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March  2020, 16(2): 835-856. doi: 10.3934/jimo.2018181

Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization

1. 

School of Mathematical Sciences, Jiangsu Key Labratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China

2. 

School of Economics and Management, Southeast University, Nanjing, 210096, China

Received  January 2018 Revised  July 2018 Published  December 2018

We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be indefinite which plays a vital role in accelerating numerical performance. In this paper, we are devoted to deriving the optimal lower bound of the proximal parameter and result in the generalized ADMM with optimal indefinite proximal term. The global convergence and the $ O(1/t) $ convergence rate measured by the iteration complexity of the proposed method are proved. Moreover, some preliminary numerical experiments on LASSO and total variation-based denoising problems are presented to demonstrate the efficiency of the proposed method and the advantage of the optimal lower bound.

Citation: Fan Jiang, Zhongming Wu, Xingju Cai. Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization. Journal of Industrial & Management Optimization, 2020, 16 (2) : 835-856. doi: 10.3934/jimo.2018181
References:
[1]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[2]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends® Mach. Learn., 3 (2011), 1-122.   Google Scholar

[3]

E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), 717-772.  doi: 10.1007/s10208-009-9045-5.  Google Scholar

[4]

S. ChenD. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61.  doi: 10.1137/S1064827596304010.  Google Scholar

[5]

P. L. Combettes and J. C. Pesquet, A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery, IEEE J-STSP., 1 (2008), 564-574.   Google Scholar

[6]

P. L. Combettes and J. C. Pesquet, Proximal splitting methods in signal processing. In Fixed-point algorithms for inverse problems in science and engineering, Springer New York, 49 (2011), 185-212.  doi: 10.1007/978-1-4419-9569-8_10.  Google Scholar

[7]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forwardbackward splitting, SIAM J. Multiscale Model. Simul., 4 (2005), 1168-1200.  doi: 10.1137/050626090.  Google Scholar

[8]

J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), 293-318.  doi: 10.1007/BF01581204.  Google Scholar

[9]

J. Eckstein and Y. Wang, Understanding the convergence of the alternating direction method of multipliers: Theoretical and computational perspectives, Pac. J. Optim., 11 (2015), 619-644.   Google Scholar

[10]

M. J. Fadili and J. L. Starck, Monotone operator splitting for optimization problems in sparse recovery, IEEE ICIP., (2009), 1461-1464.   Google Scholar

[11]

E. X. FangB. S. HeH. Liu and X. M. Yuan, Generalized alternating direction method of multipliers: new theoretical insights and applications, Math. Prog. Comp., 7 (2015), 149-187.  doi: 10.1007/s12532-015-0078-2.  Google Scholar

[12]

M. A. T. Figueiredo and J. M. Bioucas-Dias, Restoration of poissonian images using alternating direction optimization, IEEE T. Image Process., 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

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D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2 (1976), 17-40.   Google Scholar

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B. Gao and F. Ma, Symmetric ADMM with positive-indefinite proximal regularization for linearly constrained convex optimization, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[15]

R. Glowinski, On alternating direction methods of multipliers: A historical perspective, Model. Simul. Optim. Sci. Technol. Comput. Methods Appl. Sci., 34 (2014), 59-82.  doi: 10.1007/978-94-017-9054-3_4.  Google Scholar

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R. Glowinski and A. Marrocco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Fr. Autom. Inf. Rech. Opér., Anal. Numér., 9 (1975), 41-76.   Google Scholar

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Y. Gu, B. Jiang and D. R. Han, A semi-proximal-based strictly contractive Peaceman-Rachford splitting method, Avaliable on http://www.optimization-online.org (2015). Google Scholar

[18]

B. S. He, A new method for a class of linear variational inequalities, Math. Program., 66 (1994), 137-144.  doi: 10.1007/BF01581141.  Google Scholar

[19]

B. S. He, PPA-like contraction methods for convex optimization: A framework using variational inequality approach, J. Oper. Res. Soc. China, 3 (2015), 391-420.  doi: 10.1007/s40305-015-0108-9.  Google Scholar

[20]

B. S. HeH. LiuZ. R. Wang and X. M. Yuan, A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM J. Optim., 24 (2014), 1011-1040.  doi: 10.1137/13090849X.  Google Scholar

[21]

B. S. He, F. Ma and X. M. Yuan, Linearized alternating direction method of multipliers via positive-indefinite proximal regularization for convex programming, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[22]

B. S. He, F. Ma and X. M. Yuan, Optimal linearized alternating direction method of multipliers for convex programming, Avaliable on http://www.optimization-online.org (2017). Google Scholar

[23]

B. S. He and X. M. Yuan, Improving an ADMM-like splitting method via positive-indefinite proximal regularization for three-block separable convex minimization, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[24]

B. S. He, F. Ma and X. M. Yuan, Positive-indefnite proximal augmented Lagrangian method and its application to full Jacobian splitting for multi-block separable convex minimization problems, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[25]

B. S. He and H. Yang, Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities, Oper. Res. Lett., 23 (1998), 151-161.  doi: 10.1016/S0167-6377(98)00044-3.  Google Scholar

[26]

B. S. He and X. M. Yuan, On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers, Numer. Math., 130 (2015), 567-577.  doi: 10.1007/s00211-014-0673-6.  Google Scholar

[27]

M. LiX. X. Li and X. M. Yuan, Convergence analysis of the generalized alternating direction method of multipliers with logarithmic-quadratic proximal regularization, J. Optim. Theory Appl., 164 (2015), 218-233.  doi: 10.1007/s10957-014-0567-x.  Google Scholar

[28]

M. LiD. F. Sun and K. C. Toh, A majorized ADMM with indefinite proximal term for linearly constrained convex composite optimization, SIAM J. Optim., 26 (2016), 922-950.  doi: 10.1137/140999025.  Google Scholar

[29]

X. X. LiL. L. MoX. M. Yuan and J. Z. Zhang, Linearized alternating direction method of multipliers for sparse group and fused LASSO models, Comput. Statist. Data Anal., 79 (2014), 203-221.  doi: 10.1016/j.csda.2014.05.017.  Google Scholar

[30]

X. X. Li and X. M. Yuan, A proximal strictly contractive Peaceman-Rachford splitting method for convex programming with applications to imaging, SIAM J. Imaging Sci., 8 (2015), 1332-1365.  doi: 10.1137/14099509X.  Google Scholar

[31]

B. RechtM. Fazel and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.  doi: 10.1137/070697835.  Google Scholar

[32]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[33]

G. Steidl and T. Teuber, Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vis., 36 (2010), 168-184.  doi: 10.1007/s10851-009-0179-5.  Google Scholar

[34]

M. Tao and X. M. Yuan, Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM J. Optim., 21 (2011), 57-81.  doi: 10.1137/100781894.  Google Scholar

[35]

R. J. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.   Google Scholar

[36]

J. F. Yang and X. M. Yuan, Linearized augmented lagrangian and alternating direction methods for nuclear norm minimization, Math. Comp., 82 (2013), 301-329.  doi: 10.1090/S0025-5718-2012-02598-1.  Google Scholar

[37]

X. M. Yuan, Alternating direction method for covariance selection models, J. Sci. Comput., 51 (2012), 261-273.  doi: 10.1007/s10915-011-9507-1.  Google Scholar

[38]

W. X. ZhangX. J. Cai and Z. H. Jia, A proximal alternating linearization method for minimizing the sum of two convex functions, Sci. China Math., 58 (2015), 1-20.  doi: 10.1007/s11425-015-4986-4.  Google Scholar

show all references

References:
[1]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[2]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends® Mach. Learn., 3 (2011), 1-122.   Google Scholar

[3]

E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), 717-772.  doi: 10.1007/s10208-009-9045-5.  Google Scholar

[4]

S. ChenD. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), 33-61.  doi: 10.1137/S1064827596304010.  Google Scholar

[5]

P. L. Combettes and J. C. Pesquet, A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery, IEEE J-STSP., 1 (2008), 564-574.   Google Scholar

[6]

P. L. Combettes and J. C. Pesquet, Proximal splitting methods in signal processing. In Fixed-point algorithms for inverse problems in science and engineering, Springer New York, 49 (2011), 185-212.  doi: 10.1007/978-1-4419-9569-8_10.  Google Scholar

[7]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forwardbackward splitting, SIAM J. Multiscale Model. Simul., 4 (2005), 1168-1200.  doi: 10.1137/050626090.  Google Scholar

[8]

J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), 293-318.  doi: 10.1007/BF01581204.  Google Scholar

[9]

J. Eckstein and Y. Wang, Understanding the convergence of the alternating direction method of multipliers: Theoretical and computational perspectives, Pac. J. Optim., 11 (2015), 619-644.   Google Scholar

[10]

M. J. Fadili and J. L. Starck, Monotone operator splitting for optimization problems in sparse recovery, IEEE ICIP., (2009), 1461-1464.   Google Scholar

[11]

E. X. FangB. S. HeH. Liu and X. M. Yuan, Generalized alternating direction method of multipliers: new theoretical insights and applications, Math. Prog. Comp., 7 (2015), 149-187.  doi: 10.1007/s12532-015-0078-2.  Google Scholar

[12]

M. A. T. Figueiredo and J. M. Bioucas-Dias, Restoration of poissonian images using alternating direction optimization, IEEE T. Image Process., 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

[13]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2 (1976), 17-40.   Google Scholar

[14]

B. Gao and F. Ma, Symmetric ADMM with positive-indefinite proximal regularization for linearly constrained convex optimization, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[15]

R. Glowinski, On alternating direction methods of multipliers: A historical perspective, Model. Simul. Optim. Sci. Technol. Comput. Methods Appl. Sci., 34 (2014), 59-82.  doi: 10.1007/978-94-017-9054-3_4.  Google Scholar

[16]

R. Glowinski and A. Marrocco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Fr. Autom. Inf. Rech. Opér., Anal. Numér., 9 (1975), 41-76.   Google Scholar

[17]

Y. Gu, B. Jiang and D. R. Han, A semi-proximal-based strictly contractive Peaceman-Rachford splitting method, Avaliable on http://www.optimization-online.org (2015). Google Scholar

[18]

B. S. He, A new method for a class of linear variational inequalities, Math. Program., 66 (1994), 137-144.  doi: 10.1007/BF01581141.  Google Scholar

[19]

B. S. He, PPA-like contraction methods for convex optimization: A framework using variational inequality approach, J. Oper. Res. Soc. China, 3 (2015), 391-420.  doi: 10.1007/s40305-015-0108-9.  Google Scholar

[20]

B. S. HeH. LiuZ. R. Wang and X. M. Yuan, A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM J. Optim., 24 (2014), 1011-1040.  doi: 10.1137/13090849X.  Google Scholar

[21]

B. S. He, F. Ma and X. M. Yuan, Linearized alternating direction method of multipliers via positive-indefinite proximal regularization for convex programming, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[22]

B. S. He, F. Ma and X. M. Yuan, Optimal linearized alternating direction method of multipliers for convex programming, Avaliable on http://www.optimization-online.org (2017). Google Scholar

[23]

B. S. He and X. M. Yuan, Improving an ADMM-like splitting method via positive-indefinite proximal regularization for three-block separable convex minimization, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[24]

B. S. He, F. Ma and X. M. Yuan, Positive-indefnite proximal augmented Lagrangian method and its application to full Jacobian splitting for multi-block separable convex minimization problems, Avaliable on http://www.optimization-online.org (2016). Google Scholar

[25]

B. S. He and H. Yang, Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities, Oper. Res. Lett., 23 (1998), 151-161.  doi: 10.1016/S0167-6377(98)00044-3.  Google Scholar

[26]

B. S. He and X. M. Yuan, On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers, Numer. Math., 130 (2015), 567-577.  doi: 10.1007/s00211-014-0673-6.  Google Scholar

[27]

M. LiX. X. Li and X. M. Yuan, Convergence analysis of the generalized alternating direction method of multipliers with logarithmic-quadratic proximal regularization, J. Optim. Theory Appl., 164 (2015), 218-233.  doi: 10.1007/s10957-014-0567-x.  Google Scholar

[28]

M. LiD. F. Sun and K. C. Toh, A majorized ADMM with indefinite proximal term for linearly constrained convex composite optimization, SIAM J. Optim., 26 (2016), 922-950.  doi: 10.1137/140999025.  Google Scholar

[29]

X. X. LiL. L. MoX. M. Yuan and J. Z. Zhang, Linearized alternating direction method of multipliers for sparse group and fused LASSO models, Comput. Statist. Data Anal., 79 (2014), 203-221.  doi: 10.1016/j.csda.2014.05.017.  Google Scholar

[30]

X. X. Li and X. M. Yuan, A proximal strictly contractive Peaceman-Rachford splitting method for convex programming with applications to imaging, SIAM J. Imaging Sci., 8 (2015), 1332-1365.  doi: 10.1137/14099509X.  Google Scholar

[31]

B. RechtM. Fazel and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.  doi: 10.1137/070697835.  Google Scholar

[32]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[33]

G. Steidl and T. Teuber, Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vis., 36 (2010), 168-184.  doi: 10.1007/s10851-009-0179-5.  Google Scholar

[34]

M. Tao and X. M. Yuan, Recovering low-rank and sparse components of matrices from incomplete and noisy observations, SIAM J. Optim., 21 (2011), 57-81.  doi: 10.1137/100781894.  Google Scholar

[35]

R. J. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.   Google Scholar

[36]

J. F. Yang and X. M. Yuan, Linearized augmented lagrangian and alternating direction methods for nuclear norm minimization, Math. Comp., 82 (2013), 301-329.  doi: 10.1090/S0025-5718-2012-02598-1.  Google Scholar

[37]

X. M. Yuan, Alternating direction method for covariance selection models, J. Sci. Comput., 51 (2012), 261-273.  doi: 10.1007/s10915-011-9507-1.  Google Scholar

[38]

W. X. ZhangX. J. Cai and Z. H. Jia, A proximal alternating linearization method for minimizing the sum of two convex functions, Sci. China Math., 58 (2015), 1-20.  doi: 10.1007/s11425-015-4986-4.  Google Scholar

Figure 1.  Sensitivity test on the factor $r$ when $\beta = 1$
Figure 2.  Sensitivity test on the factor $r$ when $\beta = 5$
Table 1.  Numerical results for LASSO
$r=0.3$$r=-0.3$
$n$ $m$PG-ADMMPID-SADMMIPG-ADMMPG-ADMMPID-SADMMIPG-ADMM
Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)
20050065.50.0455.10.0353.90.0366.50.0451.30.0345.00.03
30080068.00.3557.20.2955.70.2869.00.3553.30.2746.60.24
300100091.10.6676.90.5575.00.5492.00.6671.00.5162.00.46
500150080.11.7967.31.4965.91.4681.31.8062.81.3955.01.22
5002000108.03.3490.72.8288.62.77108.33.3883.62.6073.12.27
800250085.85.4072.14.5670.54.4387.05.4767.04.2158.73.73
1000300079.07.4266.36.2264.86.1780.27.5461.95.8454.15.12
1500500092.822.2277.918.6176.218.2493.922.5172.317.3763.415.22
$r=0.3$$r=-0.3$
$n$ $m$PG-ADMMPID-SADMMIPG-ADMMPG-ADMMPID-SADMMIPG-ADMM
Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)
20050065.50.0455.10.0353.90.0366.50.0451.30.0345.00.03
30080068.00.3557.20.2955.70.2869.00.3553.30.2746.60.24
300100091.10.6676.90.5575.00.5492.00.6671.00.5162.00.46
500150080.11.7967.31.4965.91.4681.31.8062.81.3955.01.22
5002000108.03.3490.72.8288.62.77108.33.3883.62.6073.12.27
800250085.85.4072.14.5670.54.4387.05.4767.04.2158.73.73
1000300079.07.4266.36.2264.86.1780.27.5461.95.8454.15.12
1500500092.822.2277.918.6176.218.2493.922.5172.317.3763.415.22
Table 2.  Numerical results for TV denoising model
$r=0.3$$r=-0.3$
$n$PG-ADMMPID-SADMMIPG-ADMMPG-ADMMPID-SADMMIPG-ADMM
Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)
500278.00.03260.70.03258.70.03401.50.05356.60.04339.90.04
1000325.90.06297.40.05294.60.05464.80.08422.00.08402.30.07
2000414.90.12380.80.11376.70.11585.30.17533.20.16497.90.15
3000449.50.20423.90.19418.90.19683.80.31602.70.27573.60.26
5000488.70.33456.00.30452.30.30730.00.48662.70.44621.70.41
6000488.10.38456.00.35452.10.35720.70.56642.00.50612.70.47
8000598.30.59558.20.54553.90.54869.50.85774.70.75729.90.72
10000609.60.72561.90.66556.70.66877.01.03791.00.93748.70.88
$r=0.3$$r=-0.3$
$n$PG-ADMMPID-SADMMIPG-ADMMPG-ADMMPID-SADMMIPG-ADMM
Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)Iter.CPU(s)
500278.00.03260.70.03258.70.03401.50.05356.60.04339.90.04
1000325.90.06297.40.05294.60.05464.80.08422.00.08402.30.07
2000414.90.12380.80.11376.70.11585.30.17533.20.16497.90.15
3000449.50.20423.90.19418.90.19683.80.31602.70.27573.60.26
5000488.70.33456.00.30452.30.30730.00.48662.70.44621.70.41
6000488.10.38456.00.35452.10.35720.70.56642.00.50612.70.47
8000598.30.59558.20.54553.90.54869.50.85774.70.75729.90.72
10000609.60.72561.90.66556.70.66877.01.03791.00.93748.70.88
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