Symmetry reduction in AM/GM-based optimization

TL;DR

This paper explores how symmetry considerations can optimize AM/GM-based methods for polynomial non-negativity, leading to reduced problem sizes and faster computations, especially under symmetric group actions.

Contribution

It introduces a symmetry-adapted framework for AM/GM techniques, including a representation theorem and size reduction methods for related optimization programs.

Findings

Size of optimization problems stabilizes with symmetry

Symmetry-based reduction speeds up computations

Numerical experiments confirm efficiency gains

Abstract

The arithmetic mean/geometric mean-inequality (AM/GM-inequality) facilitates classes of non-negativity certificates and of relaxation techniques for polynomials and, more generally, for exponential sums. Here, we present a first systematic study of the AM/GM-based techniques in the presence of symmetries under the linear action of a finite group. We prove a symmetry-adapted representation theorem and develop techniques to reduce the size of the resulting relative entropy programs. We study in more detail the complexity gain in the case of the symmetric group. In this setup, we can show in particular certain stabilization results. We exhibit several sequences of examples in growing dimensions where the size of the problem stabilizes. Finally, we provide some numerical results, emphasizing the computational speed-up.