解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.

A new alternating direction method for solving separable variational inequality problems

The alternating direction methods are attractive for solving separable variational inequality problems,which solve the original large-scale problems via solving a series of small-scale problems.However,the twosubproblems solved per iteration are still variational inequality problems,which are structurally as difficult tosolve as the original problems.In this paper,we propose a new alternating direction method,which,per iteration,solves a variational inequality problem and a system of nonlinear equations.The nonlinear equation is well-conditioned and many efficient methods,such as Newton-type methods,can be adopted directly to solve it.Moreover,from numerical point of view,nonlinear equations are much easier to solve than variational inequalityproblems in many cases.Under the same mild assumptions as those in classical alternating direction methods,we prove global convergence of the new method.We also present some preliminary numerical results,which showthat our method is efficient.