Polynomial Optimization Over Unions of Sets

TL;DR

This paper introduces a unified hierarchy of Moment-SOS relaxations to globally solve polynomial optimization problems over unions of semialgebraic sets, with convergence guarantees and applications to matrix norms.

Contribution

It develops a novel unified relaxation framework for polynomial optimization over unions of sets, with convergence analysis and specific properties for univariate cases.

Findings

Hierarchy converges asymptotically or finitely under certain conditions.

Applicable to computing matrix (p,q)-norms.

Discusses special properties in the univariate case.

Abstract

This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed.The application for computing $(p,q)$-norms of matrices is also presented.