On Two Applications of H-Differentiability to Optimization and Complementarity Problems

In a recent paper 10], Gowda and Ravindran introduced the concepts of H-diierentiability and H-diierential for a function f : R n ! R n and showed that the Fr echet derivative of a Fr echet diierentiable function, the Clarke generalized Jacobian of a locally Lips-chitzian function, the Bouligand subdiierential of a semismooth function, and the C-diierential of a C-diierentiable function are particular instances of H-diierentials. In this paper, we consider two applications of H-diierentiability. In the rst application, we derive a necessary optimality condition for a local minimum of an H-diierentiable function. In the second application, we consider a nonlinear complementarity problem corresponding to an H-diierentiable function f and show how, under appropriate conditions on an H-diierential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complemen-tarity problem. These two applications were motivated by numerous studies carried out for C 1 , convex, locally Lipschitzian, and semismooth function by various researchers.