一种非精确求解结构型变分不等式的渐近点算法

Inexact Proximal Point Algorithm for Structured Variational Inequalities

近来,交替方向法成为了学者们研究的热点。对于一类子问题能够精确求解的变分不等式,该算法是有效的。然而,在实际问题中,变分不等式的子问题是非常困难甚至是不可能精确求解的。在渐近点算法的基础上得到一种非精确的渐近点算法,使得变分不等式子问题具有显式解,通过简单的预测校正步得到子问题的解。在合理的假设下,算法的收敛性得到了证明,一些数值实验表明了所提算法的有效性。

Recently,the alternating direction method of multipliers has attracted great attention.For a class of variational inequalities,this method is efficient,when the subproblems can be solved exactly.However,the subproblems could be too difficult or impossible to be solved exactly in many practical applications.In this paper,we proposed an inexact proxi-mal point method based on proximal point method.The subproblem is simple to have a closed form solution.Instead of solving the subproblems exactly,we used the simple projection-correction method to approximate the subproblems’ real solutions.Convergence of the proposed method is proved under mild assumptions and its efficiency is also verified by some numerical experiments.