Sparse multi-term disjunctive cuts for the epigraph of a function of binary variables

TL;DR

This paper introduces a new method for generating sparse multi-term disjunctive cuts for the epigraph of functions of binary variables, improving computational efficiency and solution quality in mixed-integer linear programming.

Contribution

The paper presents a novel approach to generate multi-term disjunctive cuts using restricted variable support, enabling the use of more terms and enhancing MILP solution methods.

Findings

Sparse cuts close nearly as much gap as full disjunctive cuts.

Including these cuts significantly reduces solution time and optimality gap.

Method adapts to optimally 'tilt' valid inequalities by modifying sparse coefficients.

Abstract

We propose a new method for separating valid inequalities for the epigraph of a function of binary variables. The proposed inequalities are disjunctive cuts defined by disjunctive terms obtained by enumerating a subset $I$ of the binary variables. We show that by restricting the support of the cut to the same set of variables $I$, a cut can be obtained by solving a linear program with $2^{|I|}$ constraints. While this limits the size of the set $I$ used to define the multi-term disjunction, the procedure enables generation of multi-term disjunctive cuts using far more terms than existing approaches. We present two approaches for choosing the subset of variables. Experience on three MILP problems with block diagonal structure using $|I|$ up to size 10 indicates the sparse cuts can often close nearly as much gap as the multi-term disjunctive cuts without this restriction and in a fraction of the time. We also find that including these cuts within a cut-and-branch solution method for these MILP problems leads to significant reductions in solution time or ending optimality gap for instances that were not solved within the time limit. Finally, we describe how the proposed approach can be adapted to optimally "tilt" a given valid inequality by modifying the coefficients of a sparse subset of the variables.