具有可分离结构的线性约束凸优化问题的迫近正则收缩算法

A contraction method with proximal regularization for linearly constrained convex optimization problem with separable structures

对具有可分离结构的线性约束凸优化问题(也就是目标函数是有2个算子和形式的可分离凸优化问题)展开研究,考虑在一定的假设条件下,通过选取合适的迫近正则参数矩阵G,拟利用可实现的迫近正则收缩法求解具有可分离结构的线性约束凸优化问题.将与原问题等价的变分不等式作为理论研究框架,通过将原问题转化为一系列容易求解的子问题,达到降低原问题求解难度的目的,下一个迭代点的获取通过求解子问题生成.最后,提出一种新的迫近正则收缩算法,并且应用变分不等式等相关理论对文中给出的迫近正则收缩算法进行了收敛性分析.

In this paper, we study the linearly constrained convex optimization problem with separable structures （i. e. , the convex optimization problem whose objective function is the sum of two opera- tors）. By selecting the appropriate proximal regularization parameter G,we can imitate a contraction method with implementable proximal regularization to solve the linearly constrained convex optimiza- tion problem with separable structure, and we take variational inequality as theoretical framework which is equivalent to original problem,and transform the original problem into a series of easy sub- problems to reduce the difficulty in solving original problem. The next iterate point is obtained by solving subproblems. Finally, a new proximal regularization algorithm is proposed and its conver- gence is analyzed by using relevant variational inequality theories.