Strong duality in Lasserre's hierarchy for polynomial optimization

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.