A New Hybrid Inexact Logarithmic-Quadratic Proximal Method for Nonlinear Complementarity Problems

A New Hybrid Inexact Logarithmic-Quadratic Proximal Method for Nonlinear Complementarity Problems

Ying Zhou (Iowa State University, USA) and Lizhi Wang (Iowa State University, USA)
DOI: 10.4018/joris.2010070101

Abstract

In this paper, the authors present and analyze a new hybrid inexact Logarithmic-Quadratic Proximal method for solving nonlinear complementarity problems. Each iteration of the new method consists of a prediction and a correction step. The predictor is produced using an inexact Logarithmic-Quadratic Proximal method, which is then corrected by the Proximal Point Algorithm. The new iterate is obtained by combining predictor and correction point at each iteration. In this paper, the authors prove the convergence of the new method under the mild assumptions that the function involved is continuous and monotone. Comparison to another existing method with numerical experiments on classical NCP instances demonstrates its superiority.
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Introduction

The nonlinear complementarity problem (NCP) is to determine a vector joris.2010070101.m01 such that

joris.2010070101.m02
(1) wherejoris.2010070101.m03is a nonlinear mapping. Throughout this paper we assume that joris.2010070101.m04 is continuous and monotone with respect to joris.2010070101.m05 and the solution set of (1) is nonempty. NCP has many important applications in engineering, economics, military operations planning, finance, medical treatment, supply chain management etc, (Auslender & Haddou, 1995; Das, 2009; Castagnoli & Favero, 2008; Ferris & Pang, 1997; Harker & Pang, 1990; Yan & Wang, 1997). Many numerical methods for solving NCP have been developed (Auslender & Haddou, 1995; Auslender et al., 1999; Bnouhachem & Noor, 2001; Burachik & Svaiter, 2001; Censor et al., 1994; Eckstein, 1998; Fischer, 1997; Guler, 1991; Iusem, 1998; Pang, 1995; Qi & Yang, 2002; Sun & Qi, 1999; Zhou, 2009).

NCP can be alternatively formulated as finding the zero point of an appropriate maximal monotone operator

joris.2010070101.m06
(2) i.e., finding joris.2010070101.m07 such that joris.2010070101.m08, where joris.2010070101.m09 is the normal cone operator to joris.2010070101.m10 defined by

joris.2010070101.m11
(3)

A well known method to find the zero point of a maximal monotone operator joris.2010070101.m12 is the proximal point algorithm (PPA), which starts with any vector joris.2010070101.m13 and joris.2010070101.m14 and iteratively updates joris.2010070101.m15 conforming the following problem:

joris.2010070101.m16
(4)

In order to obtain the new pointjoris.2010070101.m17, the subproblem (1.4) of PPA is equivalent to the following variational inequality problem:

Find joris.2010070101.m18such that

joris.2010070101.m19
(5)

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