Exploiting Term Sparsity in Symmetry-Adapted Basis for Polynomial Optimization

TL;DR

This paper introduces a method that combines symmetry exploitation and term sparsity to significantly reduce the computational complexity of polynomial optimization problems with group invariance.

Contribution

It extends the term sparsity hierarchy by integrating symmetry-adapted bases, enabling more efficient semidefinite programming for invariant polynomial optimization.

Findings

Reduces computational cost in polynomial optimization with symmetry.

Demonstrates efficiency gains on benchmark problems with dihedral, cyclic, and symmetric groups.

Extends previous sparsity-based hierarchies to incorporate symmetry-adapted bases.

Abstract

Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function and constraint polynomials are invariant under the action of a finite group. The present paper simultaneously exploits group symmetry and term sparsity in order to reduce the computational cost of the hierarchy. We first exploit symmetry by writing the semidefinite matrices in a symmetry-adapted basis according to an isotypic decomposition. The matrices in such a basis are block diagonal. Secondly, we exploit term sparsity on each block to further reduce the optimization matrix variables. This is a non-trivial extension of the term sparsity-based hierarchy related to sign symmetry that was introduced by two of the authors. Our method is compared with existing techniques via benchmarks on quartics with dihedral, cyclic and symmetric group symmetry.