An Improved Two-Step Method for Generalized Variational Inequalities

We propose an improved two-step extragradient algorithm for pseudomonotone generalized variational inequalities. It requires two projections at each iteration and allows one to take different stepsize rules. Moreover, from a geometric point of view, it is shown that the new method has a long stepsize, and it guarantees that the distance from the next iterative point to the solution set has a large decrease. Under mild conditions, we show that the method is globally convergent, and then the -linearly convergent property of the method is proven if a projection-type error bound holds locally. 1. Introduction Let be a multivalued mapping from into with nonempty values, where is a Euclidean space. Let be a nonempty, closed, and convex subset of the Euclidean space . The generalized variational inequality, abbreviated as GVI, is to find a vector such that there exists satisfying where stands for the inner product of vectors in . The solution set of problem (1) is denoted by . If the multivalued mapping is a single-valued mapping from to , then the GVI collapses to the classical variational inequality problem [1–4]. For the problem GVI, we all know that it plays a significant role in economics and transportation equilibrium, engineering sciences, and so forth, and it has received considerable attention in the past decades [1, 2, 5–11]. Solution methods for GVI have been studied extensively. They can be roughly categorized into two popular approaches to attack the solution existence problem of the GVI. The first is analytic approaches. Instead of solving problem directly, the analytic approach reformulates the GVI as a well-studied mathematical problem first and then invokes an existence theorem for the latter problem [12]. The second is a constructive approach in which the existence can be verified by the behavior of the proposed method which will be considered in this paper. To the best of our knowledge, the extragradient method [2, 13] is a popular constructive approach which was proposed by Korpelevich [13]. It has been proved that the method has a contract property; that is, the generated sequence by the method satisfies that for any solution of the GVI. It should be noted that the proximal point algorithm also possesses this property [14]. In [15], the authors proposed a new type extragradient projection method for variational inequalities (VI). The method proposed in [15] required only two projections at each iteration and allowed one to take different stepsize rules. Moreover, it was shown that this method had a long stepsize, and it guaranteed that the