Finite convergence of the Moment-SOS hierarchy under hidden convexity

Abstract

We consider polynomial optimization problems with compact feasible set $\mathbf{\Omega }\subset {\mathbb{R}}^d$ defined by SOS-concave polynomials $g_j$ of arbitrary degree, and whose objective function $f$ is not necessarily convex on ${\mathbb{R}}^d$. We show that, if $f$ is Hessian-$Q$-module convex over $\mathbf{\Omega }$ in the sense that its Hessian admits a specific quadratic-module representation, then the standard Moment-SOS hierarchy converges in finitely many steps without prior knowledge of this hidden (local) convexity. Strong convexity of $f$ on $\mathbf{\Omega }$ is a sufficient condition for the required Hessian representation. In addition, we give an explicit relaxation order at which exactness occurs. This demonstrates that a general-purpose hierarchy can adapt to favorable hidden properties of a specific instance without being informed of them, yielding certified global minimizers.