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An application of $$H$$-differentiability to nonnegative and unrestricted generalized complementarity problems. (English) Zbl 1147.90405
Summary: This paper deals with nonnegative nonsmooth generalized complementarity problem, denoted by GCP$$(f, g)$$. Starting with $$H$$-differentiable functions $$f$$ and $$g$$, we describe $$H$$-differentials of some GCP functions and their merit functions. We show how, under appropriate conditions on $$H$$-differentials of $$f$$ and $$g$$, minimizing a merit function corresponding to $$f$$ and $$g$$ leads to a solution of the generalized complementarity problem. Moreover, we generalize the concepts of monotonicity, $${\mathbf P}_{0}$$-property and their variants for functions and use them to establish some conditions to get a solution for generalized complementarity problem. Our results are generalizations of such results for nonlinear complementarity problem when the underlying functions are $$C^{1}$$, semismooth, and locally Lipschitzian.

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 49J50 Fréchet and Gateaux differentiability in optimization
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