Continuous cutting plane algorithms in integer programming

TL;DR

This paper introduces a novel continuous optimization approach for generating cutting planes in MILPs, framing the problem as a neural network training task to improve solution quality.

Contribution

It proposes a new continuous optimization framework for cutting plane generation, leveraging neural network parametrization, and demonstrates empirical improvements over traditional methods.

Findings

Empirical gains in MILP solving performance

Effective optimization of Gomory inequalities

Versatile approach across different MILP classes

Abstract

Cutting planes for mixed-integer linear programs (MILPs) are typically computed in rounds by iteratively solving optimization problems, the so-called separation. Instead, we reframe the problem of finding good cutting planes as a continuous optimization problem over weights parametrizing families of valid inequalities. This problem can also be interpreted as optimizing a neural network to solve an optimization problem over subadditive functions, which we call the subadditive primal problem of the MILP. To do so, we propose a concrete two-step algorithm, and demonstrate empirical gains when optimizing generalized Gomory mixed-integer inequalities over various classes of MILPs. Code for reproducing the experiments can be found at https://github.com/dchetelat/subadditive.