Strong duality in Lasserre’s hierarchy for polynomial optimization

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F16%3A00237256" target="_blank" >RIV/68407700:21230/16:00237256 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s11590-015-0868-5" target="_blank" >http://dx.doi.org/10.1007/s11590-015-0868-5</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11590-015-0868-5" target="_blank" >10.1007/s11590-015-0868-5</a>

Result language
angličtina
Original language name
Strong duality in Lasserre’s hierarchy for polynomial optimization
Original language description
A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set $$K$$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, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set $$K$$K is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that $$K$$K has a strictly feasible point.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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