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arxiv: 2606.07403 · v1 · pith:O25YTLLUnew · submitted 2026-06-05 · 🧮 math.OC · cs.LG

The Proxy Benders Decomposition

Changkun Guan , El Mehdi Er Raqabi , Mathieu Tanneau , Pascal Van Hentenryck This is my paper

Pith reviewed 2026-06-27 21:09 UTC · model grok-4.3

classification 🧮 math.OC cs.LG
keywords Benders decompositionoptimization proxiesmixed-integer programmingfacility locationnetwork designdecomposition methodscertified proxies
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The pith

A projection-and-completion layer lets optimization proxies generate valid Benders cuts without exact subproblem solves.

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

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.

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.

What carries the argument

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

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.

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.

Figures

Figures reproduced from arXiv: 2606.07403 by Changkun Guan, El Mehdi Er Raqabi, Mathieu Tanneau, Pascal Van Hentenryck.

Figure 1
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. 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. 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 ↗
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] view at source ↗
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] view at source ↗
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] view at source ↗
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] view at source ↗
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] view at source ↗
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] view at source ↗
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 the left. Proof. The primal (A.1) is feasible and the dual (A.2) has a nonempty compact feasible region, so strong duality yields d(¯y) ⊤λ ⋆ =… view at source ↗
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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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 no independent evidence
    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.

pith-pipeline@v0.9.1-grok · 5816 in / 1395 out tokens · 17020 ms · 2026-06-27T21:09:47.600665+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

88 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    2005 , publisher=

    Integer Programming: State of the art and recent advances , author=. 2005 , publisher=

    2005

  2. [2]

    50 Years of

    J. 50 Years of. 2009 , publisher=

    2009

  3. [3]

    Optima , volume=

    A celebration of 50 years of integer programming , author=. Optima , volume=

  4. [4]

    , title =

    Nair, Vinod and Hinton, Geoffrey E. , title =. Proceedings of the 27th International Conference on International Conference on Machine Learning , pages =. 2010 , isbn =

    2010

  5. [5]

    Operations Research , volume=

    Benders decomposition for production routing under demand uncertainty , author=. Operations Research , volume=. 2015 , publisher=

    2015

  6. [6]

    Decomposition based hybrid

    Behnamian, Javad , journal=. Decomposition based hybrid. 2014 , publisher=

    2014

  7. [7]

    Benders decomposition,

    Boschetti, Marco and Maniezzo, Vittorio , journal=. Benders decomposition,. 2009 , publisher=

    2009

  8. [8]

    International Journal of Mathematics in Operational Research , volume=

    A stochastic programming approach for sawmill production planning , author=. International Journal of Mathematics in Operational Research , volume=. 2013 , publisher=

    2013

  9. [9]

    Application of

    Canto, Salvador Perez , journal=. Application of. 2008 , publisher=

    2008

  10. [10]

    Computers & operations research , volume=

    A survey on benders decomposition applied to fixed-charge network design problems , author=. Computers & operations research , volume=. 2005 , publisher=

    2005

  11. [11]

    Benders decomposition and an

    Lin, Sifeng and Lim, Gino J and Bard, Jonathan F , journal=. Benders decomposition and an. 2016 , publisher=

    2016

  12. [12]

    1981 , publisher=

    Magnanti, Thomas L and Wong, Richard T , journal=. 1981 , publisher=

    1981

  13. [13]

    Numerische mathematik , volume=

    Partitioning procedures for solving mixed-variables programming problems , author=. Numerische mathematik , volume=

  14. [14]

    Rahmaniani, Ragheb and Crainic, Teodor Gabriel and Gendreau, Michel and Rei, Walter , journal=. The. 2017 , publisher=

    2017

  15. [15]

    Operations Research , volume=

    The Benders dual decomposition method , author=. Operations Research , volume=. 2020 , publisher=

    2020

  16. [16]

    1997 , publisher=

    Introduction to linear optimization , author=. 1997 , publisher=

    1997

  17. [17]

    Combining

    Zeighami, Vahid and Soumis, Fran. Combining. Transportation Science , volume=. 2019 , publisher=

    2019

  18. [18]

    Transportation science , volume=

    Benders decomposition for simultaneous aircraft routing and crew scheduling , author=. Transportation science , volume=. 2001 , publisher=

    2001

  19. [19]

    Combinatorial

    Codato, Gianni and Fischetti, Matteo , journal=. Combinatorial. 2006 , publisher=

    2006

  20. [20]

    A note on the selection of

    Fischetti, Matteo and Salvagnin, Domenico and Zanette, Arrigo , journal=. A note on the selection of. 2010 , publisher=

    2010

  21. [21]

    On generating maximal nondominated

    Sherali, Hanif D and Lunday, Brian J , journal=. On generating maximal nondominated. 2013 , publisher=

    2013

  22. [22]

    Improving

    Poojari, Chandra A and Beasley, John E , journal=. Improving. 2009 , publisher=

    2009

  23. [23]

    Strengthened

    Bodur, Merve and Dash, Sanjeeb and G. Strengthened. INFORMS Journal on Computing , volume=. 2017 , publisher=

    2017

  24. [24]

    Accelerating

    Wentges, Paul , journal=. Accelerating. 1996 , publisher=

    1996

  25. [25]

    RAIRO-Operations Research , volume=

    Solving the simple plant location problem by genetic algorithm , author=. RAIRO-Operations Research , volume=. 2001 , publisher=

    2001

  26. [26]

    European Journal of Operational Research , volume=

    An algorithm for solving large capacitated warehouse location problems , author=. European Journal of Operational Research , volume=. 1988 , publisher=

    1988

  27. [27]

    Mathematical programming , volume=

    An analysis of approximations for maximizing submodular set functions—I , author=. Mathematical programming , volume=. 1978 , publisher=

    1978

  28. [28]

    Operations Research , volume=

    A dual-based procedure for uncapacitated facility location , author=. Operations Research , volume=. 1978 , publisher=

    1978

  29. [29]

    Solving large nonconvex water resources management models using generalized

    Cai, Ximing and McKinney, Daene C and Lasdon, Leon S and Watkins Jr, David W , journal=. Solving large nonconvex water resources management models using generalized. 2001 , publisher=

    2001

  30. [30]

    Operations research , volume=

    Simultaneous assignment of locomotives and cars to passenger trains , author=. Operations research , volume=. 2001 , publisher=

    2001

  31. [31]

    Operations research , volume=

    A priori optimization of the probabilistic traveling salesman problem , author=. Operations research , volume=. 1994 , publisher=

    1994

  32. [32]

    Solving an electricity generating capacity expansion planning problem by generalized

    Bloom, Jeremy A , journal=. Solving an electricity generating capacity expansion planning problem by generalized. 1983 , publisher=

    1983

  33. [33]

    Operations research , volume=

    Benders decomposition for large-scale uncapacitated hub location , author=. Operations research , volume=. 2011 , publisher=

    2011

  34. [34]

    Management Science , volume=

    Elements of large-scale mathematical programming Part I: Concepts , author=. Management Science , volume=. 1970 , publisher=

    1970

  35. [35]

    Operations research , volume=

    Selected topics in column generation , author=. Operations research , volume=. 2005 , publisher=

    2005

  36. [36]

    Incremental

    Er Raqabi, El Mehdi and Himmich, Ilyas and El Hachemi, Nizar and El Hallaoui, Issmaïl and Soumis, François , journal =. Incremental. 2023 , issn =. doi:https://doi.org/10.1016/j.omega.2022.102821 , url =

    work page doi:10.1016/j.omega.2022.102821 2023

  37. [37]

    Stabilized

    Baena, Daniel and Castro, Jordi and Frangioni, Antonio , journal=. Stabilized. 2020 , publisher=

    2020

  38. [38]

    2023 , publisher=

    Himmich, Ilyas and Er Raqabi, El Mehdi and El Hachemi, Nizar and El Hallaoui, Issmaïl and Metrane, Abdelmoutalib and Soumis, François , journal=. 2023 , publisher=

    2023

  39. [39]

    Les Cahiers du GERAD , volume =

    Er Raqabi, El Mehdi and Wu, Yong and El Hallaoui, Issmail and Soumis, François , pages =. Les Cahiers du GERAD , volume =. 2023 , month = aug, month_numeric =

    2023

  40. [40]

    INFORMS Journal on Computing , volume=

    Val. INFORMS Journal on Computing , volume=. 2005 , publisher=

    2005

  41. [41]

    Operations Research , volume=

    Ben Amor, Hatem and Desrosiers, Jacques and Val. Operations Research , volume=. 2006 , publisher=

    2006

  42. [42]

    Vanderbeck, Fran. On. Operations research , volume=. 2000 , publisher=

    2000

  43. [43]

    Mathematical programming , volume=

    On the solution of highly degenerate linear programmes , author=. Mathematical programming , volume=. 1988 , publisher=

    1988

  44. [44]

    Transportation Research Part A: Policy and Practice , volume=

    An integrated model of facility location and transportation network design , author=. Transportation Research Part A: Policy and Practice , volume=. 2001 , publisher=

    2001

  45. [45]

    Degeneracy in the multi-source

    Brimberg, Jack and Mladenovic, Nenad , journal=. Degeneracy in the multi-source

  46. [46]

    Mathematical programming , volume=

    A projection method for the uncapacitated facility location problem , author=. Mathematical programming , volume=. 1990 , publisher=

    1990

  47. [47]

    Mathematical Programming , volume=

    An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts , author=. Mathematical Programming , volume=. 2008 , publisher=

    2008

  48. [48]

    Redesigning

    Fischetti, Matteo and Ljubi. Redesigning. Management Science , volume=. 2017 , publisher=

    2017

  49. [49]

    1998 , publisher=

    Integer programming , author=. 1998 , publisher=

    1998

  50. [50]

    Mathematical Programming , volume=

    Safe bounds in linear and mixed-integer linear programming , author=. Mathematical Programming , volume=. 2004 , publisher=

    2004

  51. [51]

    Operations Research , volume=

    Vessel service planning in seaports , author=. Operations Research , volume=. 2022 , publisher=

    2022

  52. [52]

    On-demand delivery from stores:

    Liu, Sheng and Luo, Zhixing , journal=. On-demand delivery from stores:. 2023 , publisher=

    2023

  53. [53]

    Manufacturing & Service Operations Management , volume=

    Budgeting in international humanitarian organizations , author=. Manufacturing & Service Operations Management , volume=. 2022 , publisher=

    2022

  54. [54]

    Manufacturing & Service Operations Management , volume=

    On the values of vehicle-to-grid electricity selling in electric vehicle sharing , author=. Manufacturing & Service Operations Management , volume=. 2021 , publisher=

    2021

  55. [55]

    2021 , publisher=

    Gleixner, Ambros and Hendel, Gregor and Gamrath, Gerald and Achterberg, Tobias and Bastubbe, Michael and Berthold, Timo and Christophel, Philipp and Jarck, Kati and Koch, Thorsten and Linderoth, Jeff and others , journal=. 2021 , publisher=

    2021

  56. [56]

    The irace package:

    L. The irace package:. Operations Research Perspectives , volume=. 2016 , publisher=

    2016

  57. [57]

    International Business Machines Corporation , volume=

    V12.1:. International Business Machines Corporation , volume=. 2009 , publisher=

    2009

  58. [58]

    2023 , howpublished=

    2023

  59. [59]

    Computers and Electronics in Agriculture , volume=

    Precision farming: The environmental challenge , author=. Computers and Electronics in Agriculture , volume=. 2001 , publisher=

    2001

  60. [60]

    US Geological Survey: Reston, VA, USA , volume=

    Mineral commodity summaries , author=. US Geological Survey: Reston, VA, USA , volume=

  61. [61]

    State of the

    Gorman, Michael F and Clarke, John-Paul and Gharehgozli, Amir Hossein and Hewitt, Michael and de Koster, Rene and Roy, Debjit , journal=. State of the. 2014 , publisher=

    2014

  62. [62]

    Mathematical

    Zhen, Lu and Li, Haolin and Xiao, Liyang and Lin, Dayu and Wang, Shuaian , journal=. Mathematical. 2024 , publisher=

    2024

  63. [63]

    Interfaces , volume=

    Dispatch optimization in bulk tanker transport operations , author=. Interfaces , volume=. 2018 , publisher=

    2018

  64. [64]

    2016 , publisher=

    Ding, Yi and Jia, Shuai and Gu, Tianyi and Li, Chung-Lun , journal=. 2016 , publisher=

    2016

  65. [65]

    Interfaces , volume=

    Medium-term rail scheduling for an iron ore mining company , author=. Interfaces , volume=. 2014 , publisher=

    2014

  66. [66]

    Journal of Heuristics , volume=

    Proximity Benders: a decomposition heuristic for stochastic programs , author=. Journal of Heuristics , volume=. 2016 , publisher=

    2016

  67. [67]

    Mathematical Programming , volume=

    Simplicial decomposition in nonlinear programming algorithms , author=. Mathematical Programming , volume=. 1977 , publisher=

    1977

  68. [68]

    European Journal of Operational Research , volume=

    Benders decomposition for very large scale partial set covering and maximal covering location problems , author=. European Journal of Operational Research , volume=. 2019 , publisher=

    2019

  69. [69]

    Er Raqabi, El Mehdi and El Hallaoui, Issmail and Soumis, François , pages =. The. Les Cahiers du GERAD , volume =. 2023 , month = aug, month_numeric =

    2023

  70. [70]

    2025 , publisher=

    Er Raqabi, El Mehdi and Beljadid, Ahmed and Bennouna, Mohammed Ali and Bennouna, Rania and Boussaadi, Latifa and El Hachemi, Nizar and El Hallaoui, Issmail and Fender, Michel and Jamali, Mohamed Anouar and Si Hammou, Nabil and others , journal=. 2025 , publisher=

    2025

  71. [71]

    Operations Research Letters , volume=

    Measuring the impact of primal heuristics , author=. Operations Research Letters , volume=. 2013 , publisher=

    2013

  72. [72]

    Column generation , pages=

    A primer in column generation , author=. Column generation , pages=. 2005 , publisher=

    2005

  73. [73]

    Advances in Neural Information Processing Systems , volume=

    Dual lagrangian learning for conic optimization , author=. Advances in Neural Information Processing Systems , volume=

  74. [74]

    Electric Power Systems Research , volume=

    Real-time risk analysis with optimization proxies , author=. Electric Power Systems Research , volume=. 2024 , publisher=

    2024

  75. [75]

    Operations Research , volume=

    Deepest cuts for Benders decomposition , author=. Operations Research , volume=. 2025 , publisher=

    2025

  76. [76]

    European Journal of Operational Research , volume=

    A branch-and-benders-cut algorithm for the crew scheduling and routing problem in road restoration , author=. European Journal of Operational Research , volume=. 2019 , publisher=

    2019

  77. [77]

    Transportation Science , volume=

    A Branch-and-Benders Cut Algorithm for a Stochastic Service Network Design with Crowdsourced Capacity , author=. Transportation Science , volume=. 2026 , publisher=

    2026

  78. [78]

    Discrete Applied Mathematics , volume=

    Bundle-based relaxation methods for multicommodity capacitated fixed charge network design , author=. Discrete Applied Mathematics , volume=. 2001 , publisher=

    2001

  79. [79]

    European Journal of Operational Research , volume=

    Benders decomposition without separability: A computational study for capacitated facility location problems , author=. European Journal of Operational Research , volume=. 2016 , publisher=

    2016

  80. [80]

    and Amos, Brandon and Kolter, J

    Donti, Priya L. and Amos, Brandon and Kolter, J. Zico , title=. Advances in Neural Information Processing Systems (NeurIPS) , year=

Showing first 80 references.