Benders Decomposition using Graph Modeling and Multi-Parametric Programming

Abstract

Benders decomposition is a widely used method for solving large optimization problems, but its performance is often hindered by the repeated solution of subproblems. We propose a flexible and modular algorithmic framework for accelerating Benders decomposition by embedding multi-parametric programming (mp) surrogates for optimization subproblems. Our approach leverages the OptiGraph abstraction in Plasmo$.$jl to model and decompose graph-structured problems. By solving the subproblems associated with the graph nodes once using mp, we can extract explicit piecewise affine mappings for primal and dual variables which replace the expensive subproblem solves with efficient look-ups and function evaluations during the iterative Benders process. We formally show the equivalence between classical Benders cuts and those derived from the mp solution and implement this integration in the open-source PlasmoBenders$.$jl software package. We apply it to a two-stage stochastic programming problem, which aims to make optimal capacity expansion decisions under uncertainty in product demand/prices and availability of raw materials. We evaluate single-cut and multi-cut variants of Benders and show that the mp surrogate approach achieves substantial speedups in subproblem solve time while preserving the convergence guarantees of Benders. Furthermore, we highlight advantages in the solution analysis and interpretability that is enabled by mp critical region tracking. Our results demonstrate that combining mp programming with graph modeling offers a promising and extensible foundation for structure-exploiting decomposition. By decomposing the problem into tractable subproblems, the proposed approach also aims to overcome scalability issues of mp, and the use of mp surrogates provides a unifying modeling framework to represent heterogeneous graph subproblems as common modeling objects.