Real Algebraic and Analytic Geometry
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289. Tim Netzer:
On semidefinite representations of sets.

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Submission: 2009, August 25.

Abstract:
Spectrahedra are sets defined by \textit{linear matrix inequalities}. Projections of spectrahedra are called \textit{semidefinite representable sets}. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. In this work we develop some new methods to prove semidefinite representability of sets. We examine \textit{partial linear matrix inequalities}, i.e. conditions stating that a linear matrix polynomial is conditional semidefinite (instead of positive semidefinite, as in the definition of a spectrahedron). For certain classes we prove that those conditions produce semidefinite representable sets. We then examine non-closed sets, which seem to have gained no attention at all so far. The interior of a semidefinite representable set is shown to be semidefinite representable. More general, one can remove faces of a semidefinite representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.

Mathematics Subject Classification (2000): 90C22, 15A48, 14P10, 13J30, 11E25.

Keywords and Phrases: semidefinite representations of sets, semidefinite programming, positive matrices.

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