Homogenization for polynomial optimization with unbounded sets

TL;DR

This paper develops a homogenization approach and a Moment-SOS hierarchy for polynomial optimization on unbounded sets, proving finite convergence under certain conditions and extending Positivstellensatz results.

Contribution

It introduces a homogenization formulation and a hierarchy of Moment-SOS relaxations with finite convergence guarantees for unbounded polynomial optimization.

Findings

Hierarchy of Moment-SOS relaxations has finite convergence under specific conditions.

Extended Positivstellensatz results for polynomials on unbounded sets.

Positive resolution of a conjecture related to Moment-SOS hierarchies with denominators.

Abstract

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficiency conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work.