The Proxy Benders Decomposition

2026-06-05

AI Summary

This work addresses the slow convergence of classical Benders decomposition in large-scale mixed-integer optimization, which stems from repeatedly solving similar subproblems. The authors propose a surrogate Benders decomposition framework that replaces exact subproblem solutions with certifiable optimization surrogates. A self-supervised predict-project-complete mechanism generates dual-feasible solutions to construct valid Benders cuts, ensuring theoretical correctness without relying on prediction accuracy. The approach dramatically improves computational efficiency: on 2000×2000 uncapacitated facility location instances, it achieves a median optimality gap below 0.5%, a median speedup of 161×, and reduces the number of generated cuts by over 240× in the largest cases.

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.

Problem

• Benders decomposition
• large-scale optimization
• slow convergence
• subproblem redundancy
• mixed-integer programming

Innovation

• Proxy Benders Decomposition
• Certified Optimization Proxies
• Self-supervised Prediction
• Benders Cuts
• Large-scale Mixed-integer Optimization