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Linear Matrix Inequalities meet Positivstellenstze

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Let L be a linear pencil, and consider its linear matrix inequality (LMI) L(x)>0. In this talk we describe a natural relaxation of an LMI , based on substituting matrices X for the variables x. With this relaxation, we prove that a ``perfect’’ convex Positivstellensatz: a noncommutative polynomial p is positive semidefinite on the matricial LMI domain L(X)>0 if and only if it has a weighted sum of squares representation with optimal degree bounds. We shall also give two corollaries of this theorem. First, we present an LMI domination theorem, whose special case is the SDP relaxation of the matrix cube problem due to Nemirovskii and Ben-Tal. A second consequence is the real radical duality theory for semidefinite programming.

This talk is part of the Isaac Newton Institute Seminar Series series.

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