Exploiting Sign Symmetries in Minimizing Sums of Rational Functions

TL;DR

This paper introduces a novel hierarchy of sum of squares relaxations for minimizing sums of rational functions, exploiting sign symmetries and sparsity to improve efficiency and convergence analysis.

Contribution

It develops a dual SOS hierarchy with convergence rate analysis and sign symmetry adaptation, incorporating sparsity for enhanced computational performance.

Findings

Hierarchical SOS relaxations effectively minimize sums of rational functions.

Exploiting sign symmetries leads to block-diagonal semidefinite relaxations.

Numerical experiments demonstrate improved efficiency and applicability.

Abstract

This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum of squares (SOS) relaxations that is dual to the generalized moment problem approach due to Bugarin, Henrion, and Lasserre. The investigation of the dual SOS aspect offers two benefits: 1) it allows us to conduct a convergence rate analysis for the hierarchy; 2) it leads to a sign symmetry adapted hierarchy consisting of block-diagonal semidefinite relaxations. When the problem possesses correlative sparsity as well as sign symmetries, we propose sparse semidefinite relaxations by exploiting both structures. Various numerical experiments are performed to demonstrate the efficiency of our approach. Finally, an application to maximizing sums of generalized Rayleigh quotients is presented.