Lagrange multiplier expressions for matrix polynomial optimization and tight relaxations

TL;DR

This paper introduces explicit Lagrange multiplier matrix expressions for matrix polynomial optimization, leading to a strengthened, finite-converging Moment-SOS hierarchy with demonstrated efficiency.

Contribution

It provides a novel method to derive Lagrange multiplier matrices and develops a tightened hierarchy for solving matrix polynomial optimization problems.

Findings

The strengthened hierarchy achieves finite convergence.

Numerical experiments confirm the hierarchy's efficiency.

Methods for detecting tightness and extracting optimizers are proposed.

Abstract

This paper studies matrix constrained polynomial optimization. We investigate how to get explicit expressions for Lagrange multiplier matrices from the first order optimality conditions. The existence of these expressions can be shown under the nondegeneracy condition. Using Lagrange multiplier matrix expressions, we propose a strengthened Moment-SOS hierarchy for solving matrix polynomial optimization. Under some general assumptions, we show that this strengthened hierarchy is tight, or equivalently, it has finite convergence. We also study how to detect tightness and how to extract optimizers. Numerical experiments are provided to show the efficiency of the strengthened hierarchy.