Type of Publication:  Journal article 
URI (citable link):  http://nbnresolving.de/urn:nbn:de:bsz:352opus129239 
Author:  Netzer, Tim; Plaumann, Daniel; Schweighofer, Markus 
Year of publication:  2010 
Published in:  SIAM Journal on Optimization ; 20 (2010), 4.  pp. 19441955.  ISSN 10526234 
DOI (citable link):  https://dx.doi.org/10.1137/090750196 
Summary: 
A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affinelinear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinitely representable sets. Part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, such as one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinitely representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can work only if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton, and Nie.

Subject (DDC):  510 Mathematics 
Keywords:  convex set, semialgebraic set, linear matrix inequality, spectrahedron, semidefinite programming, Lasserre relaxation, sums of squares, quadrat 
Link to License:  In Copyright 
Bibliography of Konstanz:  Yes 
NETZER, Tim, Daniel PLAUMANN, Markus SCHWEIGHOFER, 2010. Exposed Faces of Semidefinitely Representable Sets. In: SIAM Journal on Optimization. 20(4), pp. 19441955. ISSN 10526234. Available under: doi: 10.1137/090750196
@article{Netzer2010Expos12347, title={Exposed Faces of Semidefinitely Representable Sets}, year={2010}, doi={10.1137/090750196}, number={4}, volume={20}, issn={10526234}, journal={SIAM Journal on Optimization}, pages={19441955}, author={Netzer, Tim and Plaumann, Daniel and Schweighofer, Markus} }
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