The augmented Lagrangian method with full Jacobian decomposition and logarithmic-quadratic proximal regularization for multiple-block separable convex programming
The SMAI journal of computational mathematics, Volume 4 (2018) , pp. 81-120.

We consider a separable convex minimization model whose variables are coupled by linear constraints and they are subject to the positive orthant constraints, and its objective function is in form of $m$ functions without coupled variables. It is well recognized that when the augmented Lagrangian method (ALM) is applied to solve some concrete applications, the resulting subproblem at each iteration should be decomposed to generate solvable subproblems. When the Gauss-Seidel decomposition is implemented, this idea has inspired the alternating direction method of multiplier (for $m=2$) and its variants (for $m\ge 3$). When the Jacobian decomposition is considered, it has been shown that the ALM with Jacobian decomposition in its subproblem is not necessarily convergent even when $m=2$ and it was suggested to regularize the decomposed subproblems with quadratic proximal terms to ensure the convergence. In this paper, we focus on the multiple-block case with $m\ge 3$. We consider implementing the full Jacobian decomposition to ALM’s subproblems and using the logarithmic-quadratic proximal (LQP) terms to regularize the decomposed subproblems. The resulting subproblems are all unconstrained minimization problems because the positive orthant constraints are all inactive; and they are fully eligible for parallel computation. Accordingly, the ALM with full Jacobian decomposition and LQP regularization is proposed. We also consider its inexact version which allows the subproblems to be solved inexactly. For both the exact and inexact versions, we comprehensively discuss their convergence, including their global convergence, worst-case convergence rates measured by the iteration-complexity in both the ergodic and nonergodic senses, and linear convergence rates under additional assumptions. Some preliminary numerical results are reported to demonstrate the efficiency of the ALM with full Jacobian decomposition and LQP regularization.

Published online: 2018-03-28
DOI: https://doi.org/10.5802/smai-jcm.30
Classification: 90C25,  90C33,  65K05
Keywords: convex programming, splitting methods, augmented Lagrangian method, logarithmic-quadratic proximal, parallel computation, convergence rate
@article{SMAI-JCM_2018__4__81_0,
author = {Min Li and Xiaoming Yuan},
title = {The augmented Lagrangian method with full Jacobian decomposition and logarithmic-quadratic proximal regularization for multiple-block separable convex programming},
journal = {The SMAI journal of computational mathematics},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {4},
year = {2018},
pages = {81-120},
doi = {10.5802/smai-jcm.30},
language = {en},
url={smai-jcm.centre-mersenne.org/item/SMAI-JCM_2018__4__81_0/}
}
Li, Min; Yuan, Xiaoming. The augmented Lagrangian method with full Jacobian decomposition and logarithmic-quadratic proximal regularization for multiple-block separable convex programming. The SMAI journal of computational mathematics, Volume 4 (2018) , pp. 81-120. doi : 10.5802/smai-jcm.30. https://smai-jcm.centre-mersenne.org/item/SMAI-JCM_2018__4__81_0/

[1] A. Auslender; M. Teboulle Entropic proximal decomposition methods for convex programs and variational inequalities, Math. Program., Volume 91 (2001), pp. 33-47

[2] A. Auslender; M. Teboulle Interior projection-like methods for monotone variational inequalities, Math. Program., Volume 104 (2005), pp. 39-68

[3] A. Auslender; M. Teboulle; S. Ben-Tiba A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl., Volume 12 (1999), pp. 31-40

[4] D. P. Bertsekas Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982

[5] S. Boyd; N. Parikh; E. Chu; B. Peleato; J. Eckstein Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, Volume 3 (2010), pp. 1-122

[6] C. H. Chen; B. S. He; Ye Y. Y.; X. M. Yuan The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program., Ser A, Volume 155 (2016), pp. 57-79

[7] C. H. Chen; M. Li; X. M. Yuan Further study on the convergence rate of alternating direction method of multipliers with logarithmic-quadratic proximal regularization, J. Optim. Theory Appli., Volume 166 (2015), pp. 906-929

[8] P. L. Combettes; J. C. Pesquet Proximal splitting methods in signal processing, Fixed-Point Algorithms for Inverse Problems in Science and Engineering (H. H. Bauschke; R. S. Burachik; P. L. Combettes; V. Elser; D. R. Luke; H. Wolkowicz, eds.), Springer, 2011, pp. 185-212

[9] D. Davis; W. T. Yin Convergence rate analysis of several splitting schemes, Splitting Methods in Communication, Imaging, Science, and Engineering (R. Glowinski; S. J. Osher; W. T. Yin, eds.), Springer, 2017, pp. 115-163

[10] W. Deng; M. J. Lai; Z. M. Peng; W. T. Yin Parallel multi-block ADMM with $o\left(1/k\right)$ convergence, J. Sci. Comput., Volume 71 (2017), pp. 712-736

[11] B. C. Eaves On the basic theorem of complementarity, Math. Program., Volume 1 (1971), pp. 68-75

[12] J. Eckstein; D. P. Bertsekas On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., Volume 55 (1992), pp. 293-318

[13] J. Eckstein; W. Yao Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results (2012) (RUTCOR Research Report RRR 32-2012)

[14] F. Facchinei; J. S. Pang Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I. Springer Series in Operations Research, Springer-Verlag, New York, 2003

[15] R. Glowinski Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984

[16] R. Glowinski On alternating direction methods of multipliers: A historical perspective, Modeling, Simulation and Optimization for Science and Technology, Volume 34 of the series Computational Methods in Applied Sciences, Springer, 2014, pp. 59-82

[17] R. Glowinski; A. Marrocco Approximation par éléments finis d’ordre un et résolution par pénalisation-dualité d’une classe de problèmes non linéaires, R.A.I.R.O., Volume R2 (1975), pp. 41-76

[18] R. Glowinski; P. Le Tallec Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, Philadelphia, PA, 1989

[19] E. G. Gol’shtein; N. V. Tret’yakov Modified Lagrangian in convex programming and their generalizations, Math. Program. Studies, Volume 10 (1979), pp. 86-97

[20] D. R. Han; X. M. Yuan; W. X. Zhang An augmented-Lagrangian-based parallel splitting method for separable convex programming with applications to image processing, Math. Comput., Volume 83 (2014), pp. 2263-2291

[21] P. C. Hansen; J. G. Nagy; D. P. O’Leary Deblurring Images: Matrices, Spectra, and Filtering, SIAM, Philadelphia, 2006

[22] B. S. He; L. S. Hou; X. M. Yuan On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, SIAM J. Optim., Volume 25 (2015), pp. 2274-2312

[23] B. S. He; H. Liu; Wang Z. R.; X. M. Yuan A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM J. Optim., Volume 24 (2014), pp. 1101-1140

[24] B. S. He; M. Tao; X. M. Yuan Alternating direction method with Gaussian back substitution for separable convex programming, SIAM J. Optim., Volume 22 (2012), pp. 313-340

[25] B. S. He; H. K. Xu; X. M. Yuan On the proximal Jacobian decomposition of ALM for multiple-block separable convex minimization problems and its relationship to ADMM, J. Sci. Comput., Volume 66 (2016), pp. 1204-1217

[26] B. S. He; X. M. Yuan On the O($1/n$) convergence rate of Douglas-Rachford alternating direction method, SIAM J. Numer. Anal., Volume 50 (2012), pp. 700-709

[27] B. S. He; X. M. Yuan On nonergodic convergence rate of Douglas-Rachford alternating direction method of multipliers, Numer. Math., Volume 130 (2015), pp. 567-577

[28] M. Hong; Z. Q. Luo On the linear convergence of the alternating direction method of multipliers, Math. Program., Volume 162 (2017), pp. 165-199

[29] M. Li; L.-Z. Liao; X. M. Yuan Inexact alternating direction method of multipliers with logarithmic-quadratic proximal regularization, J. Optim. Theory Appli., Volume 159 (2013), pp. 412-436

[30] M. Li; D. F. Sun; K.-C. Toh A majorized ADMM with indefinite proximal terms for linearly constrained convex composite optimization, SIAM J. Optim., Volume 26 (2016), pp. 922-950

[31] M. Li; X. M. Yuan A strictly contractive Peaceman-Rachford splitting method with logarithmic-quadratic proximal regularization for convex programming, Math. Oper. Res., Volume 40 (2015), pp. 842-858

[32] T. Y. Lin; S. Q. Ma; S. Z. Zhang On the global linear convergence of the ADMM with multiblock variables, SIAM J. Optim., Volume 25 (2015), pp. 1478-1497

[33] T. Y. Lin; S. Q. Ma; S. Z. Zhang On the sublinear convergence rate of multi-block ADMM, J. Oper. Res. Soc. China, Volume 3 (2015), pp. 251-274

[34] B. Martinet Regularization d’inequations variationelles par approximations successives, Revue Francaise d’Informatique et de Recherche Opérationelle, Volume 4 (1970), pp. 154-159

[35] J. G. Melo; R. D. C. Monteiro Iteration-complexity of a Jacobi-type non-Euclidean ADMM for multi-block linearly constrained nonconvex programs (2017) (arXiv:1705.07229v1)

[36] A. S. Nemirovsky; D. B. Yudin Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, New York, 1983

[37] Y. E. Nesterov A method for unconstrained convex minimization problem with the rate of convergence O($1/{k}^{2}$), Doklady AN SSSR, Volume 269 (1983), pp. 543-547

[38] Y. E. Nesterov Gradient methods for minimizing composite objective function, Math. Program., Ser. B, Volume 140 (2013), pp. 125-161

[39] M. Patriksson A survey on the continuous nonlinear resource allocation problem, European J. Oper. Res., Volume 185 (2008), pp. 1-46

[40] Y. G. Peng; A. Ganesh; J. Wright; W. L. Xu; Y. Ma Robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE Tran. Pattern Anal. Mach. Intel., Volume 34 (2012), pp. 2233-2246

[41] M. J. D. Powell A method for nonlinear constraints in minimization problems, Optimization (R. Fletcher, ed.), Academic Press, New York, 1969, pp. 283-298

[42] R. T. Rockafellar Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., Volume 1 (1976), pp. 97-116

[43] M. Tao; X. M. Yuan Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM J. Optim., Volume 21 (2011), pp. 57-81

[44] M. Tao; X. M. Yuan On the O$\left(1/t\right)$ convergence rate of alternating direction method with logarithmic-quadratic proximal regularization, SIAM J. Optim., Volume 22 (2012), pp. 1431-1448

[45] H. Uzawa Market mechanisms and mathematical programming, Econometrica, Volume 28 (1960), pp. 872-881

[46] X. M. Yuan; M. Li An LQP-based decomposition method for solving a class of variational inequalities, SIAM J. Optim, Volume 21 (2011), pp. 1309-1318