Volume 39 Issue 12
. 
Turn off MathJax
Article Contents
Abstract
References
Article Navigation >  Applied Mathematics and Mechanics  > 2018  >  39(12): 1410-1425
WANG Xin, GUO Ke. Convergence of the Generalized Alternating Direction Method of Multipliers for a Class of Nonconvex Optimization Problems[J]. Applied Mathematics and Mechanics, 2018, 39(12): 1410-1425. doi: 10.21656/1000-0887.380334
Citation: WANG Xin, GUO Ke. Convergence of the Generalized Alternating Direction Method of Multipliers for a Class of Nonconvex Optimization Problems[J]. Applied Mathematics and Mechanics, 2018, 39(12): 1410-1425. doi: 10.21656/1000-0887.380334
shu

Convergence of the Generalized Alternating Direction Method of Multipliers for a Class of Nonconvex Optimization Problems

doi: 10.21656/1000-0887.380334

  • School of Mathematics and Information, China West Normal University,Nanchong, Sichuan 637002, P.R.China

Funds:  The National Natural Science Foundation of China(11571178; 11801455)
  • Received Date: 2017-12-27
  • Rev Recd Date: 2018-10-18
  • Publish Date: 2018-12-01
    Abstract
  • The generalized alternating direction method of multipliers (GADMM) for the minimization problems of the sum of 2 functions with linear constraints was considered, where one function was convex and the other can be expressed as the difference of 2 convex functions. For each subproblem in the GADMM, the linearization technique in the convex function difference algorithm was employed. Under the assumptions that the associated functions satisfy the Kurdyka-ojasiewicz inequality, the sequence generated with the GADMM converges to a critical point of the augmented Lagrangian function, while the penalty parameter in the augmented Lagrangian function is sufficiently large. At last, the convergence rate of the algorithm was established.
    • FullText(HTML)
    • References(23)
  • [1]
    GLOWINSKI R, MARROCCO A. Sur l’approximation par éléments finis d’ordre un et la résolution par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires[J]. Journal of Equine Veterinary Science,1975,2: 41-76.
    [2]
    YANG J F, YUAN X M. Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization[J]. Mathematics of Computation,2013,82(281): 301-329.
    [3]
    HE B S, YUAN X M. On non-ergodic convergence rate of douglas-rachford alternating direction method of multipliers[J]. Numerische Mathematik,2015,130(3): 567-577.
    [4]
    HE B S, YUAN X M. On the o(1/n) convergence rate of the Douglas-Rachford alternating direction method[J]. Society for Industrial and Applied Mathematics,2012,50(2): 700-709.
    [5]
    HONG M Y, LUO Z Q. On the linear convergence of the alternating direction method of multipliers[J]. Mathematical Programming,2017,162(2): 165-199.
    [6]
    GUO K, HAN D R, WU T T. Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints[J]. International Journal of Computer Mathematic,2017,94(8): 1653-1669.
    [7]
    GUO K, HAN D R, WANG Z W, et al. Convergence of ADMM for multi-block nonconvex separable optimization models[J]. Frontiers of Mathematics in China,2017,12(5): 1139-1162.
    [8]
    LI G Y, PONG T K. Global convergence of splitting methods for nonconvex composite optimization[J]. SIAM Journal on Optimization,2015,25(4): 2434-2460.
    [9]
    YU W, YIN W T, ZENG J S. Global convergence of ADMM in nonconvex nonsmooth optimization[J/OL]. Journal of Scientific Computing,2015. [2017-12-27]. https: // doi.org/ 10.1007/s10915-018-0757-z.
    [10]
    GABAY D. Applications of the method of multipliers tovariational inequalities[C]// In Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Amsterdam, Holland, 1983.
    [11]
    LIONS P L, MERCIER B. Splitting algorithms for the sum of two nonlinear operators[J]. Siam Journal on Numerical Analysis,1979,16(6): 964-979.
    [12]
    ECKSTEIN J, BERTSEKAS D. On the Douglas-Rachford splitting method and proximal point algorithm for maximal monotone operators[J]. Mathematical Programming,1992,55(1): 293-318.
    [13]
    MARTIENT B. Regularisation d’inéquations variations par approximations succesives[J]. Rev Franaise Informat Recherche Opérationnelle,1970,4(4): 154-159.
    [14]
    YIN P A, XIN J. Iterative l1-minimization for non-convex compressed sensing[J]. Journal of Computational Mathematics,2017,35(4): 439-451.
    [15]
    LOU Y F, YIN P H, XIN J. Point source super-resolution via non-convex l1-based methods[J]. Journal of Scientific Computing,2016,68(3): 1082-1100.
    [16]
    YIN P H, LOU Y F, HE Q, et al. Minimization of 1-2 for compressed sensing[J]. SIAM Journal on Scientific Computing,2015,37(1): A536-A563.
    [17]
    SUN T, YIN P H, CHENG L Z, et al. Alternating direction method of multipliers with difference of convex functions[J]. Advances in Computational Mathematics,2017,44(3): 1-22.
    [18]
    ROCKAFELLAR R T, WETS R J. Variational Analysis [M]. Berlin: Springer-verlag, 1998.
    [19]
    ROCKAFELLAR R T. Convex Analysis [M]. Princeton: Princeton University Press, 1970.
    [20]
    ATTOUCH H, BOLTE J, REDONT P, et al. Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-ojasiewicz inequality[J]. Mathematics of Operations Research,2010,35(2): 438-457.
    [21]
    BOLTE J, SABACH S, TEBOULLE M. Proximal alternatinglinearized minimization for nonconvex and nonsmooth problems[J]. Mathematical Programming,2014,146(1): 459-494.
    [22]
    NESTEROV Y. Introductory Lectures on Convex Optimization: A Basic Course [M]. Boston: Kluwer Academic Publishers, 2004.
    [23]
    ATTOUCH H, BOLTE J. On the convergence of the proximal algorithm for nonsmooth functions involving analytic features[J]. Mathematical Programming,2009,116(1): 5-16.
    • Relative Articles

  • Relative Articles

    [1]WANG Yannan, ZENG Jiaqin, HUANG Nanjing. Iterative Methods for Random Generalized Quasi Variational Inequalities With Applications[J]. Applied Mathematics and Mechanics, 2023, 44(11): 1378-1388. doi: 10.21656/1000-0887.440199
    [2]XIAO Wenke, CHEN Xingding. A Modified Multi-Splitting Iterative Method With the Restarted GMRES to Solve the PageRank Problem[J]. Applied Mathematics and Mechanics, 2022, 43(3): 330-340. doi: 10.21656/1000-0887.420210
    [3]LI Yuanfei. Convergence Results on Heat Source for 2D Viscous Primitive Equations of Ocean Dynamics[J]. Applied Mathematics and Mechanics, 2020, 41(3): 339-352. doi: 10.21656/1000-0887.400176
    [4]LIU Chun-mei, ZHONG Liu-qiang, SHU Shi, XIAO Ying-xiong. Convergence of an Adaptive Finite Element Method for 2D Elasticity Problems[J]. Applied Mathematics and Mechanics, 2014, 35(9): 969-978. doi: 10.3879/j.issn.1000-0887.2014.09.003
    [5]YE Chao, LUO Xian-nan, WEN Li-ping. High-Order Numerical Methods of the Fractional Order Stokes’ First Problem for a Heated Generalized Second Grade Fluid[J]. Applied Mathematics and Mechanics, 2012, 33(1): 61-75. doi: 10.3879/j.issn.1000-0887.2012.01.006
    [6]LI Ai-bing, ZHANG Li-feng, ZANG Zeng-liang, ZHANG Yun. Iterative and Adjusting Method for Computing Stream Function and Velocity Potential in Limited Domains and Its Convergence Analysis[J]. Applied Mathematics and Mechanics, 2012, 33(6): 651-662. doi: 10.3879/j.issn.1000-0887.2012.06.002
    [7]TANG Guo-ji, HUANG Nan-jing. A Projected Subgradient Method for Non-Lipschitz Set-Valued Mixed Variational Inequalities[J]. Applied Mathematics and Mechanics, 2011, 32(10): 1254-1264. doi: 10.3879/j.issn.1000-0887.2011.10.011
    [8]LUO Xue-ping, HUANG Nan-jing. Generalized H-η-Accretive Operators in Banach Spaces With an Application to Variational Inclusions[J]. Applied Mathematics and Mechanics, 2010, 31(4): 472-480. doi: 10.3879/j.issn.1000-0887.2010.04.009
    [9]ZHUANG Ping-hui, LIU Qing-xia. An Effective Numerical Method of the Rayleigh-Stokes Problem for a Heated Generalized Second Grade Fluid With Fractional Derivative[J]. Applied Mathematics and Mechanics, 2009, 30(12): 1440-1452. doi: 10.3879/j.issn.1000-0887.2009.12.005
    [10]LIU Jian-jun, HE Guo-qiang, KANG Chuan-gang. Nonlinear Implicit Iterative Method for Solving Nonlinear Ill-Posed Problems[J]. Applied Mathematics and Mechanics, 2009, 30(9): 1107-1116. doi: 10.3879/j.issn.1000-0887.2009.09.013
    [11]KANG Chuan-gang, HE Guo-qiang. A Mixed Newton-Tikhonov Method for Nonlinear Ⅲ-Posed Problems[J]. Applied Mathematics and Mechanics, 2009, 30(6): 690-700. doi: 10.3879/j.issn.1000-0887.2009.06.008
    [12]WU Wei, XU Dong-po, LI Zheng-xue. Convergence of Gradient Method for Elman Networks[J]. Applied Mathematics and Mechanics, 2008, 29(9): 1117-1123.
    [13]YIN Chang-ming, WHANG Han-xing, ZHAO Fei. Risk-Sensitive Reinforcement Learning Algorithms With Generalized Average Criterion[J]. Applied Mathematics and Mechanics, 2007, 28(3): 369-378.
    [14]HUANG Ruo-yu, ZHENG Chang-liang, ZHONG Wan-xie, YAO Wei-an. A New Quadrilateral Thin Plate Element Based on the Membrane-Plate Similarity Theory[J]. Applied Mathematics and Mechanics, 2002, 23(3): 239-248.
    [15]HUANG Ting-zhu, WANG Guang-bin. Convergence Theorems for the AOR Method[J]. Applied Mathematics and Mechanics, 2002, 23(11): 1183-1187.
    [16]XIU Nai-hua, GAO Zi-you. Convergence of a Modified SLP Algorithm for the Extended Linear Complementarity Problem[J]. Applied Mathematics and Mechanics, 2001, 22(5): 534-540.
    [17]Chen Zengqiang, Lin Maoqiong, Yuan Zhuzhi. Convergence and Stability of Recursive Damped Least Square Algorithm[J]. Applied Mathematics and Mechanics, 2000, 21(2): 209-214.
    [18]Bai Zhongzhi. Parallel Interval Matrix Multisplitting AOR Methods and Their Convergence[J]. Applied Mathematics and Mechanics, 1999, 20(2): 169-174.
    [19]Gao Yan. A Class of Wilson Arbitrary Quadrilateral Elements for an Axisymmetric Problem[J]. Applied Mathematics and Mechanics, 1998, 19(12): 1112-1117.
    [20]Huang Zheng-ming. On the Coupled Vibration of an Ideal Fluid with a Linear Elastic Structure[J]. Applied Mathematics and Mechanics, 1992, 13(10): 911-924.
    • Supplements(0)
    • Cited By

  • Cited by

    Periodical cited type(6)

    1. 袁欣，张守贵. 无摩擦弹性接触问题的自适应交替方向乘子法. 应用数学和力学. 2023(08): 989-998 . 本站查看
    2. 赵耿威，黄敬频. 求解鞍点问题的修正MSOR-like方法. 重庆理工大学学报(自然科学). 2022(11): 249-256 .
    3. 祝德春，周珺，曹满霞，黄尉. 基于l_2/l_q(q=2/3)最小化模型的块稀疏信号恢复. 应用数学和力学. 2021(09): 989-998 . 本站查看
    4. 薛中会，胡惠晴，党亚峥. 非凸不可分离问题的广义交替方向乘子法的收敛性. 上海理工大学学报. 2021(06): 580-588 .
    5. 李远飞. 海洋动力学中二维黏性原始方程组解对热源的收敛性. 应用数学和力学. 2020(03): 339-352 . 本站查看
    6. 李远飞. 在一个半无穷柱体上的非标准Stokes流体方程的二择一问题. 应用数学和力学. 2020(04): 406-419 . 本站查看

    Other cited types(6)

  • 加载中
    Created with Highcharts 5.0.7Chart context menuAccess Class DistributionFULLTEXT: 19.1 %FULLTEXT: 19.1 %META: 79.3 %META: 79.3 %PDF: 1.6 %PDF: 1.6 %FULLTEXTMETAPDF
    Created with Highcharts 5.0.7Chart context menuAccess Area Distribution其他: 5.3 %其他: 5.3 %其他: 0.1 %其他: 0.1 %China: 1.0 %China: 1.0 %Saint-Sylvain-d'Anjou: 0.2 %Saint-Sylvain-d'Anjou: 0.2 %Vancouver: 0.1 %Vancouver: 0.1 %上海: 0.5 %上海: 0.5 %保定: 0.6 %保定: 0.6 %凉山: 0.2 %凉山: 0.2 %北京: 1.3 %北京: 1.3 %南京: 0.4 %南京: 0.4 %南充: 0.7 %南充: 0.7 %南宁: 0.3 %南宁: 0.3 %南昌: 0.1 %南昌: 0.1 %台州: 0.1 %台州: 0.1 %合肥: 0.4 %合肥: 0.4 %哈尔滨: 0.7 %哈尔滨: 0.7 %哥伦布: 0.1 %哥伦布: 0.1 %嘉兴: 0.2 %嘉兴: 0.2 %天津: 0.1 %天津: 0.1 %宜宾: 0.4 %宜宾: 0.4 %宣城: 0.2 %宣城: 0.2 %平顶山: 0.1 %平顶山: 0.1 %广安: 0.1 %广安: 0.1 %广州: 0.4 %广州: 0.4 %张家口: 1.3 %张家口: 1.3 %徐州: 0.6 %徐州: 0.6 %恩施: 0.2 %恩施: 0.2 %成都: 0.1 %成都: 0.1 %扬州: 0.2 %扬州: 0.2 %承德: 0.2 %承德: 0.2 %新乡: 0.2 %新乡: 0.2 %昆明: 0.3 %昆明: 0.3 %杭州: 0.2 %杭州: 0.2 %格兰特县: 0.1 %格兰特县: 0.1 %森尼韦尔: 0.1 %森尼韦尔: 0.1 %武威: 0.2 %武威: 0.2 %武汉: 0.3 %武汉: 0.3 %沈阳: 0.1 %沈阳: 0.1 %洛杉矶: 0.2 %洛杉矶: 0.2 %济南: 0.1 %济南: 0.1 %深圳: 0.5 %深圳: 0.5 %温州: 0.3 %温州: 0.3 %湘潭: 0.1 %湘潭: 0.1 %漯河: 0.9 %漯河: 0.9 %石家庄: 0.4 %石家庄: 0.4 %秦皇岛: 0.2 %秦皇岛: 0.2 %芒廷维尤: 8.4 %芒廷维尤: 8.4 %芝加哥: 1.2 %芝加哥: 1.2 %苏州: 0.1 %苏州: 0.1 %衢州: 0.1 %衢州: 0.1 %西宁: 67.7 %西宁: 67.7 %西安: 0.2 %西安: 0.2 %运城: 0.2 %运城: 0.2 %郑州: 0.7 %郑州: 0.7 %重庆: 0.4 %重庆: 0.4 %长沙: 0.3 %长沙: 0.3 %长治: 0.1 %长治: 0.1 %青岛: 0.2 %青岛: 0.2 %其他其他ChinaSaint-Sylvain-d'AnjouVancouver上海保定凉山北京南京南充南宁南昌台州合肥哈尔滨哥伦布嘉兴天津宜宾宣城平顶山广安广州张家口徐州恩施成都扬州承德新乡昆明杭州格兰特县森尼韦尔武威武汉沈阳洛杉矶济南深圳温州湘潭漯河石家庄秦皇岛芒廷维尤芝加哥苏州衢州西宁西安运城郑州重庆长沙长治青岛

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索
    Get Citation
    PDF
    XML

    Article Metrics

    Article views (1586) PDF downloads(874) Cited by(12)
    Proportional views
    Related
    Contact Us

    Copyright © Editorial Department of Applied Mathematics and Mechanics渝ICP备11007697号-3

    address：P.O. Box 90, Chongqing Jiaotong University, No.66 Xuefu Road, Nan'an District, Chongqing, 400074, P.R.ChinaPostcode：400074

    Phone/Fax: 023-62652450　

    Email: amm1980@vip.163.com； amm1980@163.com； applmathmech@cqjtu.edu.cn

    Journal Online
    Email alert RSS

    Supported by: Beijing Renhe Information Technology Co. Ltd  

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return