一种推广的求解可分离凸优化问题的黄金比率邻近ADMM算法  

AN EXTENDED GOLDEN RATIO PROXIMAL ALTERNATING DIRECTION METHOD OF MULTIPLIERS FOR SEPARABLE CONVEX OPTIMIZATION

作　　者：闫喜红[1] 李浩 王川龙[1] 陈红梅 杨俊锋[2] Yan Xihong;Li Hao;Wang Chuanlong;Chen Hongmei;Yang Junfeng(Taiyuan Normal University,Jinzhong 030619,China;Southeast University,Nanjing 211189,China)

机构地区：[1]太原师范学院数学与统计学院,晋中030619 [2]南京大学数学系,南京210093

出　　处：《计算数学》2024年第1期1-16,共16页Mathematica Numerica Sinica

基　　金：国家自然科学基金(11901424);山西省回国留学人员科研教研资助项目(2022-170);山西省科技创新人才团队专项资助项目(202204051002018)资助。

摘　　要：ADMM算法是求解可分离凸优化问题的经典算法之一,但其无法保证原始迭代序列的收敛性且其子问题计算量很大.为了保证该算法所有迭代点列的全局收敛性及提高计算效率,采用凸组合技术的黄金比率邻近ADMM算法被提出,其中凸组合因子Ψ是关键参数.本文在黄金比率邻近ADMM算法的基础上,扩大了凸组合因子Ψ的取值范围,提出了收敛步长范围更广的推广黄金比率邻近ADMM算法.并在一定的假设下,证明了算法的全局收敛性及函数值残差和约束违反度在遍历意义下的O(1/N)次线性收敛速度.以及,当目标函数中任意一个函数强凸时,证明了算法在遍历意义下的O(1/N2)收敛率.最后,本文通过数值试验表明推广算法的有效性.Alternating direction method of multipliers(ADMM) is one of the classical algorithms for solving separable convex optimization problems,but it cannot guarantee the convergence of primal iterates and its subproblems can be computationally demanding.In order to ensure convergence and improve computational efficiency,the golden ratio proximal ADMM using convex combination technique is proposed,where the convex combination factor Ψ is the key parameter.Based on the golden ratio proximal ADMM,we enlarge parameter Ψ and propose an extended golden ratio proximal ADMM(EgrpADMM).Under very mild assumptions,we establish the global convergence and O(1/N) ergodic sublinear convergence rate in terms of function value residual and constraint violation of EgrpADMM.Furthermore,the algorithm can achieve O(1/N~2) ergodic convergence when any of the separable subfunctions of the objective function is strongly convex.Finally,we demonstrate the performance of the proposed algorithms via numerical experiments.

关 键 词：可分离凸优化模型 ADMM算法 黄金比率邻近ADMM算法 

分 类 号：O224[理学—运筹学与控制论]