On two applications of H-differentiability to optimization and complementarity problems

In a recent paper, Gowda and Ravindran (Algebraic univalence theorems for n onsmooth functions, Research Report, Department of Mathematics and Statisti cs, University of Maryland, Baltimore, MD 21250, March 15, 1998) introduced the concepts of H-differentiability and H-differential for a function f : R-n --> R-n and showed that the Frechet derivative of a Frechet differentia ble function, the Clarke generalized Jacobian of a locally Lipschitzian fun ction, the Bouligand subdifferential of a semismooth function, and the C-di fferential of a C-differentiable function are particular instances of H-dif ferentials. In this paper, we consider two applications of H-differentiability. In the first application, we derive a necessary optimality condition for a local m inimum of an H-differentiable function. In the second application, we consi der a nonlinear complementarity problem corresponding to an H-differentiabl e function f and show how, under appropriate conditions on an H-differentia l of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. These two applications were motiv ated by numerous studies carried out for C-1, convex, locally Lipschitzian, and semismooth function by various researchers.