Benders Cut Filtering for Affine Potential-Based Flow Problems with Robustness Scenarios and Topology Switching

TL;DR

This paper investigates Benders cut filtering strategies to improve large-scale optimization, proposing violation-based, diversity-based, and hybrid methods, which significantly reduce solution times in complex flow problems.

Contribution

It introduces novel Benders cut filtering techniques, including a hybrid approach, and demonstrates their effectiveness in reducing computational time for affine potential-based flow problems.

Findings

All filtering strategies solve more instances than the baseline.

Hybrid filtering achieves the largest reduction in solve time.

Filtering reduces geometric mean solve time by over 55%.

Abstract

Many large-scale optimization problems decompose into a master problem and scenario subproblems, a structure that can be exploited by Benders decomposition. In Benders decomposition, each iteration may generate many cuts from scenario subproblems, and adding all of them as constraints then causes the master problem to grow rapidly. These are constraints that may need to be added to the master problem to guarantee optimality and feasibility of solutions, but we can avoid adding those constraints that are never violated. Adding fewer cuts per iteration can reduce the number of cuts added in total, but increase the number of iterations. In contrast, the cuts filtered for regular cut selection in mixed-integer programming solvers are optional and added exclusively to improve runtime behavior. We study Benders cut filtering: given the Benders cuts produced in an iteration, which subset should be added to the master problem? To our knowledge, few prior works have studied this question. We propose violation-based filtering (retaining the most-violated cuts), diversity-based filtering via k-medoids clustering on pairwise cosine distances (adding an original cut closest to the cluster centroid), and a hybrid that selects a most-violated cut per cluster. Each strategy can be augmented with an aggregated cut that retains discarded information. Computational experiments on 149 instances of an affine potential-based flow problem with topology switching and robustness scenarios -- solved via Benders decomposition -- show that all informed filtering strategies solve at least 125 instances (vs. 91 for the unfiltered baseline), reducing shifted geometric mean solve time by 55-57%. The hybrid strategy attains the best geometric mean (271.89 s vs. 629.34 s, a 57% reduction, p < 0.001).