Sparse Copositive Polynomial Optimization

TL;DR

This paper introduces sparse Moment-SOS relaxations for copositive polynomial optimization, improving computational efficiency while providing tightness conditions and demonstrating effectiveness through numerical experiments.

Contribution

It proposes a novel sparse relaxation approach for copositive polynomial optimization and establishes tightness conditions under cop-SOS convexity.

Findings

Sparse Moment-SOS relaxations are more computationally efficient than dense ones.

Necessary and sufficient conditions for relaxation tightness are established.

Numerical experiments confirm the efficiency of the proposed method.

Abstract

This paper studies the copositive optimization problem whose objective is a sparse polynomial, with linear constraints over the nonnegative orthant. We propose sparse Moment-SOS relaxations to solve it. Necessary and sufficient conditions are shown for these relaxations to be tight. In particular, we prove they are tight under the cop-SOS convexity assumption. Compared to the traditional dense ones, the sparse Moment-SOS relaxations are more computationally efficient. Numerical experiments are given to show the efficiency.