Semiderivative Functions and Reformulation Methods for Solving Complementarity and Variational Inequality Problems

Abstract

Two major reformulation methods, the nonsmooth method and the smoothing method, for solving nonlinear complementarity problems and variational inequality problems, have been rapidly developed in recent years. Superlinear convergence of these methods is linked with semismoothness, which is based on generalized Jacobians of locally Lipschitz functions. However, the definition of generalized Jacobians relies on the Rademacher theorem, and the exact calculus rules do not hold for generalized Jacobians. These pose some restrictions and difficulties for these methods. In this paper, we define a semiderivative function G for a continuous function F and show that this concept is indeed an extension of the concept of the derivative function. The semiderivative function G is single-valued and satisfies exact calculus rules. Hence it is relatively easy to calculate. Several common nonsmooth equation reformulations of nonlinear complementarity problems and variational inequality problems can be regarded as componentwise compositions of some smooth functions and some simple generalized semismooth functions, such as the plus function, the Fischer-Burmeister function and the median function. The derivatives of several well-known smoothing functions, such as the Chen-Harker-Kanzow-Smale function, the Chen-Mangasarian function and the Gabrial-Moré function, converge to semiderivative functions of the corresponding nonsmooth reformulation func-tions as the smoothing parameter goes to zero. Based upon this new concept, superlinear convergence conditions for both the nonsmooth method and the smoothing method are established.Keywordsnonsmooth reformulationgeneralized Newton methodssuperlin-ear convergencegeneralized semismooth functions.