Deepest Cuts for Benders Decomposition

TL;DR

This paper introduces a geometric approach to Benders cut selection, focusing on the deepest cuts based on norms, which improve convergence and computational efficiency in large-scale mixed-integer problems.

Contribution

It proposes a novel unifying framework for Benders cut selection using geometric interpretation, including deepest cuts, and develops algorithms leveraging problem structure and duality.

Findings

Deepest cuts reduce computation time and iterations.

They produce high-quality bounds early in the process.

The approach unifies various cut selection strategies.

Abstract

Since its inception, Benders Decomposition (BD) has been successfully applied to a wide range of large-scale mixed-integer (linear) problems. The key element of BD is the derivation of Benders cuts, which are often not unique. In this paper, we introduce a novel unifying Benders cut selection technique based on a geometric interpretation of cut ``depth'', produce deepest Benders cuts based on $\ell_p$-norms, and study their properties. Specifically, we show that deepest cuts resolve infeasibility through minimal deviation (in a distance sense) from the incumbent point, are relatively sparse, and may produce optimality cuts even when classical Benders would require a feasibility cut. Leveraging the duality between separation and projection, we develop a Guided Projections Algorithm for producing deepest cuts while exploiting the combinatorial structure and decomposability of problem instances. We then propose a generalization of our Benders separation problem, which not only brings several well-known cut selection strategies under one umbrella, but also, when endowed with a homogeneous function, enjoys several properties of geometric separation problems. We show that, when the homogeneous function is linear, the separation problem takes the form of the Minimal Infeasible Subsystems (MIS) problem. As such, we provide systematic ways of selecting the normalization coefficients of the MIS method, and introduce a Directed Depth-Maximizing Algorithm for deriving these cuts. Inspired by the geometric interpretation of distance-based cuts and the repetitive nature of two-stage stochastic programs, we introduce a tailored algorithm to further facilitate deriving these cuts. Our computational experiments on various benchmark problems illustrate effectiveness of deepest cuts in reducing both computation time and number of Benders iterations, and producing high quality bounds at early iterations.