Real Algebraic and Analytic Geometry |

Exposed faces of semidefinitely representable sets.

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Submission: 2009, December 18.

*Abstract:
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
semidefinite representable sets. Part of the interest in
spectrahedra and semidefinite 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 semidefinite 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 condition complements sufficient
conditions recently proved by Lasserre, Helton and Nie.*

Mathematics Subject Classification (2000): 13J30, 14P10, 52-99, 90C22, 11E25, 15A48, 52A27.

Keywords and Phrases: convex set, semialgebraic set, linear matrix inequality, spectrahedron, semidefinite programming, Lasserre relaxation, sums of squares, quadratic module, preordering.

**Full text**, 12p.:
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