Optimization

Motivation

A 0–1 integer program asks for the best 0–1 vector satisfying a system of linear inequalities. Its linear relaxation is easy to optimize over, but the relaxation is usually much larger than the convex hull of the 0–1 solutions. Lift-and-project methods close this gap systematically: they lift the relaxation to a higher-dimensional space, add constraints that every 0–1 point satisfies there, and project back, obtaining a tighter relaxation that still contains every 0–1 solution.

L. Lovász and A. Schrijver introduced one of the two standard lift-and-project hierarchies in Cones of matrices and set-functions and 0–1 optimization (SIAM J. Optim., 1991). Their operators $N$ and $N_+$ represent a 0–1 point $x$ by the matrix $xx^{\mathsf T}$, impose linear (and for $N_+$ semidefinite) constraints on such matrices, and project back to $\mathbb R^{n+1}$. The same paper applies the operators to the stable set polytope, where one round already produces the odd hole, odd wheel, clique and odd antihole constraints. The Lovász–Schrijver hierarchy, the Sherali–Adams hierarchy (1990) and Lasserre's semidefinite hierarchy (2001) are the three reference lift-and-project methods; their rank lower bounds are a standard tool for proving that a relaxation cannot solve a combinatorial problem in few rounds.

This mission formalizes the first structural fact about the operator $N$: iterating it $n$ times on any relaxation in $n$ variables yields exactly the 0–1 hull (Theorem 1.4 of the paper).

Setting

Vectors live in $\mathbb R^{n+1}$ with coordinates $x_0, x_1, \dots, x_n$; the space $\mathbb R^n$ of the original problem is the hyperplane $x_0 = 1$, and polytopes are replaced by the convex cones they generate.

• A convex cone is a nonempty set closed under addition and nonnegative scaling. For a set $S$, $\operatorname{cone}(S)$ is the set of nonnegative combinations of finitely many vectors of $S$.
• The polar cone of $K$ is $K^* = \{u : u^{\mathsf T}x \ge 0 \text{ for all } x \in K\}$.
• A 0–1 vector has every coordinate, $x_0$ included, equal to $0$ or $1$. The cube cone $Q$ is the cone spanned by the 0–1 vectors with $x_0 = 1$; it is the cone over the unit cube.
• For a convex cone $K$, $K^\circ$ is the cone spanned by the 0–1 vectors in $K$. For $K \subseteq Q$ this is the cone over the convex hull of the 0–1 points of the relaxation.

For convex cones $K_1, K_2 \subseteq Q$, the matrix cone $M(K_1, K_2)$ consists of the $(n+1)\times(n+1)$ real matrices $Y = (y_{ij})$ such that

1. $Y$ is symmetric;
2. $y_{ii} = y_{0i}$ for $1 \le i \le n$ (the diagonal equals the 0th column);
3. $u^{\mathsf T} Y v \ge 0$ for every $u \in K_1^*$ and $v \in K_2^*$.

$M_+(K_1, K_2)$ adds the condition that $Y$ is positive semidefinite. The projections are $N(K_1, K_2) = \{Ye_0 : Y \in M(K_1, K_2)\}$ and $N_+(K_1, K_2) = \{Ye_0 : Y \in M_+(K_1, K_2)\}$, where $e_0$ is the 0th unit vector. The cut operator is $N(K) = N(K, Q)$, and its iterates are $N^0(K) = K$, $N^t(K) = N(N^{t-1}(K))$.

Two families of hyperplanes appear in the proofs: $H_i = \{x : x_i = 0\}$ and $G_i = \{x : x_i = x_0\}$, the hyperplanes through the two opposite facets of $Q$ in direction $i$.

Formalization targets

Goal: Theorem 1.4

For every closed convex cone $K \subseteq Q$,

$$N^n(K) = K^\circ .$$

The statement is uniform in $n$ and in $K$: no polyhedrality, no bound on the number of constraints, and no assumption that $K$ contains a 0–1 point.

Milestones

1. Condition (iii″). For a closed convex cone $K \subseteq Q$ and a symmetric $Y$ with $y_{ii} = y_{0i}$: $Y \in M(K, Q)$ if and only if every column of $Y$ is in $K$ and the difference of the first column and any other column is in $K$.
2. Lemma 1.1. For closed convex cones $K_1, K_2 \subseteq Q$,

$$(K_1 \cap K_2)^\circ \subseteq N_+(K_1, K_2) \subseteq N(K_1, K_2) \subseteq K_1 \cap K_2 .$$

3. Lemma 1.3. For a closed convex cone $K \subseteq Q$ and every $1 \le i \le n$,

$$N(K) \subseteq (K \cap H_i) + (K \cap G_i).$$

4. Claim (4) in the proof of Theorem 1.4. For every set $T$ of $t \ge 1$ coordinates, with $\bar F$ the union of the faces of the unit cube that fix the coordinates in $T$ to $0$ or $1$,

$$N^t(K) \subseteq \operatorname{cone}(K \cap \bar F).$$

5. The remark after Lemma 1.1. $N(K_1 \cap K_2, K_1 \cap K_2) \subseteq N(K_1, K_2) \subseteq N(K_1 \cap K_2, Q)$.

Significance

Theorem 1.4 is what makes $N$ a hierarchy rather than a single cut: the relaxations $K \supseteq N(K) \supseteq N^2(K) \supseteq \dots$ reach the 0–1 hull after at most $n$ rounds, so the $N$-rank of a valid inequality (the least $t$ with the inequality valid for $N^t(K)$) is a well-defined number between $0$ and $n$. The rest of the paper measures combinatorial constraints by this rank: odd hole constraints have rank one on the stable set polytope, and the rank of a stable set inequality is bounded by its defect. Rank lower bounds for lift-and-project hierarchies, in the literature that followed, all presuppose this finite convergence.

The theorem is proved in the paper; to the best of available knowledge none of the Lovász–Schrijver operators has been formalized in a proof assistant. A formalization provides machine-checked definitions of the matrix cones and the cut operators that later missions in this series (odd holes, the defect bound, the $N_+$ constraints) state their results against, and a checked proof of the column characterization (iii″) that all of those proofs use.

Difficulty

The inclusion $K^\circ \subseteq N^n(K)$ follows from Lemma 1.1 once each $N^t(K)$ is known to be a convex cone. The reverse inclusion is the content. A first attempt shows that one round of $N$ forces one coordinate to be integral, and then iterates; but $N(K)$ is not contained in the union of $K \cap H_i$ and $K \cap G_i$, only in their Minkowski sum (Lemma 1.3), so a point of $N(K)$ is not itself integral in any coordinate. The induction must carry a statement about cones spanned by intersections with unions of cube faces, and it needs each iterate $N^t(K)$ to again be a closed convex cone inside $Q$ so that Lemma 1.3 can be reapplied. Closedness of the projection $N(K)$ is not automatic: a linear image of a closed cone need not be closed.

Formalization scope

• Coordinates of $\mathbb R^{n+1}$ are indexed by Option ι for a finite type ι; none is $x_0$ and some i is $x_i$, and $n$ is the cardinality of ι, which may be $0$.
• $\operatorname{cone}(S)$ is Mathlib's PointedCone.hull ℝ S; $Q$ and $K^\circ$ are defined as spans of 0–1 vectors, as on the page, not by the inequality description $0 \le x_i \le x_0$.
• $M(K_1, K_2)$ is defined by condition (iii) through the polar cones; the column form (iii″) is a milestone, not the definition.
• The operators $N$, $N_+$ and the iterates are defined on arbitrary sets; the hypotheses (convex cone, contained in $Q$, closed) are carried by the theorems.
• Closedness. The paper tacitly takes its cones closed (they are polyhedral in all its applications), and the rewriting (iii′) on p. 169 needs it. Every statement here assumes the cones closed. Without this the goal is false: for $K = \{x : 0 < x_1 < x_0\} \cup \{0\}$ in $\mathbb R^2$, $K^\circ = \{0\}$ while $N(K) = Q$.
• In the proof of Theorem 1.4 the page places the cube $Q'$ in the hyperplane "$x_0 = 0$"; this is a misprint for $x_0 = 1$, and claim (4) is formalized with $x_0 = 1$.
• Not formalized in this mission: Lemma 1.2 (the dual description of $N(K)^*$), Lemma 1.5 (the $N_+$ analogue of Lemma 1.3, part of a later mission), and the algorithmic results of Section 1.c.

Contributions welcome: proofs that $N(K)$ is a closed convex cone contained in $Q$ whenever $K$ is, a proof of $Q^* = \operatorname{cone}\{e_i, e_0 - e_i\}$, and lemmas on cones spanned by the intersection of a generating set with a supporting hyperplane; these are reusable by the other missions of the series.

Selected references

• L. Lovász and A. Schrijver, Cones of matrices and set-functions and 0–1 optimization, SIAM Journal on Optimization 1(2) (1991) 166–190. https://doi.org/10.1137/0801013
• H. D. Sherali and W. P. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM Journal on Discrete Mathematics 3(3) (1990) 411–430. https://doi.org/10.1137/0403036
• J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM Journal on Optimization 11(3) (2001) 796–817. https://doi.org/10.1137/S1052623400366802
• M. Laurent, A comparison of the Sherali–Adams, Lovász–Schrijver, and Lasserre relaxations for 0–1 programming, Mathematics of Operations Research 28(3) (2003) 470–496. https://doi.org/10.1287/moor.28.3.470.16391

Motivation

David S. Johnson's 1974 paper Approximation Algorithms for Combinatorial Problems (J. Comput. System Sci. 9 (1974) 256–278) was one of the first systematic worst-case analyses of polynomial-time heuristics for NP-complete optimization problems. Its Section 5 proves the harmonic bound $\sum_{j=1}^k 1/j$ for the greedy algorithm on minimum-cardinality set cover, a result that still underlies the standard $\ln n$ approximation guarantee.

Section 6, the subject of this mission, asks what happens when the cost of a cover is its total size rather than its number of sets. This problem, SET COVERING II (EC), is the optimization version of the EXACT COVER recognition problem of Karp's list (Karp 1972): a family has a disjoint subcover exactly when the optimum equals the number of covered points. Johnson shows that the change of measure breaks the cardinality greedy but that a greedy rule based on an overlap ratio recovers essentially the same guarantee. The same accounting (paying for each newly covered point) later became the standard analysis of greedy weighted set cover (Chvátal 1979).

Setting

An input is a finite family $F = \{S_1, \dots, S_p\}$ of finite sets. Its covered set is $T = \bigcup_{S \in F} S$. A subcover is a subfamily $F' \subseteq F$ with $\bigcup_{S\in F'} S = T$, and its measure is

$$m_{EC}(F') = \sum_{S \in F'} |S|.$$

The optimum $F^*$ is the least measure of a subcover; since every subcover has measure at least $|T|$, an optimal subcover is one with the least possible overlapping. The subproblem EC$(k)$ admits only families in which every set has at most $k$ points.

Algorithm C2 keeps a subfamily SUB (initially empty), the unused sets LEFT (initially $F$) and the uncovered points UNCOV (initially $T$). While UNCOV is nonempty it chooses $S' \in$ LEFT minimizing

$$\mathrm{Ratio}(S) = \frac{|S - \mathrm{UNCOV}|}{|S \cap \mathrm{UNCOV}|},$$

the number of already-covered points of $S$ per newly covered point, and moves $S'$ from LEFT to SUB, removing its points from UNCOV. When several sets tie, any of them may be chosen; a subcover is choosable by C2 if some sequence of admissible choices returns it.

The overlap of a chosen set is $|S' - \mathrm{UNCOV}|$ at the moment it is chosen, and the cumulative overlap $\mathrm{OV}(F_1)$ of a run returning $F_1$ is the sum of these overlaps.

Formalization targets

Goal: Theorem 6 (p. 271)

For all $k \ge 1$ and $n > 0$,

$$R[C2, EC(k)](n) \le 1 + \ln(k) \le \sum_{j=1}^k \frac1j + \frac12,$$

and for all sufficiently large $n$, $R[C2, EC(k)](n) \ge \sum_{j=1}^k (1/j)$. In the size-free form used here, for every $k \ge 1$:

1. every subcover $M$ choosable by C2 on an input of EC$(k)$ satisfies $m_{EC}(M) \le (1 + \ln k)\,F^*$;
2. $1 + \ln k \le \sum_{j=1}^k 1/j + 1/2$;
3. some input of EC$(k)$ with $F^* > 0$ has a choosable subcover with $m_{EC}(M) \ge \big(\sum_{j=1}^k 1/j\big) F^*$.

Milestones (proof of Theorem 6, pp. 271–272)

• the measure of the output is $|T| + \mathrm{OV}(F_1)$;
• if C2 may choose a set with $\mathrm{Ratio}(S') \ge y$, then $(y+1)\,|\mathrm{UNCOV}| \le F^*$;
• with $a = F^*/|T|$ and $x = |T - \mathrm{UNCOV}|/|T|$, the next chosen set has $\mathrm{Ratio}(S') \le a/(1-x) - 1$;
• on EC$(k)$, $\mathrm{OV}(F_1) \le |T|\,(a[\ln(k) + 1] - 1)$;
• the analytic inequality $1 + \ln(k) \le \sum_{j=1}^k 1/j + 1/2$;
• the lower-bound input (Fig. 1 of the paper with every set of $F_1$ filled out to exactly $k$ points) on which C2 may pay $\sum_{j=1}^k 1/j$ times the optimum.

Significance

The result. The measure $\sum|S|$ penalizes overlap, and the paper notes (without proof, p. 270) that an algorithm returning an optimal cover for the cardinality measure can be a factor $k$ from optimal for this one. Theorem 6 shows that the ratio rule C2 is within $1 + \ln k$ of the least-overlap cover, and the lower bound shows that no analysis of C2 can beat $\sum_{j=1}^k 1/j$. The two bounds differ by less than $1/2$ for every $k$. The theorem was an early instance of a logarithmic guarantee for a weighted covering problem, where each set's cost is its size.

Formalizing it. The theorem has been proved since 1974; no machine-checked proof is known to exist. The mission produces a formal model of the EC problem and of C2 as a nondeterministic process, the overlap identity, and the discrete form of the paper's area-under-a-curve estimate. The last is the part the paper argues informally, through a step function and an integral.

Difficulty

The cardinality argument for C1 counts the sets chosen; here the sets have different sizes, so it does not apply. The overlap C2 pays per newly covered point is not bounded by a constant: early choices can be free and late ones cost up to $k-1$ per point, and the bound on the cumulative overlap must hold against the whole run, for every sequence of tie-breaks.

In a formal proof the integral must be replaced by a sum. Covered points arrive in blocks (one block per chosen set), the charge is constant on a block but the bound depends on the covered fraction at the start of the block, and the sum has to be compared with a logarithm. The terms $a\ln a$ and $a/k$ that the paper drops using $1 \le a \le k$ must be controlled as well, and the relation $1 \le a \le k$ must itself be proved from optimality. The lower bound needs an explicit run of C2 through ties on an input with $k \cdot k!$ points, checking at every stage that the intended set is a ratio minimizer.

Formalization scope

• Inputs. A family is p : ℕ with S : Fin p → Finset α (0-based, repetitions allowed; a repeated set counts twice in the measure if both copies are chosen, which C2 never does). Subcovers and SUB, LEFT are index sets. $F^*$ is a minimum over the finite, nonempty set of subcovers (Finset.inf'), never a junk value.
• Algorithm. C2 is a step relation on states (SUB, LEFT, UNCOV). The choice at Step 3 is existential over all minimizers, so every result quantifies over every choosable output (the paper's WORST). $\mathrm{Ratio}(S)$ is $+\infty$ when $S \cap \mathrm{UNCOV} = \emptyset$; the formal rule requires the chosen set to meet UNCOV and compares ratios by cross-multiplication, with no division.
• Overlap. The cumulative overlap depends on the run, not on the output alone, so it is carried by an inductive run relation RunOV.
• No problem size. The paper's $R[A, P](n)$ maximizes over inputs of size at most $n$ in an unspecified encoding. Upper bounds are stated for every input and every choosable output; the lower bound exhibits one input and one choosable output. Given monotonicity of $R$ in $n$, these are equivalent to the paper's claims. Ratios are stated multiplicatively, so $F^* = 0$ does not create a vacuous bound.
• Numbers. Measures are natural numbers cast to $\mathbb R$; $\ln$ is Real.log; $\sum_{j=1}^k 1/j$ is Mathlib's harmonic k.
• Ruled out. A deterministic tie-break would prove a weaker upper bound and could not realize the lower-bound run, and a ratio with $x/0 = 0$ would make disjoint-from-UNCOV sets the most attractive choice. The formalization uses neither.

The overlap identity and the discrete integral comparison are reusable for any greedy covering analysis that charges cost per newly covered point. Contributions welcome include proofs of the milestones, the invariants of the C2 run relation (SUB and LEFT partition the indices; UNCOV $= T - \bigcup$ SUB), and the explicit lower-bound run.

Selected references

• D. S. Johnson, Approximation algorithms for combinatorial problems, J. Comput. System Sci. 9 (1974), 256–278. https://doi.org/10.1016/S0022-0000(74)80044-9
• R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations, Plenum, 1972, 85–103. https://doi.org/10.1007/978-1-4684-2001-2_9
• V. Chvátal, A greedy heuristic for the set-covering problem, Math. Oper. Res. 4 (1979), 233–235. https://doi.org/10.1287/moor.4.3.233

Motivation

Multistage stochastic programs are solved on a finite scenario tree: a discrete probability distribution whose support points are paths of a random process. Realistic trees have far too many scenarios for the resulting optimization problem, so practitioners reduce the tree, keeping $n$ of its $N$ scenarios and redistributing the probability of the deleted ones. The reduction should keep the reduced distribution as close as possible to the original one in a probability metric that controls the optimal value of the stochastic program (Dupačová, Gröwe-Kuska, Römisch, Math. Program. 95 (2003)).

Choosing the best $n$ scenarios is a set-covering problem and NP-hard, and the algorithms used in practice (backward reduction, fast forward selection) are heuristics without error guarantees. Heitsch and Römisch (2003) therefore derived test instances with an exactly known optimum: regular binary and ternary scenario trees, for which the minimal reduction distance has a closed form once $n$ is not too small. This mission formalizes the ternary case, Proposition 3.2 of that paper. The binary case (Proposition 3.1) is a separate mission of the same series.

Setting

Fix a depth $K \in \mathbb{N}$ and branch widths $\delta^1, \dots, \delta^K \ge 0$, with $\delta^0 = 0$. A regular ternary scenario tree has $N = 3^K$ scenarios, one for each index tuple $(i_1, \dots, i_K) \in \{1, 2, 3\}^K$, where $i_k$ is the successor chosen at level $k$. Choosing successor $i_k$ adds the increment $\delta^k_{i_k} = (i_k - 2)\,\delta^k \in \{-\delta^k, 0, \delta^k\}$, and scenario $i$ is the vector $\omega_i = (\omega_i^0, \dots, \omega_i^K) \in \mathbb{R}^{K+1}$ with

$$\omega_i^k = \sum_{j=0}^{k} \delta^j_{i_j}, \qquad k = 0, \dots, K \quad \text{(eq. (19))}.$$

All scenarios have probability $p_i = 1/N$. The distance between scenarios is the maximum norm $c(\omega_i, \omega_j) = \|\omega_i - \omega_j\|_\infty = \max_{0 \le k \le K} |\omega_i^k - \omega_j^k|$.

Deleting the scenarios of an index set $J$ and moving each deleted scenario's probability to a nearest kept scenario costs the reduction distance

$$D_J = \sum_{i \in J} p_i \min_{j \notin J} \|\omega_i - \omega_j\|_\infty \quad \text{(eq. (8))},$$

which by Theorem 2.1 of the paper is the optimal transport-type distance between the original distribution and the best distribution supported on the kept scenarios. The minimal reduction distance to $n$ scenarios is $D^{min}_n = \min\{D_J : \#J = N - n\}$.

Formalization targets

Goal: Proposition 3.2 (7/9-solution)

Let $K \ge 3$ and let $k_0 \in \arg\min_{1 \le k \le K} \delta^k$ with $k_0 \le K - 2$ and $\max\{\delta^{k_0+1}, \delta^{k_0+2}\} \le 2\delta^{k_0}$. Then any two distinct scenarios are at distance at least $\delta^{k_0}$; there is a set of $\tfrac79 N$ scenarios each paired with a scenario outside it at distance exactly $\delta^{k_0}$; and for each $n \in \mathbb{N}$ with $\tfrac29 N \le n < N$,

$$D^{min}_n = \min\{D_J : \#J = N - n\} = \frac{N - n}{N}\,\delta^{k_0} \quad \text{(eq. (21))},$$

with the minimum attained.

Milestones

1. Distinct scenarios satisfy $\|\omega_i - \omega_j\|_\infty \ge \delta^{k_0}$.
2. Every $J$ with $\#J = N - n$ has $D_J \ge \frac{N-n}{N}\delta^{k_0}$.
3. The index set $I_{**}$ of the proof has $\#I_{**} = \tfrac29 N$, and its complement $J_{**}$ has $\tfrac79 N$ elements.
4. Every $j \in J_{**}$ has a partner $i \in I_{**}$ with $\|\omega_i - \omega_j\|_\infty = \delta^{k_0}$.
5. Example 4.2: for $K = 6$ and $(\delta^1, \dots, \delta^6) = (0.7, 0.9, 1.2, 1.5, 2.6, 3.3)$, $D^{min}_n = 0.7\,\frac{N-n}{N}$ for $162 \le n < 729$.

Significance

The result gives an exact optimal value for an NP-hard reduction problem on an infinite family of instances. Heitsch and Römisch use it in their numerical section to measure how far the heuristics' reduced trees are from optimal (Examples 4.1 and 4.2 are the binary and ternary test trees of that study). A closed form of this kind is also the only way to certify that a heuristic is exactly optimal on some instances rather than only competitive with other heuristics.

The proposition is proved in the paper, but the published proof of its central step is one sentence: "Similarly as in Proposition 3.1 it can be shown that there exists an index $i \in I_{**}$ for each $j \in J_{**}$ …". A formal proof supplies that case analysis, which is absent from the literature. The formalization also settles two points the printed statement leaves loose (see Formalization scope): the count of pairs at distance $\delta^{k_0}$, and the definition of $I_{**}$ when some widths vanish. No machine-checked version of this result or of the reduction distance $D_J$ is known to exist.

Difficulty

The lower bound is routine: two distinct scenarios first differ at some level $l$, where their coordinates differ by $\delta^l$ or $2\delta^l$. The substance is attainment: one must exhibit, for every $n \ge \tfrac29 N$, a kept set of size $n$ whose every deleted scenario lies at distance exactly $\delta^{k_0}$ from some kept one. The obvious candidate, keeping the scenarios that take the middle branch at level $k_0$, has every other scenario at distance exactly $\delta^{k_0}$ from a kept one, but it keeps $\tfrac13 N$ scenarios and so covers only $n \ge \tfrac13 N$. Going down to $\tfrac29 N$ kept scenarios forces a deleted scenario and its partner to differ at more than one level, and since coordinates are running sums the differences at levels $k_0+1$ and $k_0+2$ accumulate on top of the one at level $k_0$. The paper's proof of this step is not written out, and it depends on the widths of the two levels below $k_0$: the hypothesis $\max\{\delta^{k_0+1}, \delta^{k_0+2}\} \le 2\delta^{k_0}$ is essential, and the result is false without it (for $K = 3$ and $(\delta^1, \delta^2, \delta^3) = (1, 3, 3)$ one has $D^{min}_6 = 31/27$, not $7/9$).

Formalization scope

• A scenario is an index tuple $\sigma : \mathrm{Fin}\,K \to \mathrm{Fin}\,3$; $\sigma(r)$ is the successor at paper level $r + 1$, with $\mathrm{Fin}\,3$ values $0, 1, 2$ standing for the paper's $i = 1, 2, 3$. The widths are $\delta : \mathbb{N} \to \mathbb{R}$, of which only $\delta(1), \dots, \delta(K)$ are used; the standing assumption $\delta^k \in \mathbb{R}_+$ (p. 196) is the hypothesis $\delta(k) \ge 0$ for $1 \le k \le K$. Scenarios live in $\mathrm{Fin}(K+1) \to \mathbb{R}$, whose Mathlib norm is the maximum norm. Probabilities are uniform, $1/3^K$.
• $D_J$ is defined for a general finite index set, probabilities and cost, with the inner minimum a Finset.inf' over the complement of $J$; the complement must be nonempty, so no default value arises. $D^{min}_n$ is stated as IsLeast of the set of all values $D_J$ with $\#J = N - n$: the goal asserts both the lower bound for every $J$ and attainment by some $J$. A formalization that exhibits a single $J$ with $D_J = \frac{N-n}{N}\delta^{k_0}$, or states an infimum without attainment, or drops any hypothesis on $k_0$, is a different (and in the last case false) statement.
• $\tfrac29 N \le n$ is written $2 \cdot 3^K \le 9n$, and $\tfrac79 N$ as $7 \cdot 3^{K-2}$.
• Pairs. As printed, "there are $\tfrac79 N$ distinct pairs of scenarios such that the distance between the members of each pair is exactly $\delta^{k_0}$" is false as an exact count (for $K = 3$ and $\delta = (1,1,1)$ there are 130 such pairs, not 21). The goal states what the proof constructs: a set $J_{**}$ of $\tfrac79 N$ scenarios, each paired with a scenario outside $J_{**}$ at distance exactly $\delta^{k_0}$.
• $I_{**}$. The paper defines $I_{**}$ by testing whether the increments $\delta^k_{i_k}$ vanish. When one of $\delta^{k_0}, \delta^{k_0+1}, \delta^{k_0+2}$ is $0$ this no longer identifies the middle branch and the count $\tfrac29 N$ fails, although the proposition remains true. The formalization defines $I_{**}$ by branch indices (middle branch versus outer branches), which agrees with the paper whenever these three widths are positive. No positivity hypothesis is added to the goal.
• Welcome contributions: a reusable library for regular scenario trees (first differing level, distance of paths), and the case analysis of milestone 4. The binary mission of this series needs the same lower-bound argument with the constant $2\delta^{k_0}$.

Selected references

• H. Heitsch, W. Römisch, Scenario Reduction Algorithms in Stochastic Programming, Computational Optimization and Applications 24 (2003), 187–206. https://doi.org/10.1023/A:1021805924152
• J. Dupačová, N. Gröwe-Kuska, W. Römisch, Scenario reduction in stochastic programming: An approach using probability metrics, Mathematical Programming 95 (2003), 493–511. https://doi.org/10.1007/s10107-002-0331-0

Why exact reduction distances matter

Multistage stochastic programs are solved on a finite scenario tree, a discrete probability measure whose atoms are paths of a stochastic process. The size of the deterministic equivalent grows with the number of scenarios, so practitioners replace the original measure $P=\sum_{i=1}^N p_i\delta_{\omega_i}$ by a measure supported on $n<N$ of its scenarios. Stability theory for stochastic programs (Dupačová, Gröwe-Kuska and Römisch, Math. Program. 95 (2003), doi:10.1007/s10107-002-0331-0) bounds the change of the optimal value by a probability metric between the two measures, which leads to the optimal scenario reduction problem: choose which $N-n$ scenarios to delete so that this distance is smallest.

That problem is a set-covering problem and is NP-hard, and the algorithms that Heitsch and Römisch study in the same paper (backward reduction, fast forward selection) are heuristics without error guarantees. To test them one needs original measures whose optimal reduction distance is known exactly. Section 3 of Heitsch and Römisch, Scenario Reduction Algorithms in Stochastic Programming, Comput. Optim. Appl. 24 (2003) (doi:10.1023/A:1021805924152) supplies such instances: regular binary and ternary scenario trees, for which the minimal distance to any reduced tree with at least a fixed fraction of the scenarios is an explicit formula. This mission formalizes the binary case, Proposition 3.1.

Setting

Fix a horizon $K\in\mathbb N$ and level parameters $\delta^1,\dots,\delta^K\ge0$, with $\delta^0=0$. A regular binary scenario tree has $N=2^K$ scenarios. A scenario is determined by a branch $i_k\in\{1,2\}$ at every level $k=1,\dots,K$, and is the vector $\omega_i=(\omega_i^0,\dots,\omega_i^K)\in\mathbb R^{K+1}$ with

$$\omega_i^k=\sum_{j=0}^k\delta^j_{i_j},\qquad \delta^j_{i_j}=(2i_j-3)\,\delta^j\in\{-\delta^j,+\delta^j\}.\tag{19}$$

All scenarios start at the root $\omega_i^0=0$ and carry probability $p_i=1/N$. Scenarios are compared in the maximum norm $\|\omega-\tilde\omega\|_\infty=\max_{k=0,\dots,K}|\omega^k-\tilde\omega^k|$.

Deleting the scenarios with indices in $J\subset\{1,\dots,N\}$ and moving each deleted scenario's probability to a nearest kept scenario costs the reduction cost

$$D_J=\sum_{i\in J}p_i\min_{j\notin J}\|\omega_i-\omega_j\|_\infty,\tag{8}$$

which by Theorem 2.1 of the paper is the minimal Kantorovich-type distance between $P$ and a measure supported on the kept scenarios. The minimal reduction distance for $n$ kept scenarios is $D^{min}_n=\min\{D_J:\#J=N-n\}$.

In the Lean development, scenarios are indexed by σ : Fin K → Fin 2 (Fin-index $r$ is tree level $r+1$, value $0$ is the branch $-\delta$, value $1$ is $+\delta$), lev σ k is the branch at level $k$, scenario δ σ : Fin (K+1) → ℝ is $\omega_\sigma$, and redCost δ J hJ is $D_J$.

Formalization targets

Goal: Proposition 3.1 (3/4-solution)

Let $K\ge3$, $k_0\in\arg\min_{1\le k\le K}\delta^k$, $k_0\le K-2$ and $\max\{\delta^{k_0+1},\delta^{k_0+2}\}\le2\delta^{k_0}$. Then any two distinct scenarios are at distance at least $2\delta^{k_0}$; there is a set $J_*$ of $\frac34N$ scenarios each of which has a partner outside $J_*$ at distance exactly $2\delta^{k_0}$; and for every $n\in\mathbb N$ with $\frac N4\le n<N$

$$D^{min}_n=\min\{D_J:\#J=N-n\}=\frac{N-n}{N}\,2\delta^{k_0}.\tag{20}$$

Milestones

1. Two scenarios that first differ at level $l$ are at distance $\ge2\delta^l\ge2\delta^{k_0}$.
2. $D_J\ge\frac{N-n}{N}2\delta^{k_0}$ for every $J$ with $\#J=N-n$.
3. The index set $I_*$ (branch at $k_0$ opposite to the common branch at $k_0+1,k_0+2$) has $\#I_*=N/4$, so $\#J_*=\frac34N$.
4. Every $j\in J_*$ has a partner $i\in I_*$ with $\|\omega_i-\omega_j\|_\infty=2\delta^{k_0}$.
5. Example 4.1: for $K=10$ and the paper's parameters, Proposition 3.1 applies with $k_0=1$ and $D^{min}_n=\frac{N-n}{N}$ for $256\le n<1024$.

Significance

The result. Proposition 3.1 gives the exact optimum of an NP-hard combinatorial problem on an explicit, parametrized family of instances of every size $N=2^K$. Section 4 of the paper uses it (Example 4.1, $N=1024$) as ground truth for the relative accuracy of backward reduction and fast forward selection. Without it, the quality of a heuristic reduction on a large tree could only be compared with other heuristics or with lower bounds.

The formalization. The proposition is proved in the paper; to our knowledge no machine-checked version exists. A formal proof certifies the benchmark values, and the statement also corrects the printed text in two places. First, "there are $\frac34N$ distinct pairs … at distance exactly $2\delta^{k_0}$" is false as an exact count (for $K=3$, $\delta=(1,1,1)$ there are twenty such pairs, not six), so the goal states "at least", in the form the proof exhibits. Second, the paper's sign-based definition of $I_*$ degenerates when some $\delta^k=0$, although the proposition still holds; the mission defines $I_*$ by branch indices.

Difficulty

The lower bound is a direct computation. The content is the matching upper bound: a set $J$ of the prescribed size for which every deleted scenario has a kept scenario at the minimal possible distance. A natural first attempt pairs scenarios that differ only at level $k_0$. That handles only half of the scenarios with a single partner each, and it cannot reach $\frac34N$ deleted scenarios. Once scenarios differ at more than one level, their maximum-norm distance is a maximum of several partial sums, and keeping all of them at most $2\delta^{k_0}$ is exactly where the hypothesis $\max\{\delta^{k_0+1},\delta^{k_0+2}\}\le2\delta^{k_0}$ enters. Without it, eq. (20) fails: for $K=3$, $\delta=(1,3,3)$ and $n=2$ the true minimum is $\frac52$, not $\frac32$. The passage from $n=N/4$ to general $n\ge N/4$ also needs care: the deleted set must shrink while each remaining deleted scenario keeps its partner among the kept ones.

Formalization scope

• The index type is Fin K → Fin 2, which has exactly $2^K$ elements. The paper's $(K+1)$-tuple has a level-0 entry with no choice, so it is dropped, and the vector $\omega$ keeps its $K+1$ coordinates with $\omega^0=0$.
• The parameters are δ : ℕ → ℝ, with $\delta^k\ge0$ and the arg min stated for $k=1,\dots,K$ only. $\delta^k>0$ is not assumed, because the paper allows $\delta^k\in\mathbb R_+$ and the proposition holds with zeros.
• The cost is Mathlib's norm on Fin (K+1) → ℝ, which is the maximum norm, and $p_i=1/2^K$ is written out.
• $D_J$ requires a nonempty set of kept scenarios, and its inner minimum is a finite Finset.inf'.
• $D^{min}_n$ is stated with IsLeast over the set of attained values $D_J$, $\#J=N-n$, so both the lower bound and attainment are part of the goal. A proof of $D_{J}\le\frac{N-n}{N}2\delta^{k_0}$ for a single exhibited $J$, or a real infimum without attainment, does not prove the goal.
• "$\frac N4\le n$" is written $2^K\le4n$, and $\frac34N$ is $3\cdot2^{K-2}$.
• The pairs claim is "at least $\frac34N$ pairs", expressed as a set $J_*$ of that size with a partner outside $J_*$ for every member.
• All hypotheses on $k_0$ appear in the goal; dropping any of them makes (20) false.

Useful infrastructure: sup-norm lemmas for Fin n → ℝ (pi_norm_le_iff_of_nonneg, norm_le_pi_norm), Finset.inf' lemmas, and counting functions Fin K → Fin 2 with prescribed values (Fintype.card_fun, Fintype.card_pi). The tree and reduction-cost definitions are shared in spirit with the ternary-tree mission of this series (Proposition 3.2), and a proof whose structure transfers to $d=3$ is welcome. Contributions of proofs of the milestones individually, in any order, are welcome.

Selected references

• H. Heitsch, W. Römisch, Scenario Reduction Algorithms in Stochastic Programming, Computational Optimization and Applications 24 (2003), 187–206. doi:10.1023/A:1021805924152
• J. Dupačová, N. Gröwe-Kuska, W. Römisch, Scenario reduction in stochastic programming: An approach using probability metrics, Mathematical Programming 95 (2003), 493–511. doi:10.1007/s10107-002-0331-0

Why reduce scenarios

Multistage and two-stage stochastic programs are solved numerically on a discrete probability distribution: a finite set of scenarios $\omega_1,\dots,\omega_N$ with probabilities $p_1,\dots,p_N$. The size of the resulting optimization problem grows with $N$, and scenario sets produced by sampling or by historical data are often far too large to be solved directly. Scenario reduction replaces the original distribution by one supported on a small subset of the scenarios, chosen so that the optimal value and solutions of the stochastic program change as little as possible.

Stability theory for stochastic programs (Rachev and Römisch, 2002) shows that this change is controlled by a probability distance of Fortet–Mourier type, which for discrete measures is bounded by the value of a transportation problem. Dupačová, Gröwe-Kuska and Römisch (2003) turned this into a combinatorial problem and proposed greedy backward and forward heuristics. Heitsch and Römisch (2003) gave faster versions of both heuristics; the forward version, fast forward selection, is the subject of this mission. Implementations of these reduction heuristics are distributed with the GAMS modelling system (SCENRED) and are used in energy and finance applications of stochastic programming.

Setting

Let $E$ be a finite-dimensional real vector space with a norm $\|\cdot\|$, let $\omega_0\in E$, and let $h:[0,\infty)\to[0,\infty)$ be continuous and nondecreasing with $h(0)=0$. The cost between two points of $E$ is

$$c(\omega,\tilde\omega)=\max\bigl\{1,\,h(\|\omega-\omega_0\|),\,h(\|\tilde\omega-\omega_0\|)\bigr\}\,\|\omega-\tilde\omega\| .$$

It is nonnegative, symmetric, and zero on the diagonal.

The original distribution is $P=\sum_{i=1}^N p_i\delta_{\omega_i}$ with $p_i>0$ and $\sum_i p_i=1$. Deleting the scenarios in a set $J\subset\{1,\dots,N\}$ and assigning new weights $q_j\ge 0$, $\sum_{j\notin J}q_j=1$, to the kept ones gives $Q=\sum_{j\notin J}q_j\delta_{\omega_j}$. The distance $D(J;q)$ between $P$ and $Q$ is the optimal value of the transportation problem

$$D(J;q)=\min\Bigl\{\sum_{i=1}^N\sum_{j\notin J}c(\omega_i,\omega_j)\eta_{ij}:\ \eta_{ij}\ge 0,\ \sum_{i}\eta_{ij}=q_j,\ \sum_{j\notin J}\eta_{ij}=p_i\Bigr\}.$$

The reduction cost of deleting $J$ is

$$D_J=\sum_{i\in J}p_i\min_{j\notin J}c(\omega_i,\omega_j),$$

and the optimal reduction problem (8) minimizes $D_J$ over all $J$ with $\#J=N-n$, where $n$ is the number of scenarios to keep.

Forward selection builds the kept set greedily. With $J^{[0]}=\{1,\dots,N\}$ and $J^{[i]}=\{1,\dots,N\}\setminus\{u_1,\dots,u_i\}$, it chooses

$$u_i\in\arg\min_{u\in J^{[i-1]}}D_{J^{[i-1]}\setminus\{u\}},\qquad i=1,\dots,n. \tag{16}$$

Fast forward selection (Algorithm 2.4) computes the same choices through an updated cost matrix: $c^{[1]}_{ku}=c(\omega_k,\omega_u)$, $c^{[i]}_{ku}=\min\{c^{[i-1]}_{ku},c^{[i-1]}_{ku_{i-1}}\}$, $z^{[i]}_u=\sum_{k\in J^{[i-1]}\setminus\{u\}}p_kc^{[i]}_{ku}$, and $u_i\in\arg\min_{u\in J^{[i-1]}}z^{[i]}_u$.

Formalization targets

Goal: Theorem 2.5

For $1\le n\le N$ and every run $u_1,\dots,u_n$ of Algorithm 2.4, with any tie-breaking in the arg min,

$$u_i\ \text{satisfies (16)}\quad\text{and}\quad z^{[i]}_{u_i}=D_{J^{[i]}}\qquad(i=1,\dots,n).$$

Milestones

1. Theorem 2.1 (redistribution). For $J$ with at least one kept scenario, $D_J=\min_q D(J;q)$, and the minimum is attained at $\bar q_j=p_j+\sum_{i\in J,\,j(i)=j}p_i$ for every choice of nearest kept scenarios $j(i)$.
2. Eq. (10). $D_{\{1,\dots,N\}\setminus\{u\}}=\sum_{i=1}^Np_ic(\omega_i,\omega_u)$, so (8) with $\#J=N-1$ is problem (10).
3. Eq. (12). The sum $lb$ of the $N-n$ smallest single-deletion costs $p_l\min_{j\neq l}c(\omega_l,\omega_j)$, taken in the greedy order (11), is at most $D_J$ for every $J$ with $\#J=N-n$.
4. Optimality condition (p. 191). If each $l_i$ has a nearest other scenario outside $\{l_1,\dots,l_{N-n}\}\setminus\{l_i\}$, then $\{l_1,\dots,l_{N-n}\}$ solves (8).
5. Eq. (17), unrolled recursion. For any index sequence, $c^{[i]}_{ku}=\min_{j\notin J^{[i-1]}\setminus\{u\}}c(\omega_k,\omega_j)$ for $u\in J^{[i-1]}$.
6. Eq. (17), conclusion. For any index sequence, $z^{[i]}_u=D_{J^{[i-1]}\setminus\{u\}}$ for $u\in J^{[i-1]}$.

Significance

Theorem 2.5 certifies that the cheap update of Algorithm 2.4 (one pairwise minimum per matrix entry and step) produces exactly the greedy forward selection defined through the reduction costs, and that the running objective $z^{[i]}_{u_i}$ is the reduction cost of the scenarios deleted so far. Combined with Theorem 2.1, $z^{[i]}_{u_i}$ is the optimal transportation distance between $P$ and the best measure on the kept scenarios, which is the quantity practitioners monitor to decide how many scenarios to keep. The lower bound (12) and the optimality condition give a posteriori quality certificates for any reduced set.

All results of this mission are proved in the paper or in the works it cites (Dupačová et al., 2003); none is open. To the best of a platform search, none has been machine-checked. The mission provides a verified specification of a widely deployed algorithm, a formal link between a combinatorial set-covering objective and a finite transportation problem, and definitions (reduction cost, transportation plans with a partially free target marginal, greedy runs with arbitrary tie-breaking) reusable by the regular-tree missions of this series and by later scenario-tree construction papers.

Difficulty

The mathematics is elementary; the difficulty is bookkeeping. The recursion for $c^{[i]}$ refers to the previous step's column $u_{i-1}$, which itself was updated, so unrolling it to a minimum over $\{u,u_1,\dots,u_{i-1}\}$ is an induction on $i$ in which the index sets $J^{[i]}$, the 1-based step counter and the complement structure all move together. The natural first attempt, identifying $c^{[i]}_{ku}$ with the minimum over the complement of $J^{[i]}$, is off by one step: the correct set is the complement of $J^{[i-1]}\setminus\{u\}$, which contains $u$ itself. For Theorem 2.1 the lower bound requires using that $c(\omega_i,\omega_i)=0$ for kept scenarios and that every plan ships all of $p_i$ somewhere outside $J$; the attainment part requires constructing the plan explicitly from the choice $j(\cdot)$, including scenarios for which several kept scenarios are equally near.

Formalization scope

• Scenarios are ω : Fin N → E with E a finite-dimensional real normed space; the paper's closed set $\Omega\subset\mathbb R^s$ plays no role beyond containing the scenarios and is omitted. Scenarios need not be distinct.
• $h$ is a function ℝ → ℝ with the paper's assumptions imposed on $[0,\infty)$ (IsGrowthFunction); every theorem carries them, together with $p_i>0$ and $\sum_ip_i=1$.
• The functions $f_0$, $g$ and the stochastic program (1)–(2) that motivate $c$ appear in no statement.
• $D(J;q)$ is the paper's finite transportation problem (p. 188), not the Kantorovich functional on measures. Weights $q$ and plans $\eta$ are indexed by all of $\{1,\dots,N\}$ with entries at deleted indices fixed to $0$.
• $D_J$ requires a proof that the complement of $J$ is nonempty; minima are Finset.inf', never a real infimum with a default value.
• Algorithm 2.4 is a relation on sequences u : ℕ → Fin N with 1-based steps. $c^{[i]}$ is the printed recursion, extended to all indices; runs are any sequences satisfying the arg-min conditions, so every tie-breaking rule is covered.
• The paper's standing restriction $n<N$ is relaxed to $n\le N$ in Theorem 2.5; the statement remains true at $n=N$.
• A trivializing formalization is ruled out: defining $c^{[i]}$ or $z^{[i]}$ directly as the minimum over the selected set or as $D_{J^{[i-1]}\setminus\{u\}}$ would make Theorem 2.5 hold by definition, and proving it for one fixed tie-breaking rule would prove less than the paper; neither is done.
• Proofs of the milestones, alternative proofs of Theorem 2.1 via LP duality, and a verified executable implementation of Algorithm 2.4 are all welcome.

Selected references

• H. Heitsch, W. Römisch, Scenario Reduction Algorithms in Stochastic Programming, Computational Optimization and Applications 24 (2003), 187–206. https://doi.org/10.1023/A:1021805924152
• J. Dupačová, N. Gröwe-Kuska, W. Römisch, Scenario reduction in stochastic programming: an approach using probability metrics, Mathematical Programming 95 (2003), 493–511. https://doi.org/10.1007/s10107-002-0331-0
• S. T. Rachev, W. Römisch, Quantitative stability in stochastic programming: the method of probability metrics, Mathematics of Operations Research 27 (2002), 792–818. https://doi.org/10.1287/moor.27.4.792.304

Motivation

A monotone variational inequality asks for a point at which a monotone operator and a convex function are in equilibrium. It unifies convex minimization (where $F$ is a gradient), convex–concave saddle-point problems (where $F$ is the skew gradient of a Lagrangian), Nash equilibria of monotone games, and complementarity problems in economics and traffic assignment. In operations research, first-order methods for such problems are the workhorse behind large-scale saddle-point formulations of linear and conic programs, where only one operator evaluation and one projection or proximal step per iteration are affordable.

The classical method for Lipschitz monotone operators is Korpelevich's extragradient method (1976) and its proximal variant, Tseng's forward–backward–forward method (2000); both need two evaluations of $F$ per iteration. The reflected projected gradient method of Malitsky (SIAM J. Optim., 2015) uses one evaluation of $F$ but evaluates it at $2z^k-z^{k-1}$, a point that may lie outside the domain of $g$. Malitsky's Golden Ratio Algorithm (GRAAL), introduced in Golden Ratio Algorithms for Variational Inequalities (preprint 2018; published in Mathematical Programming, doi:10.1007/s10107-019-01416-w), uses one evaluation of $F$, always at a feasible point, and one proximal step per iteration. Its fixed-step version, Theorem 1 of that paper, is the subject of this mission; the explicit, adaptive-step version (Theorem 2) is a separate mission of this series.

Setting

Let $\mathcal E$ be a finite-dimensional real inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\| = \sqrt{\langle\cdot,\cdot\rangle}$. Let $g:\mathcal E\to(-\infty,+\infty]$ and write $\operatorname{dom} g = \{x : g(x)<+\infty\}$. Let $F:\operatorname{dom} g\to\mathcal E$. The variational inequality is

$$\text{find } z^*\in\mathcal E \quad\text{such that}\quad \langle F(z^*), z-z^*\rangle + g(z)-g(z^*)\ \ge\ 0\qquad \forall z\in\mathcal E. \tag{1}$$

The standing assumptions are:

• (C1) the solution set $S$ of (1) is nonempty;
• (C2) $g$ is proper (never $-\infty$, finite somewhere), convex, and lower semicontinuous;
• (C3) $F$ is monotone: $\langle F(u)-F(v),u-v\rangle\ge0$ for all $u,v\in\operatorname{dom} g$.

The proximal operator of $g$ is $\operatorname{prox}_g(z) = \operatorname{argmin}_x\{g(x)+\tfrac12\|x-z\|^2\}$. Let $\varphi = \frac{\sqrt5+1}{2}$ be the golden ratio, so that $\varphi^2 = 1+\varphi$. For a step $\lambda>0$ and arbitrary starting points $z^1,\bar z^0\in\mathcal E$, the Golden Ratio Algorithm generates, for $k\ge1$,

$$\bar z^k = \frac{(\varphi-1)z^k + \bar z^{k-1}}{\varphi},\qquad z^{k+1} = \operatorname{prox}_{\lambda g}\big(\bar z^k - \lambda F(z^k)\big). \tag{6}$$

The first line is a convex combination of the newest iterate and the previous average; the second is a forward–backward step taken from the average rather than from $z^k$.

Formalization targets

Goal: Theorem 1

If $F$ is $L$-Lipschitz on $\operatorname{dom} g$ ($L>0$), (C1)–(C3) hold, and $\lambda\in\big(0,\frac{\varphi}{2L}\big]$, then there is $z^*\in S$ with

$$z^k\to z^*\qquad\text{and}\qquad \bar z^k\to z^*\qquad(k\to\infty).$$

Both sequences converge, to one and the same solution. The goal is stated with the paper's exact step range; no rate is claimed, as the paper claims none.

Milestones

1. Eq. (4), the prox-inequality: for proper convex lsc $g$,

$$\bar x = \operatorname{prox}_g z \iff \langle\bar x - z, x-\bar x\rangle\ge g(\bar x)-g(x)\quad\forall x\in\mathcal E.$$

2. Eq. (12), an identity using only the averaging step of (6): for every point $z^*$,

$$\|z^{k+1}-z^*\|^2 = (1+\varphi)\|\bar z^{k+1}-z^*\|^2-\varphi\|\bar z^k-z^*\|^2+\tfrac1\varphi\|z^{k+1}-\bar z^k\|^2 .$$

3. Eq. (14), the energy inequality: for $z^*\in S$ and $k\ge2$,

$$(1+\varphi)\|\bar z^{k+1}-z^*\|^2+\tfrac\varphi2\|z^{k+1}-z^k\|^2\le(1+\varphi)\|\bar z^k-z^*\|^2+\tfrac\varphi2\|z^k-z^{k-1}\|^2-\varphi\|z^k-\bar z^k\|^2 .$$

4. Lemma 1 (Bauschke–Combettes, Theorem 5.5): a sequence that is Fejér monotone with respect to a nonempty set $C$ and whose cluster points all lie in $C$ converges to a point of $C$.

Significance

The result. Theorem 1 shows that monotone variational inequalities with a Lipschitz operator can be solved with one operator evaluation and one proximal step per iteration, with $F$ evaluated only at points of $\operatorname{dom} g$, where it is defined. This matters when $F$ is expensive (a large matrix–vector product, a simulation) or undefined outside the feasible set (for instance an operator involving $\log x$ on the positive orthant). The analysis also explains the constant: the averaging weight $\varphi$ is the largest $c$ with $1/c\ge c-1$, and the step bound $\varphi/(2L)$ follows from it. The fixed-step analysis is the template for the explicit, adaptive-step EGRAAL of the same paper (Theorem 2), which needs only local Lipschitz continuity of $F$.

The formalization. The theorem has a published proof, and no machine-checked version of it or of GRAAL is known. Mathlib contains the golden ratio, Lipschitz conditions, lower semicontinuity and cluster points, but no proximal operator of an extended-valued function, no prox-inequality and no Fejér-monotonicity convergence lemma. This mission produces those pieces and a complete convergence proof for a first-order VI method, which are reusable for projected gradient, forward–backward, extragradient and reflected-gradient analyses.

Difficulty

The naive approach, to show that $\|z^k-z^*\|$ decreases, fails: GRAAL is not Fejér monotone in $z^k$, because the forward step is taken from the average $\bar z^k$ and uses $F(z^k)$ rather than $F$ at the new point. The quantity that decreases is an energy mixing $\|\bar z^k-z^*\|^2$ with the successive difference $\|z^k-z^{k-1}\|^2$, and both the averaging identity and the Lipschitz estimate must produce matching coefficients for the cross terms to cancel. The energy inequality alone gives only boundedness and vanishing successive differences; convergence of the whole sequence, and the fact that the limit solves (1) when $g$ is merely lower semicontinuous and extended-valued, is a separate step. On the formal side, $g$ takes the value $+\infty$, so the prox-inequality and the variational inequality must be handled in extended arithmetic without letting $\infty-\infty$ decide anything.

Formalization scope

• $\mathcal E$ is a type E with [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E].
• $g$ is E → EReal. (C2) is IsProperConvexLSC g: never $\bot$, somewhere finite, convex epigraph $\{(x,t)\in E\times\mathbb R: g(x)\le t\}$, and LowerSemicontinuous g on all of E. $\operatorname{dom} g$ is effDom g = {x | g x ≠ ⊤}.
• $F$ is a total function E → E; monotonicity and the Lipschitz bound $\|F(u)-F(v)\|\le L\|u-v\|$ are required on effDom g only. The step range is 0 < λ, λ ≤ φ / (2 * L) with 0 < L and φ = Real.goldenRatio.
• $S$ is solutionSet g F: points of effDom g satisfying (1) for every $z\in E$, evaluated in EReal.
• The proximal step is the argmin predicate IsProxPoint (fun x => λ * g x) w z⁺, not a choice function, so no junk value is involved. A run of (6) is IsGRAALRun g F λ z zbar on sequences ℕ → E indexed as in the paper: $z^1$ and $\bar z^0$ are free and the entry $z^0$ is unused.
• The conclusion is ∃ zs ∈ solutionSet g F, Tendsto z atTop (𝓝 zs) ∧ Tendsto zbar atTop (𝓝 zs).

The hypotheses of the goal are jointly satisfiable, so the theorem is not vacuous: for $g\equiv0$ and $F\equiv0$ every point is a solution and constant sequences form a run of (6); a formalization under which IsGRAALRun has no instances, or in which $S$ may be empty, is ruled out. Two hypotheses are added to printed statements and flagged in their notes: $C\neq\emptyset$ in Lemma 1, which is false without it, and $z^1\in\operatorname{dom} g$ in Eq. (14), needed at $k=2$ because the paper's $F$ is only defined on $\operatorname{dom} g$.

Welcome contributions: existence and uniqueness of the proximal point of a proper convex lsc function in finite dimensions; the prox-inequality; Fejér-monotonicity lemmas; the energy inequality; and the final convergence argument. The prox and Fejér infrastructure is independent of the golden ratio and is shared with the second mission of this series.

Selected references

• Y. Malitsky, Golden Ratio Algorithms for Variational Inequalities, preprint, Optimization Online 6598, 2018. https://optimization-online.org/wp-content/uploads/2018/05/6598.pdf ; published in Mathematical Programming. https://doi.org/10.1007/s10107-019-01416-w
• H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, 2011 (2nd ed. 2017). https://doi.org/10.1007/978-3-319-48311-5
• G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika i Matematicheskie Metody 12 (1976) 747–756.
• P. Tseng, A modified forward–backward splitting method for maximal monotone mappings, SIAM J. Control Optim. 38 (2000) 431–446. https://doi.org/10.1137/S0363012998338806
• Y. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim. 25 (2015) 502–520. https://doi.org/10.1137/14097238X