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

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

doi: 10.21656/1000-0887.380334
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
  • 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.
  • loading
  • [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.
  • 加载中

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

    Article Metrics

    Article views (1235) PDF downloads(867) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return