Strong duality in Lasserre's hierarchy for polynomial optimization

TL;DR

This paper proves that in Lasserre's hierarchy for polynomial optimization, adding a redundant ball constraint ensures no duality gap between primal and dual SDP relaxations when the feasible set is compact.

Contribution

It establishes strong duality in Lasserre's hierarchy for polynomial optimization problems with compact feasible sets by adding a redundant ball constraint, without requiring interior points.

Findings

No duality gap between primal and dual SDP relaxations with the added constraint.

Elementary SDP duality results suffice for the proof.

Applicable to compact semi-algebraic sets without interior points.

Abstract

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set $K$ described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J.~B.~Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, when the POP feasibility set $K$ is compact, we show that there is no duality gap between each primal and dual SDP problem in Lasserre's hierarchy, provided a redundant ball constraint is added to the description of set $K$. Our proof uses elementary results on SDP duality, and it does not assume that $K$ has an interior point.