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.

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.

In this paper, we study the equivalence characterizations of several modified fixedpoint equations to variational inequalities (VI). Based on these equations, we give some applications in constructing iterative methods for the solution of the VI. Especially, we show global convergence, the sublinear convergence, and the finite termination of a new iterative algorithm under certain conditions.

In this note, we develop some new properties of a fundamental quantity associated with a P-matrix introduced by Mathias and Pang[1]. Also, based on extensions of such a quantity, we obtain global error bounds for the vertical and horizontal linear complementarity problems.