Some geometric results in semidefinite programming
Certain geometric results are developed for the feasible regions of semidefinite programs, called spectrahedra. Firstly, a characterization of the faces of spectrahedra is developed. Specifically, given a point \(x\) in a spectrahedron \(G\), we derive an expression for the minimal face of \(G\) that contains \(x\). This yields, among other things, characterizations of extreme points and extreme rays of spectrahedra. Then an investigation of polars of spectrahedra is carried out. In particular, a notion of the algebraic polar of a spectrahedra is introduced and it is shown that the usual polar is the closure of the well described algebraic polar. By employing some of the analysis here, the first author has subsequently developed the first polynomial size gap-free dual program for semidefinite programming.
scientific article; zbMATH DE number 836800
scientific article