An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

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Abstract

We propose an analytic center cutting plane method to determine if a matrix is completely positive, and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman, Dür, and Shaked-Monderer [Electronic Journal of Linear Algebra, 2015]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia, Vera, and Zuluaga [INFORMS Journal on Computing, 2018]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like O(d2) for d×d matrices.
Original languageEnglish
Place of PublicationIthaca
PublisherCornell University Library
Number of pages16
Publication statusSubmitted - 2020

Publication series

NamearXiv
Volume2006.05319

Keywords

  • copositive optimization
  • analytic center cutting plane method
  • completely positive matrices

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