Positive semidefinite lifts and factorizations

Representing polytopes by means of linear matrix inequalities as been a highly successful strategy in combinatorial optimization. Geometrically it corresponds to writing a polytope as the projection of an affine slice of the cone of positive semidefinite (psd) matrices i.e., a spectrahedron. Efforts to understand the theoretical limits of such techniques have connected the existance of such representations to a particular type of matrix factorization, the psd factorization of a nonnegative matrix, and its corresponding notion of psd rank. In this talk we will do a brief survey of the main results in the area, its connections to matrix theory and combinatorics and some of the open problems that remain.

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