Some geometric results in semidefinite programming

The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semidefinite Programs, called hereSpectrahedra.We first develop a characterization for the faces of spectrahedra. More specifically, given a pointx in a spectrahedron, we derive an expression for the minimal face containingx. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.

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