一般混合变分不等式的捆集近似算法  

A Bundle Proximal Method for Solving General Mixed Variational Inequalities

作　　者：夏福全[1] 黄南京[2] 

机构地区：[1]四川师范大学数学与软件科学学院,成都610066 [2]四川大学数学科学学院,成都610064

出　　处：《数学物理学报（A辑）》2011年第4期866-879,共14页Acta Mathematica Scientia

基　　金：国家自然科学基金(10671135;70831005);四川省教育厅重点项目(09ZA091);四川省应用基础项目(2010JY0121);教育部博士点基金(20105134120002)资助

摘　　要：该文研究了一般混合变分不等式解的捆集近似算法.该方法综合应用Cohen所介绍的辅助原理和Kiwiel所介绍的关于非光滑凸优化的捆集Bregman近似方法,构造迭代序列{x^n}.在迭代算法的每一步,通过求解迭代子问题获得当前迭代点x^n.一方面,x^n是迭代子问题的近似极小值点(非精确极小值点);另一方面,在迭代的每一子问题中,根据非光滑凸泛函f的次梯度,构造分段光滑的凸泛函(?)_k用以替代非光滑泛函f,这两方面使得迭代算法的每个子问题都容易求解,迭代点x^n容易获得.该文首先介绍如何构造作者的迭代算法,如何判别当前迭代点的好坏以及算法的终止条件.其次,在映象T满足伪Dunn性质的条件下,证明了迭代算法产生的迭代序列{x^n}收敛于一般混合变分不等式的解.In this paper,the authors consider a bundle proximal method for solving general mixed variational inequalities.The method is based on the auxiliary problem principle due to Cohen and the bundle Bregman proximal method for convex nonsmooth optimization due to Kiwiel.The strategy is to approximate,in the subproblems,the nonsmooth convex function / by a sequence of linear convex piecewise functions f_k,which is constructed from accumulated subgradient linearizations of f.As in the bundle Bregman proximal method for nonsmooth optimization,the method generates a sequence {x^k} by taking x^k to be an approximate minimizer of subproblems.This makes the subproblems more tractable.The authors first explain how to build a new iterative scheme and a stopping criterion to determine whether the current approximation is good enough.This criterion is different from that commonly used in the special case of nonsmooth optimization.The authors also prove that the convergence of the algorithm for the case that the mapping T satisfies the pseudo-Dunn property.

关 键 词：迭代算法 近似方法 捆集方法 强凸泛函 伪Dunn性质 

分 类 号：O176.3[理学—数学] O178[理学—基础数学]