Stochastic-Combination Search Direction Method for Monotone Variational Inequality Problems

This paper proposes a stochastic-combination search direction method for monotone variational inequality (VI) problems. Existing methods are developed with regard to one or some of the specified characteristics of the VI problem, but few of them are designed to solve all types of the VI problems. To investigate a more flexible method, which may perform fast convergence for all monotone VI problems, a new stochastic search direction is proposed in this paper. Such a search direction is a stochastic combination of two profitable search directions via two random weighting parameters. At each iteration, a best search direction together with its step size is selected in order to obtain a maximal progress of such iteration. The descent proposition of the stochastic direction is proved, which is useful to guarantee the convergence. Numerical examples are provided to show the efficiency of the proposed new solution algorithm. It is shown that the stochastic search direction is better than either or both of the other two search directions among a majority of the iterations. Therefore, it has the potential to achieve a faster convergence rate.