ON THE LINEAR CONVERGENCE RATE OF GENERALIZED ADMM FOR CONVEX COMPOSITE PROGRAMMING

被引：0
作者：
Wang, Han [1 ]
Li, Peili [2 ]
Xiao, Yunhai [2 ]
机构：
[1] School of Mathematics and Statistics, Henan University, Kaifeng
[2] Center for Applied Mathematics of Henan Province, Henan University, Kaifeng
来源：
Journal of Applied and Numerical Optimization | 2024年 / 6卷 / 01期
基金：
中国国家自然科学基金;
关键词：
Alternating direction method of multipliers; Calmness; Error bound condition; Linear convergence rate; Semi-proximal terms;
D O I：
10.23952/jano.6.2024.1.07
中图分类号：
学科分类号：
摘要：
Over the fast few years, the numerical success of the generalized alternating direction method of multipliers (GADMM) proposed by Eckstein and Bertsekas [Math. Progam. 1992] has inspired intensive attention in analyzing its theoretical convergence properties. This paper is devoted to the linear convergence rate of the semi-proximal GADMM (sPGADMM) for solving linearly constrained convex composite optimization problems. The semi-proximal terms contained in each subproblem possess the abilities of handling with multi-block problems efficiently. We initially present some important inequalities for the sequence generated by the sPGADMM, and then establish the local linear convergence rate under the assumption of calmness. As a by-product, the global convergence property is also discussed. © 2024 Journal of Applied and Numerial Optimization.
引用
收藏
页码：115 / 134
页数：19
相关论文
共 40 条
  • [11] Eckstein J., Some Saddle-function splitting methods for convex programming, Optim. Meth. Softw., 4, pp. 75-83, (1994)
  • [12] Eckstein J., Bertsekas D.P., On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55, pp. 293-318, (1992)
  • [13] Eckstein J., Yao W., Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results, Rutcor Research Reports, 32, 3, (2012)
  • [14] Fang E.X., He B.S., Liu H., Yuan X.M., Generalized alternating direction method of multipliers: New theoretical insights and applications, Math Program. Comput., 7, pp. 149-187, (2015)
  • [15] Fazel M., Pong T.K., Sun D.F., Tsend P., Hankel matrix rank minimization with applications to system identification and realization, SIAM J. Matrix Anal. Appl., 34, pp. 946-977, (2013)
  • [16] Gabay D., Applications of the method of multipliers to variational inequalities, Stud. Math. Appl., 15, pp. 299-331, (1983)
  • [17] Gabay D., Mercier B., A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2, pp. 17-40, (1976)
  • [18] Glowinski R., On Alternating Direction Methods of Multipliers: A Historical Perspective, Modeling, Simulation and Optimization for Science and Technology, (2014)
  • [19] Glowinski R., Marroco A., Sur l’approximation, paréléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires, ESAIM: Math. Model. Numer. Anal., 9, pp. 41-76, (1975)
  • [20] Han D.R., Su D.F., Zhang L.W., Linear rate convergence of the alternating direction method of multipliers for convex composite programming, Math. Oper. Res., 43, pp. 622-637, (2018)
1234