Aug-PDG:带不等式约束凸优化算法的线性收敛性  

Aug-PDG:linear convergence of convex optimization with inequality constraints

作　　者：孟敏 李修贤 MENG Min;LI Xiu-xian(Department of Control Science and Engineering,School of Electronics and Information Engineering,Tongji University,Shanghai 201804,China;Shanghai Research Institute for Intelligent Autonomous Systems,Tongji University,Shanghai 201210,China;Shanghai Institute of Intelligent Science and Technology,Tongji University,Shanghai 200092,China)

机构地区：[1]同济大学电子与信息工程学院控制科学与工程系,上海201804 [2]同济大学上海自主智能无人系统科学中心,上海201210 [3]同济大学上海智能科学与技术中心,上海200092

出　　处：《控制理论与应用》2022年第10期1969-1977,共9页Control Theory & Applications

基　　金：Supported by the Shanghai Pujiang Program(21PJ1413100);the National Natural Science Foundation of China(62003243,62103305);the Shanghai Municipal Science and Technology Major Project(2021SHZDZX0100);the Shanghai Municipal Commission of Science and Technology(1951-1132101);the Young Elite Scientist Sponsorship Program by cast of China Association for Science and Technology(YESS20200136);the Fundamental Research Funds for the Central Universities(22120210096)。

摘　　要：原始-对偶梯度算法广泛应用于求解带约束的凸优化问题,大部分文献仅证明了该算法的收敛性,而没有分析其收敛速度.因此,本文研究了求解带有不等式约束凸优化的一类离散算法,即增广原始-对偶梯度算法(AugPDG),证明了Aug-PDG算法在一些较弱的假设条件下可以半全局线性收敛到最优解,并明确给出了算法中步长的上界.最后,数值算例证实了所得理论结果的有效性.The primal-dual gradient algorithm has been widely employed for solving constrained optimization problems.While the convergence of this algorithm was proved in most references,it is less investigated whether it is globally linearly convergent.Therefore,this paper studies convergence rate of its variant,i.e.,the augmented primal-dual gradient algorithm(Aug-PDG),for handling the convex optimization problem with general convex inequality constraints.Specifically,it is shown that the Aug-PDG can converge semi-globally to the optimizer at a linear rate under some mild assumptions and an explicit bound is provided for the stepsize in this algorithm.Finally,a numerical example is presented to illustrate the effectiveness of the theoretical result.

关 键 词：凸优化 非线性约束 线性收敛 增广原始-对偶梯度算法 

分 类 号：O224[理学—运筹学与控制论]