Strong duality in Lasserre's hierarchy for polynomial optimization

Cédric Josz (1, 2) , Didier Henrion (3)

1 Département Méthodes et Appui
2 POMDAPI - Environmental Modeling, Optimization and Programming Models
3 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

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.

Domaines: Optimisation et contrôle [math.OC]

Cédric Josz, Didier Henrion. Strong duality in Lasserre's hierarchy for polynomial optimization. 2014. ⟨hal-00997726v1⟩