基于投影收缩的SA方法求解随机变分不等式问题

Stochastic Approximation Approaches to the Atochastic Variational Inequality Problem Based on Projection and Contraction Method

求解变分不等式的各种算法中,投影收缩算法易于执行、稳健、而且可以处理大规模问题,因此发展迅速.何炳生教授根据变分不等式及投影算子的性质确定的三个不等式,提出了求解变分不等式的投影收缩算法,此方法简单易行,且便于实现.用随机近似方法来求解随机变分不等式和随机优化问题已经被广泛的研究,其中函数值和一阶导数不可求,但可以用近似的方法得到.将投影收缩算法应用到求解随机变分不等式当中,在一些适当的条件下,可得到全局收敛的结果.

Projection and contraction method is an important algorithm for solving variational inequalities. It is easy to implement and robust. It can handle large scale problem. Professor He put forward some projection and contraction methods based on some basic properties of the projection operator. Stochastic approximation methods have been extensively studied in the literature for solving systems of stochastic equations and stochastic optimization problems where function values and first order derivatives are not observable, but can be approximated through simulation. A new algorithm is presented in this paper. The projection and contraction method is applied to stochastic approximation method. Global convergence result of proposed method is obtained under appropriate conditions.