) on each class, with the identical statement for the M$^\natural$/L$^\natural$ variants. This is the theorem mission 10-conjugacy-i deferred, citing exactly the missing infrastructure this series has since built.

Supporting structural targets

Twelve further results build the surrounding theory. Proposition 8.2 gives the easy two-variable case of the general submodularity-preservation fact (Theorem 8.1, already a milestone of mission 10-conjugacy-i); Proposition 8.3 is the technical minimizer-difference lemma the goal's harder direction is built from. Theorem 8.5 derives the M-convex/L-convex cone polarity from the goal, and Theorem 8.6 extends the correspondence beyond the polyhedral case to general closed proper convex functions. Proposition 8.14 and Theorems 8.15-8.16 build the separation theory for M♮-/L♮-convex and concave function pairs, with integral witnesses when the functions are integer valued; Theorem 8.21 (parts 1-2) derives the Fenchel-type strong-duality equality these separation theorems make possible. Propositions 8.29-8.30 and Theorem 8.31 (plus Theorem 8.32, found by direct reading immediately after 8.31) build the basic theory of M2-convex functions: their domains and minimizer sets are M2-convex, they are integrally convex, and their global optimality reduces to a finite local check.

Significance

The goal is the theorem that retroactively explains this entire series' two-track structure: missions 20-25 (M-convex sets and functions) and 08/21/26-28 (L-convex sets and functions) are not two independent theories that happen to share techniques — they are conjugate images of each other, so every theorem proved on one side has a dual counterpart automatically available on the other via Theorem 8.4. This is made concrete immediately: Theorem 8.5's cone polarity and the diagram the book draws connecting $M_0[\mathbb R]$, $0L[\mathbb R\to\mathbb R]$, and submodular set functions $S[\mathbb R]$ (already correspondences this series proved independently, in missions 24-ch06d-mconvexfunctions and 28-ch07d-lconvexfunctions) are shown to be facets of one single conjugacy fact rather than three separate coincidences. The separation and Fenchel duality theorems (8.15, 8.16, 8.21) are the discrete analogues of the two theorems every convex optimization course opens with, and the book is explicit that they are not corollaries of the classical versions plus convex extensibility — they carry genuinely combinatorial content, specializing to Frank's discrete separation theorem and Edmonds's intersection theorem as examples the book itself gives.

None of these results are open — they are Murota's account of the duality at the heart of discrete convex analysis, the reason the theory needed two dual notions rather than one. What this mission contributes is a faithful, machine-checked formal statement of each, completing a theorem mission 10-conjugacy-i explicitly left for a future session once the necessary polyhedral apparatus existed, and including one result (Theorem 8.32) the platform's own automated extractor missed; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal's harder direction (L⇒M) would try to verify the exchange inequality for $g^\bullet$ directly from the definition of the transform; the book's actual proof instead identifies the exchange inequality with a statement about weighted minimizers of $g$ itself via Proposition 8.3 (the minimizer-difference bound), converting a claim about the conjugate function into a claim about $g$'s own combinatorial structure — a genuine change of perspective, not a direct calculation. Proposition 8.3's own proof is the hardest single argument in this block: it derives the minimizer-difference bound by a contradiction argument that constructs an explicit pair of "worse" minimizers via a join/meet perturbation and derives a strict inequality from Theorem 7.29's translation inequality — a multi-step combinatorial argument with no direct shortcut.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; convex functions are WithTop ℝ valued throughout (never EReal, except for the Legendre-Fenchel transform itself, whose defining supremum/infimum can genuinely be infinite). All thirteen numbered results found in this chunk's page range — the twelve in BRIEF.md's own table plus Theorem 8.32 — are placed, with one documented scope reduction: Theorem 8.21 states only its real-attainment parts (1)-(2), not the integer-attainment refinement of parts (3)-(4), which needs a separate argument no other result in this chunk requires — see HARD.md. Concave functions $h$ are always represented via $h_2 = -h$ and every inequality $f \ge h$ restated as $f + h_2 \ge 0$, avoiding WithTop ℝ negation entirely. This chunk's own BRIEF.md inherited the chapters-4-7 page-offset boilerplate (printed = PDF $-$ 19); chapter 8 uses offset 18, confirmed against the PDF's own footers — every citation here uses the corrected offset. This mission's base vocabulary is redeclared from missions 10-conjugacy-i, 20-ch04b-mconvexsets, 21-ch05b-lconvexsets, 23-24-ch06*-mconvexfunctions, and 26-27-ch07*-lconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the thirteen sorrys are welcome; the goal and Proposition 8.3 carry the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• K. Murota and A. Shioura, "M-convex function on generalized polymatroid," Mathematics of Operations Research, 24 (1999), pp. 95-105 [152] (the polyhedral M-/L-convex conjugacy theory this mission's real-variable results are drawn from).

Motivation

The Legendre-Fenchel transform is the single most structurally important operation in convex analysis: for a proper closed convex function $f$, its conjugate $f^\bullet(p) = \sup_x \{\langle p,x\rangle - f(x)\}$ is again proper closed convex, and the transform is an involution — $f^{\bullet\bullet} = f$. This one fact underlies duality theory across optimization: every strong-duality theorem is, at bottom, a statement about conjugate pairs. Chapters 6 and 7 of this book developed M-convex and L-convex functions as if they were two separate theories, each with its own exchange axiom, optimality criterion, and proximity theorem. Chapter 8 reveals they were never separate: the Legendre-Fenchel transform, suitably discretized, is a bijection between the two classes. This mission formalizes that discrete conjugacy theorem together with its classical real-valued precursor and a genuine function-level generalization of Edmonds's intersection theorem, completing the picture that chunks 06 through 09 built the two halves of.

Setting

Let $V$ be a finite ground set. For $f : \mathbb R^V \to \mathbb R \cup \{+\infty\}$, the Legendre-Fenchel transform is $f^\bullet(p) = \sup\{\langle p,x\rangle - f(x) : x \in \mathbb R^V\}$; $f$ is submodular if $f(x)+f(y) \ge f(x\vee y)+f(x\wedge y)$ and supermodular under the reverse inequality. For $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$, the discrete Legendre-Fenchel transform restricts the same supremum formula to $p \in \mathbb Z^V$: $f^\bullet(p) = \sup\{\langle p,x\rangle - f(x) : x \in \mathbb Z^V\}$ for $p \in \mathbb Z^V$ — a genuinely different object from the real-valued transform, since the supremum is now over integer $x$ only, and the codomain is checked back against the discrete M-/L-convexity axioms of chunks 06–09. The integer biconjugate $f^{\bullet\bullet}$ is the transform applied twice. $f$ is integer valued if every finite value it takes is an integer (the classes $M[\mathbb Z\to\mathbb Z]$, $L[\mathbb Z\to\mathbb Z]$ of the goal theorem are exactly the M-/L-convex functions with this property).

Formalization targets

Goal: Theorem 8.12 (the discrete conjugacy theorem)

(1) The classes $M[\mathbb Z\to\mathbb Z]$ and $L[\mathbb Z\to\mathbb Z]$ are in one-to-one correspondence under the discrete Legendre-Fenchel transform: for $f \in M[\mathbb Z\to\mathbb Z]$ and $g \in L[\mathbb Z\to\mathbb Z]$, $f^\bullet \in L[\mathbb Z\to\mathbb Z]$, $g^\bullet \in M[\mathbb Z\to\mathbb Z]$, $f^{\bullet\bullet}=f$, and $g^{\bullet\bullet}=g$. (2) The same correspondence holds between $M^\natural[\mathbb Z\to\mathbb Z]$ and $L^\natural[\mathbb Z\to\mathbb Z]$.

Milestones: Theorem 8.1, Proposition 8.11, Theorem 8.17

Theorem 8.1: the conjugate of a real-valued submodular function is always supermodular — the classical warm-up, and evidence that submodularity/supermodularity is not symmetric under conjugation on its own (the converse fails). Proposition 8.11: the integer biconjugate recovers $f$ at any point with a nonempty integer subdifferential — the fact that makes discrete biconjugation meaningful at all. Theorem 8.17 (the M-convex intersection theorem): a point jointly minimizes a sum of two M$^\natural$-convex functions if and only if a single linear functional separately certifies it as a minimizer of each perturbed function — the function-level generalization of chunk 04's Edmonds's intersection theorem for M-convex sets.

Significance

The result itself. The discrete conjugacy theorem is, in the book's own words, "the unifying result of the entire book": every theorem proved separately for M-convex functions (chunks 06–07) has an exact mirror for L-convex functions (chunks 08–09) precisely because the Legendre-Fenchel transform carries one class to the other. Theorem 8.17's function-level Edmonds generalization shows the payoff directly — the classical matroid-intersection-style min-max duality of chunk 04 was never really about sets; it is a special case (indicator functions) of a duality that holds for the whole class of M-convex functions.

Formalizing it. No matching item exists on the platform for conjugate functions, discrete conjugacy, or this generality of intersection theorem. This mission gives the first formal statement of the discrete conjugacy theorem, distinguishing it carefully from its real-valued (polyhedral) precursor, Theorem 8.4 — a genuinely different, harder theorem this mission does not draft (see Formalization scope), since the integer bijection needs the M-/L-proximity theorems of chunks 06–09 to control integrality under convex extension, while the real-valued case does not.

Difficulty

The obvious approach — try to prove the discrete conjugacy theorem directly by mimicking the real-valued proof (Theorem 8.4) with ℤ in place of ℝ everywhere — fails, because the real-valued proof's key step (Proposition 8.3, an infimal-convolution argument comparing arg min sets of perturbed polyhedral functions) has no immediate discrete analogue: a discrete arg min need not vary continuously with the perturbation the way a polyhedral one does. The book's actual strategy instead routes through the convex extension of the discrete function (chunk 06/08's bridge to chapter 3's integral convexity), applies the already-proved real-valued conjugacy theorem to the extension, and then must separately argue that the resulting conjugate, restricted back to integer points, is again integer-valued and satisfies the discrete exchange axiom — an argument that needs different treatment depending on whether the original function's domain is bounded or unbounded (an exhaustion argument via restriction to a growing integer interval, invoking chunk 06's proximity theorem to control convergence). Skipping this discreteness argument and treating the real-valued theorem as if it settled the integer case would silently discard exactly the chapter's own point.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq. ConvexConjugate (the discrete transform) has domain and codomain both (V → ℤ) → WithTop ℝ, obtained by taking the defining supremum in EReal (a complete lattice, so it is always total) and projecting back via a new FromEReal map — this is what lets the biconjugate f•• typecheck as an equality of functions of the same type as f. ConvexConjugateR (the real-valued transform, used only by the milestone Theorem 8.1) is a separate object with no shared code, per the explicit warning against conflating the two transforms; the two never appear in the same item.

A trivializing formalization of the goal would draft only the real-valued case (Theorem 8.4) as if it were the discrete theorem, or would silently allow WithTop ℝ's subtraction-avoidance convention to change which values are compared; neither is done. Theorem 8.4 itself (the polyhedral conjugacy theorem) is not drafted in this mission at all — it would require a fresh, otherwise-unused polyhedral M-/L-convex-function layer on Rⱽ that no other item here needs (see MODERATION_NOTES.md). The M-/L-separation theorems (8.15, 8.16) and the Fenchel-type duality theorem (8.21) are likewise left for a follow-on mission; contributions building the polyhedral bridge or the separation theorems, which depend on machinery this mission establishes, are welcome.

Selected references

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

Motivation

A differentiable convex function on $\mathbb{R}^n$ is determined, up to an additive constant, by its gradient, and a vector field is a gradient of a convex function exactly when it satisfies a monotonicity condition along closed cycles. Convex analysis and optimization need the same statement for nonsmooth and extended-valued functions on infinite-dimensional spaces: the subdifferential replaces the gradient, and the question becomes which multivalued maps from a Banach space to its dual arise as subdifferentials, and how much of the function they determine. The answer underlies the treatment of optimality conditions, variational inequalities and evolution equations governed by subdifferentials, where one works with the operator $\partial f$ and needs to recover $f$ from it.

Timeline.

• 1966: R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (DOI 10.2140/pjm.1966.17.497), studied cyclically monotone operators and stated the characterization of subdifferentials as the maximal cyclically monotone operators (its Theorem 3), together with the maximal monotonicity of subdifferentials (its Theorem 4).
• 1969: H. Brézis pointed out a gap in the 1966 proofs of maximality and uniqueness: a family of dual vectors $x_\varepsilon^*$ used in the argument might increase unboundedly in norm as $\varepsilon \to 0$.
• 1970: Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (DOI 10.2140/pjm.1970.33.209), repaired the argument for arbitrary real Banach spaces, proving Theorem A (maximal monotonicity of $\partial f$) and Theorem B (the characterization treated here).

Setting

Let $E$ be a real Banach space with dual $E^*$, and write $\langle x, x^* \rangle$ for the value of $x^* \in E^*$ at $x \in E$. A proper convex function on $E$ is a function $f : E \to (-\infty, +\infty]$, not identically $+\infty$, with $f((1-\lambda)x + \lambda y) \le (1-\lambda)f(x) + \lambda f(y)$ for all $x, y \in E$ and $0 < \lambda < 1$. It is lower semicontinuous (lsc) in the norm topology. Its subdifferential is the multivalued map

A multivalued map $T : E \to E^*$ is a cyclically monotone operator if

and maximal cyclically monotone if, in addition, its graph $\{(x, x^*) \mid x^* \in T(x)\}$ is not properly contained in the graph of any other cyclically monotone operator. The conjugate of $f$ is $f^*(x^*) = \sup_{x \in E} (\langle x, x^* \rangle - f(x))$ on $E^*$, and $j(x) = \tfrac12\|x\|^2$. In the Lean development these are ProperConvex, subdiff, IsCyclicallyMonotone, IsMaximalCyclicallyMonotone, conj and halfSqNorm, in the namespace RockafellarMaxMono.Cyclic.

Formalization targets

Goal: Theorem B (p. 210)

For every multivalued map $T : E \to E^*$ on a real Banach space $E$,

and if $f$ and $g$ are lsc proper convex with $\partial f = T = \partial g$, then $g = f + c$ for a real constant $c$. Both halves are one statement.

Milestones, in attack order

1. (3.7) For a finite continuous convex function $f$ on a real Banach space, $\partial f(x)$ is nonempty and weak* compact and $f'(x;u) = \max\{\langle u, x^* \rangle \mid x^* \in \partial f(x)\}$.
2. Finite continuous case (pp. 214–215). For finite continuous convex $f, g$ on a real Banach space, $\partial g(x) \supset \partial f(x)$ for all $x$ implies $g = f + \mathrm{const}$.
3. (3.1) $\partial(f + j)(x) = \partial f(x) + \partial j(x)$ for lsc proper convex $f$.
4. Proposition 1 $x^{**} \in \partial f^*(x^*)$ if and only if $x^{**}$ is a weak** limit of a bounded net $x_i$ with $x_i^* \in \partial f(x_i)$, $x_i^* \to x^*$ in norm.
5. (p. 213) $(f + j)^*$ is finite and continuous on $E^*$.
6. (p. 211) $f^{**}$ restricted to $E$ is $f$.
7. (3.6) For lsc proper convex $f, g$: $\partial g(x) \supset \partial f(x)$ for all $x$ implies $g = f + \mathrm{const}$.

Significance

The result. Theorem B gives an intrinsic description of subdifferential maps: an operator is the subdifferential of a closed proper convex function if and only if it satisfies the cycle inequality and cannot be enlarged without violating it. The uniqueness clause says that a closed convex function is recovered from its subdifferential up to a constant, the nonsmooth counterpart of recovering a function from its gradient. Milestone 7 is stronger than uniqueness: a one-sided inclusion $\partial f \subseteq \partial g$ already forces $g = f + c$, and this is what gives maximality.

Formalizing it. The theorem is proved in the paper, and nothing of it is formalized on the platform. Mathlib has convex functions, continuous duals, biduals and weak-* topologies, but not extended-valued subdifferentials on Banach spaces, conjugate duality in the nonreflexive setting, or monotone operator theory. This mission produces formal statements of the paper's steps, the standard max formula for directional derivatives on a Banach space, and the Fenchel conjugate facts the argument uses, each as a separate target.

Difficulty

In a reflexive space the argument is short, because $\partial f^*$ is the inverse of $\partial f$. In a nonreflexive space it is not: $\partial f^*$ maps $E^*$ into $E^{**}$, and points of $E^{**} \setminus E$ appear. The naive route, transferring the inclusion $\partial f \subseteq \partial g$ to the conjugates by inverting the maps, breaks down there, and the 1966 argument failed at a related step, where the dual vectors in an approximation could be unbounded. Relating $\partial f^*$ to $\partial f$ without reflexivity is where the difficulty sits; the boundedness of the approximating nets in Proposition 1 is essential and cannot be dropped. The finite continuous case and the max formula (3.7) are needed on an arbitrary Banach space, including the dual $E^*$, not only on $E$.

Formalization scope

$E$ is a real Banach space (NormedAddCommGroup, NormedSpace ℝ, CompleteSpace); $E^*$ is StrongDual ℝ E with the operator norm, $E^{**}$ is StrongDual ℝ (StrongDual ℝ E), and $E \hookrightarrow E^{**}$ is NormedSpace.inclusionInDoubleDual. No reflexivity, inner product or finite dimension is assumed. Explicit readings:

• Values in $(-\infty, +\infty]$ are EReal with the requirement $f(x) \ne -\infty$; convexity is the paper's inequality for $0 < \lambda < 1$ in EReal arithmetic. Lower semicontinuity is in the norm topology.
• A multivalued map is E → Set (StrongDual ℝ E); $T = \partial f$ means $T(x) = \partial f(x)$ for every $x$.
• The cycle inequality quantifies over all $n \in \mathbb{N}$ and points indexed by Fin (n + 1) with wrap-around addition, so the last term is $\langle x_n - x_0, x_n^* \rangle$. Maximality is graph inclusion among cyclically monotone operators, not among monotone operators.
• The uniqueness constant is a real number, never $\pm\infty$.
• "$\supset$" in (3.6) is non-strict inclusion, and the hypothesis is one-sided.
• A net is a nonempty directed partially ordered index type with convergence along atTop; weak** convergence is pointwise convergence on $E^*$; "bounded" is a uniform norm bound.
• "Finite and continuous" for $(f+j)^*$ means equal everywhere to a continuous real-valued function. The max in (3.7) is IsGreatest, so it is attained; the directional derivative is the limit along $\lambda \to 0^+$.
• The print's "$\partial(f + j) = \partial f(x) + \partial j(x)$" in (3.1) is read as $\partial(f+j)(x)$.

A formalization that drops lower semicontinuity, allows an extended-real constant, or replaces "maximal cyclically monotone" by "maximal monotone" states a different, and in the first two cases false or trivial, theorem; the statements here keep all three.

The proof reduces Theorem B to milestone 7 through Theorem 1 of Rockafellar (1966) and its Corollary 2, which are not stated in this paper and are not milestones; formal statements of them are welcome as supporting theorems. Contributions of reusable infrastructure are welcome: extended-valued subdifferentials and conjugates on normed spaces, the Fenchel–Moreau identity on $E$, the sum rule with a continuous function, and the max formula for directional derivatives.

Selected references

• R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. https://doi.org/10.2140/pjm.1970.33.209
• R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 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 J. Math. 14 (1964), 243–247. https://doi.org/10.2140/pjm.1964.14.243
• J.-J. Moreau, Fonctionnelles convexes, mimeographed lecture notes, Collège de France, 1967.
• R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. https://doi.org/10.1515/9781400873173

Motivation

Convexity is normally defined additively — a function's value at a mixture is bounded by the mixture of its values — but many of the properties that make convexity useful in optimization (a local minimum is global, level sets are well-behaved) survive under a much weaker, purely ordinal notion: quasi-convexity, which compares function values rather than adding them. A nondecreasing rescaling of a convex function is generally not convex, but it is always quasi-convex — so a theory built only on ordinal comparisons automatically covers every such rescaling for free, at the cost of a more delicate proof architecture (since the algebraic cancellations available to additive convexity are no longer available).

Chapter 6's second half asks exactly how far this idea extends in the discrete setting: does the M-convexity exchange axiom have an ordinal, quasi-convex relaxation that still supports the same strong minimization theory — an optimality criterion, a minimizer-cut lemma, and, most significantly, a proximity theorem with the same explicit distance bound? This mission formalizes the chapter's answer: yes, and the relevant relaxed class, functions satisfying condition (SSQM$_{\ne}$), is large enough to include every strictly increasing rescaling of an M-convex function, a class the M-convex theory of chunk 06 alone says nothing about.

Setting

Let $V$ be a finite ground set and $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain. Building on chunk 06's M-convex exchange axiom (M-EXC[Z]), this chapter introduces several ordinal relaxations. $f$ is weakly quasi M-convex, satisfying (QMw), if for every pair of distinct points $x, y \in \operatorname{dom} f$ there exist $u$ in the positive support and $v$ in the negative support of $x - y$ with $f(x - \chi_u + \chi_v) \le f(x)$ or $f(y + \chi_u - \chi_v) \le f(y)$ — an "or" where (M-EXC[Z]) demands an additive inequality. Two further conditions restrict attention to points of different function value and sharpen the conclusion to a three-way trichotomy (strictly better on one side, or exactly tied on both): (SSQM$_{\ne}$) quantifies universally over $u$ (as in (M-EXC[Z])), while (SSQM$_{\ne,w}$) quantifies existentially over both $u$ and $v$ (as in (QMw)). The linear perturbation of $f$ by $p : V \to \mathbb R$ is $f[p](x) = f(x) - \langle p, x \rangle$.

Formalization targets

Goal: Theorem 6.78 (the quasi M-proximity theorem)

Let $f$ satisfy (SSQM$_{\ne}$), $n = |V|$, $\alpha$ a positive integer. If $x_\alpha \in \operatorname{dom} f$ satisfies $f(x_\alpha) \le f(x_\alpha + \alpha(\chi_v - \chi_u))$ for all $u, v \in V$, then $\arg\min f \ne \emptyset$ and there is $x^* \in \arg\min f$ with $\|x_\alpha - x^*\|_\infty \le (n-1)(\alpha - 1)$ — verbatim the same conclusion, and the same exact bound, as chunk 06's Theorem 6.37(1), now established for the strictly larger class satisfying (SSQM$_{\ne}$) rather than the M-convex exchange axiom itself.

Milestones: Theorems 6.68(2), 6.76, 6.77

Theorem 6.68(2): $f$ satisfies (M-EXC[Z]) if and only if every linear perturbation $f[p]$ satisfies (QMw) — quantifying exactly how much weaker (QMw) is pointwise, and how the gap closes once quantified over every perturbation. Theorem 6.76 (the quasi M-optimality criterion): the direct analogue of chunk 06's Theorem 6.26 for the quasi-convexity classes — a purely pairwise local check still characterizes global (or, in the (QMw) case, strict unique) optimality. Theorem 6.77 (the quasi M-minimizer cut): chunk 06's Theorem 6.28 continues to hold verbatim when its M-convexity hypothesis is replaced by (SSQM$_{\ne}$) — the structural fact the proximity theorem's proof is built from survives the relaxation intact.

Significance

The result itself. The proximity theorem is the result algorithms actually use: a scaling algorithm for minimizing quasi-convex functions of this kind inherits exactly the same correctness guarantee, with exactly the same distance bound, as the M-convex case — this is a genuine broadening of chapter 10's algorithmic reach, not a restatement dressed in weaker hypotheses. Every strictly increasing scalar transformation of an M-convex objective (a common modeling device — re-expressing a cost in utility units, or applying a monotone risk measure) now falls under a proximity theorem, whereas prior to this chapter's relaxation such a transformation would generally destroy M-convexity itself and leave optimization theory silent on the transformed problem.

Formalizing it. No matching item exists on the platform for quasi M-convexity in any of its forms. Formalizing Theorem 6.78 requires first pinning down (SSQM$_{\ne}$) exactly (there are six closely related axiom variants in this section of the book, only three of which — (QMw), (SSQM$_{\ne}$), (SSQM$_{\ne,w}$) — are needed for this mission's chosen results), and this mission also captures, via Theorem 6.68(2), the precise sense in which these relaxed conditions are strictly weaker than plain M-convexity while remaining tightly connected to it.

Difficulty

The natural first instinct, given how close the quasi-convexity axioms look to (M-EXC[Z]), is to try to prove Theorem 6.78 by directly imitating chunk 06's proof of Theorem 6.37 line by line. This mostly works — the proof structure (fix a target coordinate, build a chain of strictly decreasing values via repeated exchange steps, bound the chain's length using the scaled hypothesis) survives verbatim — but every step that chunk 06's proof took by adding two instances of the exchange inequality together must be replaced by an ordinal argument, since (SSQM$_{\ne}$) only ever asserts a disjunction of value comparisons, never an additive inequality relating four function values simultaneously the way (M-EXC[Z])'s $f(x)+f(y) \ge f(x-\chi_u+\chi_v)+f(y+\chi_u-\chi_v)$ does. The book's proof handles this by working with strict inequalities and the trichotomy structure of (SSQM$_{\ne}$) directly rather than algebraic cancellation — the same overall architecture, but every arithmetic step rebuilt as a case analysis on which disjunct of (SSQM$_{\ne}$) fires.

Formalization scope

This mission builds directly on chunk 06's published items (CharVec, SuppPos, SuppNeg, DomZ, MExchangeAxiom, ArgMin), per the platform's textbook convention that a later chapter of the same book imports an earlier one's definitions rather than redrafting them; its own namespace DiscreteConvex.MConvexFunctions.Quasi nests under chunk 06's DiscreteConvex.MConvexFunctions accordingly. Δf(z;v,u) (Eq. (6.2)) is never reified as a separate object; every occurrence is unfolded directly into an f-value comparison, avoiding WithTop ℝ subtraction throughout, consistent with chunk 06's own convention.

A trivializing formalization of the goal would silently strengthen (SSQM$_{\ne}$) back to plain M-convexity (making this mission redundant with chunk 06's Theorem 6.37) or loosen the exact bound $(n-1)(\alpha-1)$ to an unspecified function of $n, \alpha$; neither is done. Six axiom variants appear in this section of the book ((QM), (SSQM), (QMw), (SSQM$_w$), (SSQM$_{\ne}$), (SSQM$_{\ne,w}$)); only the three actually needed by this mission's four items are drafted, and Theorem 6.68's first part (an implication chain among the other three) is left out — see MODERATION_NOTES.md. Contributions building the polyhedral M-convex-function bridge (§6.11–6.12, Theorems 6.59–6.64), the level-set characterizations (Theorems 6.72, 6.74), or the scaled quasi M-minimizer cut (Theorem 6.79, the direct generalization of Theorem 6.77 drafted here) are welcome.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• M. Avriel, W. E. Diewert, S. Schaible, I. Zang, Generalized Concavity, Plenum Press, 1988.