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P.J.C. Dickinson (peter.dickinsoncantab.net) Abstract: Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi [K.G. Murty and S.N. Kabadi, Some NPcomplete problems in quadratic and nonlinear programming, Mathematical Programming, 39, no.2:117–129, 1987] that the strong membership problem for the copositive cone, that is deciding whether or not a given matrix is in the copositive cone, is a coNPcomplete problem. The question of whether or not the strong membership problem for the dual of the copositive cone, the completely positive cone, is also an NPhard problem has so far been left open. In this paper it is proven that the strong membership problem for the completely positive cone is in fact NPhard. Furthermore, it is shown that even the weak membership problems for both of these cones are NPhard. We also present an alternative proof of the NPhardness of the strong membership problem for the copositive cone. Keywords: Copositive; Completely Positive; NPhard; Membership Category 1: Linear, Cone and Semidefinite Programming Citation: Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, The Netherlands, April 2011 Download: [PDF] Entry Submitted: 05/24/2011 Modify/Update this entry  
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