Some NP-complete problems in quadratic and nonlinear programming
Mathematical Programming: Series A and B
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Approximation of the Stability Number of a Graph via Copositive Programming
SIAM Journal on Optimization
Graph Theory With Applications
Graph Theory With Applications
The Operator $\Psi$ for the Chromatic Number of a Graph
SIAM Journal on Optimization
On the copositive representation of binary and continuous nonconvex quadratic programs
Mathematical Programming: Series A and B
On the computation of C* certificates
Journal of Global Optimization
An Adaptive Linear Approximation Algorithm for Copositive Programs
SIAM Journal on Optimization
A Variational Approach to Copositive Matrices
SIAM Review
Journal of Global Optimization
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Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi (Math. Program. 39(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 co-NP-complete problem. From this it has long been assumed that this implies that the question of whether or not the strong membership problem for the dual of the copositive cone, the completely positive cone, is also an NP-hard problem. However, the technical details for this have not previously been looked at to confirm that this is true. In this paper it is proven that the strong membership problem for the completely positive cone is indeed NP-hard. Furthermore, it is shown that even the weak membership problems for both of these cones are NP-hard. We also present an alternative proof of the NP-hardness of the strong membership problem for the copositive cone.