Restricted-Multiplier Positivstellens\"atze for Polynomial Optimization with Affine Parameters

Abstract

We develop positivity certificates for polynomials that are affine in auxiliary parameters, with sum-of-squares multipliers depending only on the principal variables. Under a restricted Archimedean order-unit condition, every strictly positive scalar polynomial admits such a certificate. We construct a convergent restricted moment-SOS hierarchy for polynomial optimization problems with affine parameters and establish a flatness criterion for global optimality detection and global minimizer recovery. The scalar theorem extends to polynomial-matrix constraints with any number of affine parameters. For matrix-valued objectives, pointwise positivity is sufficient with one parameter but can fail with two or more parameters; uniform free positivity provides a general sufficient replacement. Applications include robust linear programming, robust Lyapunov design, min-max polynomial optimization, analysis and control of input-affine dynamical systems. Numerical experiments show that the proposed approach substantially reduces semidefinite block sizes and solution times while incurring little or no degradation in bound quality for most test cases.