一类新的黄金比率原始对偶算法  

A NEW GOLDEN RATIO PRIMAL-DUAL ALGORITHM

作　　者：周丹青 常小凯[2] 杨俊锋[1] Zhou Danqing;Chang Xiaokai;Yang Junfeng(Department of Mathematics,Nanjing University,Nanjing 210093;School of Science,Lanzhou University of Technology,Lanzhou 730050)

机构地区：[1]南京大学数学系,南京210093 [2]兰州理工大学理学院,兰州730050

出　　处：《高等学校计算数学学报》2022年第1期97-106,共10页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金NSFC-11771208和NSFC-11922111;科技部重点研发计划2020YFA0713800的资助;甘肃省创新能力提升项目2020A022;国家自然科学基金NSFC-12161053的资助。

摘　　要：1引言令R^(p)和R^(q)是有限维欧几里德空间,并且每个空间的内积和诱导范数分别表示为<·,·>,||·||=√<·,·>.在本文中,我们旨在解决以下优化问题。In this paper,we propose and analyze a new golden ratio primal-dual algorithm for solving convex optimization problems involving the sum of a smooth function with Lipschitzian gradient,a nonsmooth proximable function and a nonsmooth proximable function composed with a linear term.The proposed algorithm is full-splitting in the sense that it does not rely on solving any subproblems or linear system of equations iteratively,the smooth function is handled by gradient evaluation,and the nonsmooth functions are handled by their proximity operators.Several well-known primal-dual splitting algorithms are closely related to this work,e.g.,the classical Arrow-Hurwicz method and the primal-dual algorithm of Chambolle and Pock.In particular,it is an extension of the golden ratio primal-dual algorithm recently proposed by Chang and Yang to include an extra smooth term with Lipschitzian gradient.Global iterate convergence as well as O(1/N) ergodic convergence rate results,measured by a primal-dual gap function,are established,where N denotes the iteration counter.

关 键 词：欧几里德空间 内积和 有限维 原始对偶算法 范数 优化问题 黄金比率 

分 类 号：O221.2[理学—运筹学与控制论] O224[理学—数学]