Globally Convergent Broyden-Like Methods for Semismooth Equations and Applications to VIP, NCP and MCP

In this paper, we propose a general smoothing Broyden-like quasi-Newton method for solving a class of nonsmooth equations. Under appropriate conditions, the proposed method converges to a solution of the equation globally and superlinearly. In particular, the proposed method provides the possibility of developing a quasi-Newton method that enjoys superlinear convergence even if strict complementarity fails to hld. We pay particular attention to semismooth equations arising from nonlin-ear complementarity problems, mixed complementarity problems and variational inequality problems. We show that under certain conditions, the related methods based on the perturbed Fischer-Burmeister function, Chen-Harker-Kanzow-Smale smoothing function and Gabriel-Mor e class of smoothing functions converge globally and superlinearly. 1 The work of the rst author was done while he was visiting Kyoto University. This paper is an extended version of the unpublished technical report 25] by the authors.