Type of Publication:  Journal article 
Publication status:  Published 
Author:  Kriel, TomLukas; Schweighofer, Markus 
Year of publication:  2018 
Published in:  SIAM Journal on Optimization ; 28 (2018), 2.  pp. 17961816.  ISSN 10526234.  eISSN 10957189 
ArXivID:  arXiv:1704.07231 
DOI (citable link):  https://dx.doi.org/10.1137/17M1128290 
Summary: 
Consider a finite system of nonstrict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior, and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negativesemidefiniteness on $S$ (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation. In their seminal work of 2009, Helton and Nie showed under the same conditions that $S$ is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a nonconstructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of $S$.

Subject (DDC):  510 Mathematics 
Bibliography of Konstanz:  Yes 
Refereed:  Yes 
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KRIEL, TomLukas, Markus SCHWEIGHOFER, 2018. On the Exactness of Lasserre Relaxations for Compact Convex Basic Closed Semialgebraic Sets. In: SIAM Journal on Optimization. 28(2), pp. 17961816. ISSN 10526234. eISSN 10957189. Available under: doi: 10.1137/17M1128290
@article{Kriel2018Exact38753.2, title={On the Exactness of Lasserre Relaxations for Compact Convex Basic Closed Semialgebraic Sets}, year={2018}, doi={10.1137/17M1128290}, number={2}, volume={28}, issn={10526234}, journal={SIAM Journal on Optimization}, pages={17961816}, author={Kriel, TomLukas and Schweighofer, Markus} }
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