A Smoothing Inexact Newton Method for Generalized Nonlinear Complementarity Problem

Meixia Li,Haitao Che

doi:10.1155/2012/401835

Abstract

Based on the smoothing function of penalized Fischer-Burmeister NCP-function, we propose a new smoothing inexact Newton algorithm with non-monotone line search for solving the generalized nonlinear complementarity problem. We view the smoothing parameter as an independent variable. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the generalized nonlinear complementarity problem. We also establish the local superlinear (quadratic) convergence of the proposed algorithm under the BD-regular assumption. Preliminary numerical experiments indicate the feasibility and efficiency of the proposed algorithm.

Highlights

Consider the generalized nonlinear complementarity problem, denoted by GNCP F, G, K, which is to find a vector x ∈ Rn such thatF x ∈ K, G x ∈ K◦,F x G x 0, 1.1 where F, G : Rn → Rm are continuously differentiable mappings
GNCP F, G, K finds important applications in many fields, such as engineering and economics, and is a wide class of problems that contains the classical nonlinear complementarity problem abbreviated as NCP ; see 1–3 and references therein
The conditions under which a stationary point of the reformulated optimization is a solution of the GNCP F, G, K were provided in the literature

Summary

Introduction

Consider the generalized nonlinear complementarity problem, denoted by GNCP F, G, K , which is to find a vector x ∈ Rn such that. Wang et al 6 reformulated the problem as a system of nonlinear and nonsmooth equations and proposed a nonsmooth L-M method to solve this problem and proved that the algorithm is both globally and superlinearly convergent under mild assumptions. Zhang et al 7 rearranged the GNCP over a polyhedral as a smoothing system of equations, developed a smoothing Newton-type method for solving it, and proved that their method has local superlinear quadratic convergence under certain conditions. Inexact Newton methods have been proposed for solving NCP 8–10. We propose a new smoothing inexact Newton algorithm with non-monotone line search for solving GNCP by using a new type of smoothing function. We show that any accumulation point of the generated sequence is a solution of the GNCP, and we establish the local superlinear quadratic convergence of the proposed algorithm under the BD-regular assumption. The null space of a matrix B is denoted by N B

Preliminaries

Stationary Point and Nonsingularity Conditions

Algorithm and Convergence Property

Numerical Experiments

Conclusion