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Strong duality in Lasserre's hierarchy for polynomial optimization

Cedric Josz(cedric.josz***at***inria.fr)
Didier Henrion(henrion***at***laas.fr)

Abstract: 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.

Keywords: polynomial optimization; semidefinite programming

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Global Optimization (Theory )

Citation:

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Entry Submitted: 05/28/2014
Entry Accepted: 05/28/2014
Entry Last Modified: 05/28/2014

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