A new incremental constraint projection method for solving monotone variational inequalities

In this paper, we propose a new incremental constraint projection algorithm for solving variational inequalities, where the underlying function is monotone plus and Lipschitz continuous. The algorithm consists two steps. In the first step, we compute a predictor point. This procedure requires a single random projection onto some set Xwi and employs an Armijo-type linesearch along a feasible direction. Then in the second step an iterate is obtained as the random projection of some point onto the set Xwi which we have used in the first step. The incremental constraint projection algorithm is considered for random selection and for cyclic selection of the samples wi. Accordingly, this algorithm is named random projection algorithm and cyclic projection algorithm. The method is shown to be globally convergent to a solution of the variational inequality problem in almost sure sense both random projection method and cyclic projection method. We provide some computational experiments and compare the efficiency of random projection method and cyclic projection method with some known algorithms.