Sparse Polynomial Optimization with Matrix Constraints

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.