A Modified Nonsmooth Levenberg–Marquardt Algorithm for the General Mixed Complementarity Problem

As is well known, the mixed complementarity problem is equivalent to a nonsmooth equation by using a median function. By investigating the generalized Jacobi of a composite vector-valued maximum function, a nonsmooth Levenberg–Marquardt algorithm is proposed in this paper. In the present algorithm, we adopt a new LM parameter form and discuss the local convergence rate under the local error bound condition, which is weaker than nonsingularity. Finally, the numerical experiments and the application for the real-time pricing in smart grid illustrate the effectiveness of the algorithm.

On a Nonsmooth Gauss–Newton Algorithms for Solving Nonlinear Complementarity Problems

In this paper, we propose a new version of the generalized damped Gauss–Newton method for solving nonlinear complementarity problems based on the transformation to the nonsmooth equation, which is equivalent to some unconstrained optimization problem. The B-differential plays the role of the derivative. We present two types of algorithms (usual and inexact), which have superlinear and global convergence for semismooth cases. These results can be applied to efficiently find all solutions of the nonlinear complementarity problems under some mild assumptions. The results of the numerical tests are attached as a complement of the theoretical considerations.

The Levenberg-Marquardt-Type Methods for a Kind of Vertical Complementarity Problem

Two kinds of the Levenberg-Marquardt-type methods for the solution of vertical complementarity problem are introduced. The methods are based on a nonsmooth equation reformulation of the vertical complementarity problem for its solution. Local and global convergence results and some remarks about the two kinds of the Levenberg-Marquardt-type methods are also given. Finally, numerical experiments are reported.