Certifying Convergence of Lasserre's Hierarchy via Flat Truncation

TL;DR

This paper introduces flat truncation as a universal certificate for the finite convergence of Lasserre's hierarchy in polynomial optimization, linking it to the existence of global minimizers and providing conditions for asymptotic satisfaction.

Contribution

It establishes flat truncation as a key criterion for certifying convergence of Lasserre's hierarchy, extending to various types and relaxation methods in polynomial optimization.

Findings

Flat truncation certifies finite convergence under certain conditions.

Asymptotic flat truncation holds under the archimedean condition.

Jacobian SDP relaxations always satisfy flat truncation.

Abstract

This paper studies how to certify the convergence of Lasserre's hierarchy of semidefinite programming relaxations for solving multivariate polynomial optimization. We propose flat truncation as a general certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: i) Putinar type Lasserre's hierarchy has finite convergence if and only if flat truncation holds, under some general assumptions, and this is also true for the Schmudgen type one; ii) under the archimedean condition, flat truncation is asymptotically satisfied for Putinar type Lasserre's hierarchy, and similar is true for the Schmudgen type one; iii) for the hierarchy of Jacobian SDP relaxations, flat truncation is always satisfied. The case of unconstrained polynomial optimization is also discussed.