Sparse Polynomial Optimization with Matrix Constraints

TL;DR

The paper tackles sparse matrix polynomial optimization with matrix constraints by developing and analyzing a sparse Moment-SOS hierarchy that exploits correlative sparsity. It establishes a necessary and sufficient condition for tightness in terms of a sparse sum-of-squares decomposition, enabling exact recovery of the optimum when the condition holds, and provides a practical flat-truncation method to certify tightness and extract minimizers. In the convex/SOS-convex setting, the authors prove that tightness holds for all relaxation orders under broad assumptions, extending the reach of SOS relaxations to a wider class of problems. The work is complemented by extensive numerical experiments, including explicit examples, joint minimizers, PMI center problems, multisystem $H_2$ controller synthesis, and random matrix problems, demonstrating the efficiency and scalability of the sparse approach relative to dense formulations.

Abstract

This paper studies the hierarchy of sparse matrix Moment-SOS relaxations for solving sparse polynomial optimization problems with matrix constraints. First, we prove a sufficient and necessary condition for the sparse hierarchy to be tight. Second, we discuss how to detect the tightness and extract minimizers. Third, for the convex case, we show that the hierarchy of the sparse matrix Moment-SOS relaxations is tight, under some general assumptions. In particular, we show that the sparse matrix Moment-SOS relaxation is tight for every order when the problem is SOS-convex. Numerical experiments are provided to show the efficiency of the sparse relaxations.

Theorems & Definitions (26)

• Example 1.1
• Theorem 3.1
• proof
• Example 3.2
• Example 3.3
• Theorem 3.4
• proof
• Theorem 4.1
• proof
• Remark 4.3