Exploiting Symmetry in Integer Convex Optimization using Core Points

TL;DR

This paper introduces core points to exploit symmetry in integer convex optimization, enabling problem decomposition and competitive solving performance against commercial solvers.

Contribution

It defines core points and develops algorithms leveraging symmetry, improving solution methods for symmetric integer convex problems.

Findings

Core points enable effective problem decomposition.

Algorithms based on core points compete with commercial solvers.

Successfully solved an open MIPLIB problem.

Abstract

We consider convex programming problems with integrality constraints that are invariant under a linear symmetry group. To decompose such problems we introduce the new concept of core points, i.e., integral points whose orbit polytopes are lattice-free. For symmetric integer linear programs we describe two algorithms based on this decomposition. Using a characterization of core points for direct products of symmetric groups, we show that prototype implementations can compete with state-of-the-art commercial solvers, and solve an open MIPLIB problem.