2013/05/20 by João Gouveia, Gouveia, João, Richard Z. Robinson +3
Engineering · Mathematics · #Advanced Optimization Algorithms Research #Combinatorics (math.CO) #FOS: Mathematics #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #graph theory and CDMA systems #math.CO #math.OC
paper · pdf · doi:10.48550/arxiv.1305.4600
11 pages
openalex publication_date 2013/05/20 · arxiv created 2014/05/30 · arxiv updated 2014/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(minp,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.