Newton and quasi-Newton methods for a class of nonsmooth equations and related problems

The paper presents concrete realizations of quasi-Newton methods for solving several standard problems including complementarity problems, special variational inequality problems, and the Karush-Kuhn-Tucker (KKT) system of nonlinear programming. A new approximation idea is introduced in this paper. The Q-superlinear convergence of the Newton method and the quasi-Newton method are established under suitable assumptions, in which the existence of F'(x*) is not assumed. The new algorithms only need to solve a linear equation in each step. For complementarity problems, the QR factorization on the quasi-Newton method is discussed.