Copositivity tests based on the linear complementarity problem

Copositivity tests are presented based on new necessary and sufficient conditions requiring the solution of linear complementarity problems (LCP). Methodologies involving Lemke’s method, an enumerative algorithm and a linear mixed-integer programming formulation are proposed to solve the required LCPs. A new necessary condition for (strict) copositivity based on solving a Linear Program (LP) is also discussed, which can be used as a preprocessing step. The algorithms with these three different variants are thoroughly applied to test matrices from the literature and to max-clique instances with matrices up to dimension 496 × 496. We compare our procedures with three other copositivity tests from the literature as well as with a general global optimization solver. The numerical results are very promising and equally good and in many cases better than the results reported elsewhere.