Lagrange multiplier expressions for matrix polynomial optimization and tight relaxations
Lei Huang, Jiawang Nie, Jiajia Wang, Lingling Xie
TL;DR
This work addresses matrix polynomial optimization (MPO) of the form $\min f(x)$ subject to $G(x)\succeq 0$ by deriving explicit Lagrange multiplier matrices under a nondegeneracy condition and incorporating them into a strengthened matrix Moment–SOS hierarchy. The authors show that, when a Lagrange multiplier expression $\Theta(x)$ exists (via a nonsingularity condition on a matrix polynomial $P(x)$), one can replace the multiplier variable with $\Theta(x)$ and obtain a tighter relaxation. Under general assumptions, including attainment of $f_{\min}$ at a KKT point and an Archimedean property, the strengthened hierarchy is proven to have finite convergence, with every sufficiently large relaxation order giving $f_{k,\mathrm{sos}}=f_{k,\mathrm{mom}}=f_{\min}$, and flat truncation enabling minimizer extraction. Numerical experiments demonstrate significant speedups over the standard matrix Moment–SOS relaxations, validating practical effectiveness even without compactness or strict complementarity. Overall, the paper provides a principled, scalable approach to solving MPOs by exploiting matrix KKT structures through Lagrange multiplier expressions and SOS-based certificates.
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.