New symmetries in mixed-integer linear optimization: Symmetry heuristics and complement-based symmetries

TL;DR

This paper introduces new symmetry-based heuristics and a novel class of complement-based symmetries for mixed-integer linear programming, enhancing solution quality and symmetry detection.

Contribution

It proposes symmetry heuristics using permutations and orbits, and defines complement-based symmetries for binary MILPs, expanding symmetry exploitation methods.

Findings

Heuristics improve feasible solution objective values.

Complement-based symmetries identify symmetric opposite decisions.

Techniques are illustrated with practical examples.

Abstract

We present two novel applications of symmetries for mixed-integer linear programming. First we propose two variants of a new heuristic to improve the objective value of a feasible solution using symmetries. These heuristics can use either the actual permutations or the orbits of the variables to find better feasible solutions. Then we introduce a new class of symmetries for binary MILP problems. Besides the usual permutation of variables, these symmetries can also take the complement of the binary variables. This is useful in situations when two opposite decisions are actually symmetric to each other. We discuss the theory of these symmetries and present a computational method to compute them. Examples are presented to illustrate the usefulness of these techniques.