Exploiting symmetries in SDP-relaxations for polynomial optimization

TL;DR

This paper explores methods to leverage symmetries in polynomial optimization problems using SDP relaxations, focusing on computational efficiency and block decomposition techniques for symmetric groups.

Contribution

It introduces new approaches for exploiting symmetries in polynomial optimization, including block decomposition and geometric quotient computation, enhancing SDP relaxation efficiency.

Findings

Block decomposition improves computational efficiency.

Degree principle can be exploited for symmetric groups.

Methods for computing in geometric quotients are proposed.

Abstract

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.