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Worst-case results for positive semidefinite rank. (English) Zbl 1344.90046
The positive semidefinite rank is an extension of the notion of the nonnegative rank of a matrix. The nonnegative rank of a matrix is the smallest possible integer $$k$$ such that each entry $$(i,j)$$ of the matrix can be written as the inner product of two vectors $$a_i$$ and $$b_j$$ in the cone of nonnegative vectors of size $$k$$. The positive semidefinite rank is the smallest possible integer $$k$$ such that each entry $$(i,j)$$ of the matrix can be written as the inner product of two matrices $$A_i$$ and $$B_j$$ in the cone of positive semidefinite matrices of size $$k$$. These ranks can be interpreted as a measure of complexity of a polytope, in the sense that they encode the size of a sparse representation of a polytope through a factorization of its matrix description. The paper under review reports on state-of-the-art results on lower bounds on the positive semidefinite rank, both for generic polytopes and for specific polytopes.

MSC:
 90C22 Semidefinite programming 52B11 $$n$$-dimensional polytopes 15A23 Factorization of matrices
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References:
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