Finite convergence of the Moment-SOS hierarchy for polynomial matrix optimization

TL;DR

This work advances the theory of polynomial matrix optimization by establishing finite convergence of the matrix Moment-SOS hierarchy under Archimedean QM[G] and a trio of optimality conditions (NDC, SCC, SOSC) at every minimizer. The authors develop a locally equivalent reformulation and local SOS representations to show that f − f_min lies in a finitely generated quadratic module, yielding finite convergence and enabling minimizer extraction via flat truncation once the relaxation order is large enough. They further show that minimizers of the moment relaxation possess flat truncations, providing a practical certificate for convergence and a pathway to recover actual minimizers of the original PMO. Collectively, the results connect nonlinear semidefinite optimization with Moment-SOS methods, extend scalar convergence theory to the matrix setting, and offer concrete criteria for diagnosing and certifying convergence in polynomial matrix optimization.

Abstract

This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the finite convergence of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second order sufficient condition hold at every minimizer, under the Archimedean property. A useful criterion for detecting the finite convergence 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 conditions. These results give connections between nonlinear semidefinite optimization theory and Moment-SOS methods for solving polynomial matrix optimization.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4: schhol
• Theorem 2.5: Proposition 3.4, Sch05
• Theorem 3.1
• Theorem 3.2
• Example 3.3