A hybrid Newton method for solving the variational ... - CiteSeerX

A hybrid Newton method for solving the variational inequality problem via the D-gap function J.M. Peng

2

M. Fukushima

1

3

2

State Key Laboratory of Scienti c and Engineering Computing Institute of Computational Mathematics and Scienti c/Engineering Computing Academic Sinica, Beijing, P. O. Box 2719, 100080, China 3

Department of Applied Mathematics and Physics Graduate School of Engineering, Kyoto University, Kyoto 606-01, Japan

May 5, 1997

Abstract The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the D-gap function. It is proved that the D-gap function has bounded level sets for the strongly monotone VIP. A hybrid Newton-type method is proposed for minimizing the D-gap function.

Under some conditions, it is shown that the

algorithm is globally convergent and locally quadratically convergent.

Keywords: Variational inequality problem, D-gap function, Newton's method, unconstrained optimization, global convergence, quadratic convergence. 1

Introduction

The variational inequality problem (VIP) is to nd a vector

h

( 3 ) ; y 0 x3 i  0 ;

F x

8 2 y

3

x

X;

2

X

such that (1)

where X is a nonempty closed convex subset of