Duality attainment and strict feasibility of the generalized moment problem and its relaxations

TL;DR

This paper investigates the duality and strict feasibility of the generalized moment problem (GMP) with measures supported on semialgebraic sets, providing conditions for attainment and new proofs, with applications to tensor optimization and quantum information.

Contribution

It establishes duality attainment and strict feasibility for GMPs on certain semialgebraic sets, offering two novel proofs and extending understanding of moment hierarchies.

Findings

Attainment of the dual problem under a relative interior assumption.

Existence of strictly feasible measures via two different proofs.

Application of results to tensor optimization and quantum information theory.

Abstract

The generalized moment problem (GMP) is an infinite dimensional linear problem over the cone of finite nonnegative Borel measures. When a GMP instance involves finitely many polynomial moment constraints, moment/sum-of-squares hierarchies provide a sequence of bounds converging to the optimal value. We consider GMP instances with measures supported over a compact basic semialgebraic set $X$. We study the case when $X$ has nonempty interior, and the case when $X$ is the vanishing set of prescribed polynomials forming a Gr\"obner basis of the ideal they generate, which we assume is real radical. Under a relative interior assumption, we show attainment of the infinite dimensional dual problem, and attainment of each associated finite dimensional sum-of-squares strengthening. For the latter we present two disjoint proofs. The first is obtained by adapting results regarding the closedness of quadratic modules, and the second builds on Csisz\'ar's work on exponential density constructions to find a strictly feasible measure. Finally, we discuss the special case where $X$ is the product of spheres, and applications of our results to GMP instances arising from tensor optimization and quantum information theory.