# Strong duality in Lasserre's hierarchy for polynomial optimization

3 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
Abstract : 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.
Document type :
Journal articles

Cited literature [10 references]

https://hal.archives-ouvertes.fr/hal-00997726
Contributor : Didier Henrion <>
Submitted on : Monday, December 1, 2014 - 3:59:23 PM
Last modification on : Friday, April 12, 2019 - 4:23:37 PM
Document(s) archivé(s) le : Monday, March 2, 2015 - 1:35:14 PM

### Files

nogap.pdf
Files produced by the author(s)

### Citation

Cédric Josz, Didier Henrion. Strong duality in Lasserre's hierarchy for polynomial optimization. Optimization Letters, Springer Verlag, 2016, 10 (1), pp.3-10. ⟨10.1007/s11590-015-0868-5⟩. ⟨hal-00997726v2⟩

Record views