Strong duality in Lasserre's hierarchy for polynomial optimization

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

1 Département Méthodes et Appui
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraicset $K$ 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) relaxationsof increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation ofa moment problem) and a dual SDP formulation (a sum-of-squares representationof 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 problemin Lasserre's hierarchy, provided a redundant ball constraint is added to thedescription 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. Optimization Letters, 2016, 10 (1), pp.3-10. ⟨10.1007/s11590-015-0868-5⟩. ⟨hal-00997726v2⟩