A smooth Levenberg-Marquardt method without nonsingularity condition for wLCP

In this paper we consider the weighted Linear Complementarity Problem (wLCP). By using a smooth weighted complementarity function, we reformulate the wLCP as a smooth nonlinear equation and propose a Levenberg-Marquardt method to solve it. The proposed method differentiates itself from the current Levenberg-Marquardt type methods by adopting a simple derivative-free line search technique. It is shown that the proposed method is well-defined and it is globally convergent without requiring wLCP to be monotone. Moreover, the method has local sub-quadratic convergence rate under the local error bound condition which is weaker than the nonsingularity condition. Some numerical results are reported.