# 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.
Type de document :
Article dans une revue
Optimization Letters, Springer Verlag, 2016, 10 (1), pp.3-10. 〈10.1007/s11590-015-0868-5〉

Littérature citée [10 références]

https://hal.archives-ouvertes.fr/hal-00997726
Contributeur : Didier Henrion <>
Soumis le : lundi 1 décembre 2014 - 15:59:23
Dernière modification le : mercredi 12 décembre 2018 - 15:18:11
Document(s) archivé(s) le : lundi 2 mars 2015 - 13:35:14

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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〉

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