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The Proxy Benders Decomposition

T0 review · 0 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read A projection-and-completion layer lets optimization proxies generate valid Benders cuts without exact subproblem solves.

desk verdict Proxy-BD swaps exact Benders subproblem solves for a self-supervised proxy plus projection-completion certification that keeps cuts valid regardless of prediction quality, with reported 161x speedups and 240x fewer cuts on large UFL instances. read the letter →

arxiv 2606.07403 v1 pith:O25YTLLU submitted 2026-06-05 math.OC cs.LG

Changkun Guan , El Mehdi Er Raqabi , Mathieu Tanneau , Pascal Van Hentenryck This is my paper · ORCID
classification math.OCcs.LG
keywords Bendersdecompositionoptimizationproxiesmixed-integerprogrammingfacilitylocationnetworkdesignmethodscertified
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces Proxy-BD to replace the repeated exact solves of subproblems in classical Benders decomposition with a self-supervised proxy that predicts solutions, then projects and completes them into dual-feasible points. The certification layer guarantees that every generated cut remains valid regardless of how accurate the proxy prediction is. A reader would care because classical Benders often slows down on large problems due to solving highly similar subproblems many times and zigzagging behavior. If the approach holds, decomposition methods could handle substantially larger instances in facility location and network design without losing correctness. Experiments on up to 2000 by 2000 instances report median optimality gaps below 0.5 percent along with large reductions in time and cut count.

What carries the argument

The projection-and-completion certification layer that converts proxy predictions into dual-feasible solutions for provably valid Benders cuts.

What would settle it

An instance in which the projection-and-completion layer fails to produce any dual-feasible solution from the proxy output, so that no valid cut can be generated and the algorithm cannot proceed correctly.

Watch

Extended reading notes

Core claim

The Proxy Benders decomposition framework replaces subproblem optimization with certified optimization proxies that follow a predict-project-and-complete mechanism to produce dual-feasible solutions for generating provably valid Benders cuts, preserving the theoretical validity of the decomposition independently of prediction quality.

Load-bearing premise

The projection-and-completion certification layer can always convert an arbitrary proxy prediction into a dual-feasible solution that produces a valid Benders cut, regardless of how poor the initial proxy output is.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces Proxy Benders Decomposition (Proxy-BD), replacing exact repeated solves of Benders subproblems with certified optimization proxies that follow a self-supervised predict-project-and-complete mechanism. This produces dual-feasible solutions and provably valid Benders cuts via a projection-and-completion certification layer whose validity holds independently of proxy prediction quality, supported by a formal characterization of proxy-induced cuts. The framework extends naturally to branch-and-Benders-cut schemes. Experiments on large-scale uncapacitated facility location instances (up to 2000×2000) and network design problems report median optimality gaps below 0.5%, median speedups up to 161×, and more than 240× fewer cuts on the largest instances, with gains scaling with recourse complexity.

Significance. If the formal characterization and validity proof hold, the work offers a meaningful advance by decoupling Benders cut validity from proxy accuracy, enabling safe use of learned proxies to accelerate decomposition without compromising guarantees. The reported empirical gains on large instances, together with the explicit scaling behavior and extension to modern variants, indicate practical utility. The parameter-free validity preservation by construction is a notable strength.

minor comments (1)
  1. [Abstract] The abstract and introduction could more explicitly distinguish the self-supervised training of the proxy from any post-hoc adjustments in the experimental section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive assessment of the formal validity results and empirical scaling behavior, and the recommendation to accept. No major comments were raised, so we have no revisions to address.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper introduces Proxy-BD as an algorithmic framework that replaces exact subproblem solves with a predict-project-and-complete proxy mechanism. Validity of the generated Benders cuts is established by a separate projection-and-completion certification layer whose formal characterization and proof of dual feasibility are presented as independent of the proxy's prediction accuracy. No equations or claims reduce a reported performance metric (speedups, cut counts, optimality gaps) to quantities defined by fitted parameters inside the paper; the experimental outcomes on UFL instances are treated as empirical consequences of the algorithm rather than tautological outputs. No load-bearing self-citations, imported uniqueness theorems, or ansatzes smuggled via prior work are invoked to justify the core derivation. The chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

Review performed on abstract only; therefore the ledger records only the high-level assumptions visible in the provided text. The proxy model itself is treated as an invented mechanism whose parameters are not enumerated.

assumptions (1)
  • domain assumption Benders cuts generated from dual-feasible solutions remain valid optimality cuts even when the dual solution is obtained via projection and completion rather than exact optimization.
    Stated in abstract paragraph 2 as the mechanism that preserves theoretical validity independently of prediction quality.
invented entities (1)
  • certified optimization proxy
    purpose: Replace repeated exact subproblem solves while still producing dual-feasible points for valid Benders cuts
    The proxy is the central new component introduced by the paper; no independent evidence of its existence or performance is supplied beyond the abstract claims.

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Cite this review

Pith. "Pith review of The Proxy Benders Decomposition." pith.science (2026). https://pith.science/paper/O25YTLLU

@misc{pith2026260607403,
  author       = {Pith},
  title        = {Pith review of: The Proxy Benders Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O25YTLLU}},
  note         = {Machine review of arXiv:2606.07403}
}
read the original abstract

Benders decomposition is a fundamental framework for solving large-scale mixed-integer optimization problems with complicating variables that, when fixed, yield significantly easier subproblems. However, classical Benders decomposition repeatedly solves highly similar subproblems and often exhibits zigzagging behavior across iterations, leading to slow convergence in large-scale settings. Motivated by the repetitive structure and parametric nature of Benders subproblems, this paper introduces the proxy Benders decomposition (Proxy-BD), a new decomposition framework in which subproblem optimization is replaced by certified optimization proxies rather than repeated exact solves. The proposed proxy follows a self-supervised predict-project-and-complete mechanism that produces dual-feasible solutions for generating provably valid Benders cuts. The framework preserves the theoretical validity of the decomposition independently of prediction quality through a projection-and-completion certification layer. A formal characterization of proxy-induced cuts is established, and the framework naturally extends to modern decomposition schemes, including branch-and-Benders-cut algorithms. Computational experiments on large-scale facility location and network design problems demonstrate that Proxy-BD substantially reduces the computational effort of subproblems while maintaining near-optimal solution quality. On large-scale uncapacitated facility location instances up to 2000x2000, Proxy-BD achieves median optimality gaps below 0.5%, yields up to 161x median speedups, and reduces the number of generated cuts by more than 240x on the largest instances. The computational gains consistently increase with recourse complexity, indicating that proxy-based inference scales substantially more favorably than repeated exact subproblem optimization in large-scale decomposition settings.

Figures

Figures reproduced from arXiv: 2606.07403 by the authors.

Figure 1
Figure 1. Slice-normalized geometry: the slice certificate [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Offline pipeline for Proxy-BD. Stage 1: Oracle Benders is run on training instances to [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Online Proxy-BD loop. At each iteration, the RMP is solved, and the proxy generates a [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Online B&BC Proxy-BD loop. The master MIP is solved once. The proxy is invoked as a [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Per-shape distribution of the proxy’s true optimality gap on held-out CAP instances, [PITH_FULL_IMAGE:figures/full_fig_p033_5.png]
Figure 6
Figure 6. Figure 6: Held-out SOTA proxy on UFL. Top: per-shape distribution of the proxy’s true optimality [PITH_FULL_IMAGE:figures/full_fig_p035_6.png]
Figure 7
Figure 7. Figure 7: Convergence under the SOTA deployment on a representative (median-speedup) instance [PITH_FULL_IMAGE:figures/full_fig_p036_7.png]
Figure 8
Figure 8. Figure 8: Scaling of solve time and optimality gap with instance size for CAP: exact outer-loop [PITH_FULL_IMAGE:figures/full_fig_p038_8.png]
Figure 9
Figure 9. Figure 9: Convergence of the optimality gap against the number of cuts separated for CAP. The [PITH_FULL_IMAGE:figures/full_fig_p039_9.png]
Figure 10
Figure 10. Figure 10: Left: the recourse-feasible cone K in d-space and the induced (non-facet) cut d1 +d2 ≤ 0 obtained from λ box = (1, 1). Right: the certificate cone R and the box-normalized feasible region R(1); maximizing d(¯y) ⊤λ yields λ box = (1, 1), which maps to the red cut on th…

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Forward citations

Cited by 1 Pith paper

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