Generalized truncated moment problems with unbounded sets

TL;DR

This paper develops a hierarchy of Moment-SOS relaxations to solve generalized truncated moment problems over unbounded sets, enabling the computation of finitely atomic measures or certificates of nonexistence.

Contribution

It introduces a convergent hierarchy of Moment-SOS relaxations for unbounded sets, extending the applicability of moment problem solutions.

Findings

Hierarchical Moment-SOS relaxations effectively approximate cones.

Method can find finitely atomic representing measures.

Numerical experiments demonstrate practical effectiveness.

Abstract

This paper studies generalized truncated moment problems with unbounded sets. First, we study geometric properties of the truncated moment cone and its dual cone of nonnegative polynomials. By the technique of homogenization, we give a convergent hierarchy of Moment-SOS relaxations for approximating these cones. With them, we give a Moment-SOS method for solving generalized truncated moment problems with unbounded sets. Finitely atomic representing measures, or certificates for their nonexistence, can be obtained by the proposed method. Numerical experiments and applications are also given.