A Characterization for Tightness of the Sparse Moment-SOS Hierarchy

TL;DR

This paper characterizes when the sparse Moment-SOS hierarchy yields exact solutions for sparse polynomial optimization, linking tightness to the structure of the objective as a sum of specific nonnegative polynomials.

Contribution

It provides a necessary and sufficient condition for the hierarchy's tightness and offers practical criteria for ensuring exactness in sparse polynomial optimization.

Findings

Hierarchy is tight iff the objective is a sum of sparse nonnegative polynomials in the sum of ideal and quadratic module.

Several sufficient conditions for tightness are identified, including convexity and optimality conditions.

The results guide when the sparse Moment-SOS hierarchy can be reliably used for exact solutions.

Abstract

This paper studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.