Symmetry detection for quadratic optimization using binary layered graphs

Abstract

Symmetry in mathematical optimization may create multiple, equivalent solutions. In nonconvex optimization, symmetry can negatively affect algorithm performance, e.g., of branch-and-bound when symmetry induces many equivalent branches. This paper develops detection methods for symmetry groups in quadratically-constrained quadratic optimization problems. Representing the optimization problem with adjacency matrices, we use graph theory to transform the adjacency matrices into binary layered graphs. We enter the binary layered graphs into the software package nauty that generates important symmetric properties of the original problem. Symmetry pattern knowledge motivates a discretization pattern that we use to reduce computation time for an approximation of the point packing problem. This paper highlights the importance of detecting and classifying symmetry and shows that knowledge of this symmetry enables quick approximation of a highly symmetric optimization problem.