Worst-case results for positive semidefinite rank

• 成果类型: Article
• 署名作者: Gouveia, Joo; Robinson, Richard Z.; Thomas, Rekha R.
• 署名单位: Universidade de Coimbra; University of Washington; University of Washington Seattle
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-015-0867-4
• 发表日期: 2015
• 页码: 201-212
• 关键词: nonnegative rank; factorizations; complexity; sets

摘要

We present 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 -dimensional polytope with vertices is at least improving on previous lower bounds. For polygons with vertices, we show that psd rank cannot exceed which in turn shows that the psd rank of a matrix of rank three is at most . In general, a nonnegative matrix of rank has psd rank at least and we pose the problem of deciding whether the psd rank is exactly . Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when is fixed.