Some Results on Subanalytic Variational Inclusions

Abstract

This chapter deals with variational inclusions of the form 0 ∈ f (x) + g(x) + F(x) where f is a locally Lipschitz and subanalytic function, g is a Lipschitz function, F is a set-valued map, acting all in ℝn and n is a positive integer. The study of the previous variational inclusion depends on the properties of the function g. The behaviour as been examinated in different cases: when g is the null function, when g possesses divided differences and when g is not smooth and semismooth. We recall and give a summary of some known methods and the last section is very original and is unpublished. In this last section we combine a Newton type method (applied to f) with a secant type method (applied to g) and we obtain superlinear convergence to a solution of the variational inclusion. Our study in the present chapter is in the context of subanalytic functions, which are semismooth functions and the usual concept of derivative is replaced here by the the concept of Clarke's Jacobian. © Springer-Verlag Berlin Heidelberg 2013.