Further Application of H-Differentiability to Generalized Complementarity Problems Based on Generalized Fisher-Burmeister Functions

Abstract

We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP($f,g$) where $f$ and $g$ are $H$-differentiable. We describe $H$-differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on the $H$-differentials of $f$ and $g$, we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP$(f,g)$ to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved for $C^1$, semismooth, and locally Lipschitzian.