Finite convergence of Moment-SOS relaxations with non-real radical ideals

TL;DR

This paper investigates conditions under which the Moment-SOS hierarchy converges finitely for polynomial optimization problems, including cases with non-real radical ideals and unbounded sets, extending previous results and resolving a conjecture.

Contribution

It establishes finite convergence criteria for Moment-SOS relaxations with non-real radical ideals and introduces a homogenized hierarchy for unbounded sets, confirming a prior conjecture.

Findings

Finite convergence under classical optimality conditions.

Finite convergence of homogenized Moment-SOS hierarchy for unbounded sets.

Proof of finite convergence for Moment-SOS hierarchy with denominators.

Abstract

We consider the linear conic optimization problem with the cone of nonnegative polynomials. Its dual optimization problem is the generalized moment problem. Moment-SOS relaxations are powerful for solving them. This paper studies finite convergence of the Moment-SOS hierarchy when the constraining set is defined by equations whose ideal may not be real radical. Under the archimedeanness, we show that the Moment-SOS hierarchy has finite convergence if some classical optimality conditions hold at every minimizer of the optimal nonnegative polynomial for the linear conic optimization problem. When the archimedeanness fails (this is the case for unbounded sets), we propose a homogenized Moment-SOS hierarchy and prove its finite convergence under similar assumptions. Furthermore, we also prove the finite convergence of the Moment-SOS hierarchy with denominators. In particular, this paper resolves a conjecture posed in the earlier work.