Mathematical Programming 24 (1982)284- 313 North-Holland Publishing Company ITERATIVE METHODS FOR VARIATIONAL AND COMPLEMENTARITY PROBLEMS* J.S. PANG** and D. CHAN Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. Received 4 May 1981 Revised manuscript received 9 November 1981 In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods. Key words: Variational Inequality, Complementarity. Iterative Methods, Convergence, Traffic Equilibria. 1. Introduction Given a subset K of R" and a mapping f from R" into itself, the variational inequality problem VI(K, f) is to find a vector x* E K such that (y - x*)Tf(x *) >- 0 for all y ~ K. In the important special case where the set K is the nonnegative orthant R+, the problem VI(K, f) is equivalent to the nonlinear complementarity problem: Find a vector x E R" so that [30] x >- O, f(x) >- 0 and xTf(x) = 0, (1) The variational inequality problem has been called variously as the stationary point problem by Eaves [16] and the generalized equation by Robinson [44]. In this paper, we shall use the term variational inequality. The theory as well as the applications of both the variational inequality and the nonlinear complementarity problems has been well documented in the * This research was based on work supported by the National Science Foundation under grant ECS-7926320. ** Present address: School of Management and Administration, The University of Texas at Dallas, Richardson, TX 75080, U.S.A. 284