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We present copositivity tests based on new necessary and sufficient conditions which require the solution of linear complementarity problems (LCP). We propose methodologies involving Lemke's method, an enumerative algorithm and a linear mixed-integer programming formulation to solve the required LCPs. Moreover, we discuss a new necessary condition for (strict) copositivity based on solving a linear program, 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 of order up to 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.

Polynomial interpolation or regression models are an important tool in Derivativefree Optimization, acting as surrogates of the real function. In this work, we propose the use of these models in the multiobjective framework of directional direct search, namely the one of Direct Multisearch. Previously evaluated points are used to build quadratic polynomial models, which are minimized in an attempt of generating nondominated points of the true function, defining a search step for the algorithm. Numerical results state the competitiveness of the proposed approach.

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