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Worst-Case Results For Positive Semidefinite Rank

Jo\~ao Gouveia(jgouveia***at***mat.uc.pt)
Richard Z. Robinson(rzr***at***uw.edu)
Rekha R. Thomas(rrthomas***at***uw.edu)

Abstract: This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^(1/4) improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4ceil(v/6) which in turn shows that the psd rank of a p by q matrix of rank three is at most 4ceil(min{p,q}/6). In general, a nonnegative matrix of rank (k+1 choose 2) has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed.

Keywords: psd rank, nonnegative rank, extended formulations

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Linear, Cone and Semidefinite Programming (Other )

Citation: May 2013

Download: [PDF]

Entry Submitted: 05/20/2013
Entry Accepted: 05/20/2013
Entry Last Modified: 05/20/2013

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