Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

Georgia Kouyialis,Ruth Misener,Xiaoyu Wang

doi:10.3390/pr7110838

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.

Highlights

When the optimization variables can be permuted without changing the structure of the underlying optimization problem, we say that the formulation group of an optimization problem is symmetric [1,2]
Given an integer n > 0, the circle packing problem asks: what is the largest radius r for which n non-overlapping circles can be placed in the unit square? Costa et al [3] show that the formulation group, i.e., a subgroup of symmetry group generated by permuting variables and constraints, is isomorphic to a symmetry group created by permuting the variable indices and switching the two coordinates in a unit square (C2 × Sn )
The recent computational comparison of Pfetsch and Rehn [18] indicates that these state-of-the-art symmetry handling methods expedite the solution process for the MIPLIB 2010 instances and enable more instances to be solved in a time limit

Summary

Introduction

When the optimization variables can be permuted without changing the structure of the underlying optimization problem, we say that the formulation group of an optimization problem is symmetric [1,2]. A number of authors have considered a range of symmetry detection methods, e.g., for constraint programming [4], integer programming [1,5,6,7,8], and mixed-integer nonlinear optimization [2] These automatic symmetry detection methods can be used to mitigate the computational difficulties caused by symmetries, e.g., with symmetry-breaking constraints [9,10,11], objective perturbation [12], specialized branching strategies [13,14], cutting planes [15,16], and extended formulations [17]. The recent computational comparison of Pfetsch and Rehn [18] indicates that these state-of-the-art symmetry handling methods expedite the solution process for the MIPLIB 2010 instances and enable more instances to be solved in a time limit

Results

Discussion

Conclusion