凸优化分裂收缩算法统一框架的新进展——从好不容易凑出一个方法到并不费劲构造一族算法  

NEW PROGRESS IN A UNIFIED FRAMEWORK OF SPLITTING AND CONTRACTION METHODS FOR CONVEX OPTIMIZATION-FROM STRUGGLING TO PUT TOGETHER A METHOD TO EFFORTLESSLY CONSTRUCTING A FAMILY OF ALGORITHMS

作　　者：何炳生[1] He Bingsheng(Department of Mathematics,Nanjing University,Nanjing 210093)

机构地区：[1]南京大学数学系,南京210093

出　　处：《高等学校计算数学学报》2024年第1期1-22,共22页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金委NSFC Grant 11871029资助项目。

摘　　要：1引言求解线性约束凸优化问题,我们发表了一个预测-校正的算法统一框架,在[1,2-4]分别做了介绍.这个框架主要是用来验证一些算法的收敛性,借助这个框架也构造了一些算法[5,13].最近,我们发现了算法框架收敛性条件的等价表示[14],据此可以实现“从好不容易凑出一个方法到并不费劲构造一族算法”.Many of the convex optimization problems in scientific and engineering computation have linear constraints.After introducing multipliers,the problem is reduced to finding the saddle point of the Lagrange function.The saddle point is like the equilibrium point between two parties in a conflict of interest,and its equivalent mathematical expression is a monotone variational inequality.Based on this consideration,in the past ten years,we have proposed a unified framework of contraction methods using the basic knowledge of calculus and linear algebra,as well as general optimization principles.In addition to proving convergence,the algorithmic framework can also be used to construct some algorithms.Based on our recently discovered equivalent representation of the convergence conditions,this article points out that it can help us effortlessly design a class of contraction methods.Every method we have explored in the past is a special case in the current class.Finally,according to the equivalent framework,some methods are proposed for solving the multi-block separable convex optimization problems with linear equality(or inequality)constraints.

关 键 词：算法的收敛性 等价表示 统一框架 凸优化 收敛性条件 约束凸优化问题 算法框架 收缩算法 

分 类 号：O221.2[理学—运筹学与控制论] O224[理学—数学]