In this paper, we use a unified approach to analyze the local convergence behavior of a wide class of projection-type methods for solving variational inequality problems. Under certain conditions, it is shown that, in a finite number of iterations, either the sequence of iterates terminates at a solution of the concerned problem or all iterates enter and remain in the relative interior of the optimal face and, hence, the subproblem reduces to a simpler form.

This paper concerns about the possibility of identifying the active set in a noninterior continuation method for solving the standard linear complementarity problem based on the algorithm and theory presented by Burke and Xu (J. Optim. Theory Appl. 112 (2002) 53). It is shown that under the assumptions of P-matrix and nondegeneracy, the algorithm requires at most O (ρ log (β0μ0/τ)) iterations to find the optimal active set, where β0 is the width of the neighborhood which depends on the initial point, μ0 > 0 is the initial smoothing parameter, ρ is a positive number which depends on the problem and the initial point, and τ is a small positive number which depends only on the problem.

This paper develops convergence theory of the gradient projection method by Calamai andMore (Math. Programming, vol. 39, 93-116, 1987) which, for minimizing a continuously differentiable optimization problem min{f .(x): x ∈ Ω} where Ω is a nonempty closed convex set, generates a sequence xk+1=(ak-ak)▽f (xk))where the stepsize ak > 0 is chosen suitably. It is shown that, when f (x) is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either xk→x* and x* is a minimizer (stationary point); or ‖xk‖→arg min{f (x) : x ∈ Ω}= and f (xk) inf{ f (x): x ∈ Ω}.

A noninterior continuation method is proposed for nonlinear complementarity problems. It improves the noninterior continuation methods recently studied by Burke and Xu [Math. Oper. Res., 23 (1998), pp. 719{734} and Xu [The Global Linear Convergence of an Infeasible Non-Interior Path-following Algorithm for Complementarity Problems with Uniform P-functions, Preprint, Department of Mathematics, University of Washington, Seattle, 1996]; the interior point neighborhood technique is extended to a broader class of smoothing functions introduced by Chen and Mangasarian [Comput. Optim. Appl., 5 (1996), pp. 97{138}. The method is shown to be globally linearly convergent following the methodology established by Burke and Xu. In addition, a local acceleration step is added to the method so that it is also locally quadratically convergent under suitable assumptions.

Projection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems.In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods.