Lagrange Multiplier Expressions for Matrix Polynomial Optimization and Tight Relaxations
Abstract
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. Given 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.
Fields of Study & Subjects
Matrix Theory and Algorithms
Advanced Optimization Algorithms Research
Tensor decomposition and applications
Publication details
- Journal / source
- SIAM Journal on Optimization
- Volume
- 36
- Issue
- 2
- Pages
- 1017–1042
- Publication date
- 15 June 2026
- Language
- EN