Tightness of the matrix Moment-SOS hierarchy

This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the tightness (i.e., the finite convergence) of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second order sufficient condition hold at every minimizer, under the usual archimedeanness assumption. A useful criterion for detecting tightness is the flat truncation. Our second result is to show that every minimizer of the moment relaxation must have a flat truncation when the relaxation order is big enough, under the above mentioned optimality assumptions. These results give connections between nonlinear semidefinite optimization theory and Moment-SOS methods for solving polynomial matrix optimization.