ABSTRACT

In this paper, we propose a new incremental constraint projection method (containing random projection method and cyclic projection method) for solving variational inequality problems in $R^n$, where the underlying function is Lipschitz continuous and monotone plus. We focus on special structures that lend themselves to sampling, such as when X is the intersection of a large number of sets, and/or F is an expected value or is the sum of a large number of component functions. Our method requires only two projections onto a suitable halfspace which replaces the projections onto constrained set $X_k$. We prove the sequence generated by our method is globally convergent to a solution of the variational inequalities in almost sure sense both random projection method and cyclic projection method. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.

摘要

在本文中，我们提出了一种新的增量约束投影方法（包含随机投影法和循环投影法）来解决变分不等式问题。$R^n$，其中基础函数是 Lipschitz 连续和单调加。我们专注于适合采样的特殊结构，例如当X是大量集合的交集，和/或F是预期值或大量组件函数的总和时。我们的方法只需要在合适的半空间上进行两个投影，从而将投影替换到约束集上$X_{ķ}$. 我们证明了我们的方法生成的序列在随机投影方法和循环投影方法几乎肯定意义上都全局收敛于变分不等式的解。最后，我们提供了数值实验来展示所提出算法的效率和优势。