Exposed Faces of Semidefinitely Representable Sets

13 years 2 months ago

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A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are aﬃne linear 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, like 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 only work if all faces of the considered set are exposed. This necessary ...

Tim Netzer, Daniel Plaumann, Markus Schweighofer

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