A Newton-like method for nonsmooth variational inequalities (Q960909)

Linear and nonlinear complementarity problems as well as variational inequalities can be formulated in form of generalized equations \[ 0\in F(x) + G(x) \] where \(F\) is a continuous mapping from an open subset \(\mathcal D \) of a Banach space \(\mathcal X\) into a Banach space \(\mathcal Y\), and \(G\) is a set-valued map from \(\mathcal X\) to the subsets of \(\mathcal Y\) with closed graph. The authors consider a Newton-like method for finding the solution \(x^*\) this equation \[ 0 \in F(x_n) + A(x_n) (x_{n+1} - x_n) + G(x_{n+1}) \] approximately, where \(A(x_n)\in {\mathcal L}({\mathcal X}, {\mathcal Y})\) replaces \(F'(x_n)\) in the Newton method. It is proved that the Newton-like method converges linearly using a Lipschitz-like concept of set-valued mappings and a slant differentiability hypothesis of the operator \(F\) at \(x^*\).