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April  2019, 15(2): 723-737. doi: 10.3934/jimo.2018067

A proximal alternating direction method for multi-block coupled convex optimization

1. 

Institute of Electromagnetics and Acoustics, Department of Electronic Science, Xiamen University, Xiamen, 361005, China

2. 

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

* Corresponding author

Received  August 2016 Revised  December 2017 Published  June 2018

Fund Project: The second author is supported by the National Natural Science Foundation of China [Grant No. 11401314].

In this paper, we propose a proximal alternating direction method (PADM) for solving the convex optimization problems with linear constraints whose objective function is the sum of multi-block separable functions and a coupled quadratic function. The algorithm generates the iterate via a simple correction step, where the descent direction is based on the PADM. We prove the convergence of the generated sequence under some mild assumptions. Finally, some familiar numerical results are reported for the new algorithm.

Citation: Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067
References:
[1]

A. AgarwalS. Negahban and M. J. Wainwright, Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions, Ann. Appl. Stat., 40 (2012), 1171-1197.  doi: 10.1214/12-AOS1000.  Google Scholar

[2]

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

[3]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, FnT Mach. Learn., 3 (2010), 1-122.  doi: 10.1561/2200000016.  Google Scholar

[4]

X. CaiD. Han and X. Yuan, On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function, Comput. Optim. Appl., 66 (2017), 39-73.  doi: 10.1007/s10589-016-9860-y.  Google Scholar

[5]

C. ChenB. HeX. Yuan and Y. Ye, The direct extension of ADMM for Muti-block convex minimization problems is not necessarily convergent, Math. Program., 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar

[6]

C. ChenM. LiX. Liu and Y. Ye, Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: Convergence analysis and insights, Mathematics, 65 (2017), 1231-1249.  doi: 10.1287/opre.2017.1615.  Google Scholar

[7]

C. Chen, Y. Shen and Y. You, On the convergence analysis of the alternating direction method of multipliers with three blocks, Abstr. Appl. Anal., 2013 (2013), Art. ID 183961, 7 pp.  Google Scholar

[8]

Y. CuiX. LiD. Sun and K. C. Toh, On the convergence properties of a majorized alternating direction method of multipliers for linearly constrained convex optimization problems with coupled objective functions, J. Optim. Theory Appl., 169 (2016), 1013-1041.  doi: 10.1007/s10957-016-0877-2.  Google Scholar

[9]

D. Davis and W. Yin, Convergence rate analysis of several splitting schemes, UCLA CAM Report, (2014), 14-51.   Google Scholar

[10]

W. Deng and W. Yin, On the global and linear convergence of the generalized alternating direction method of multipliers, J Sci. Comput., 66 (2016), 889-916.  doi: 10.1007/s10915-015-0048-x.  Google Scholar

[11]

J. Eckstein and M. Fukushima, Some reformulation and applications of the alternating direction method of multipliers, Scale Optim., (1994), 115-134.   Google Scholar

[12]

J. Eckstein and D. 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

[13]

F. Facchinei and C. Kanzow, Penalty methods for the solution of generalized Nash equilibrium problems, SIAM J. Control. Optim., 20 (2010), 2228-2253.  doi: 10.1137/090749499.  Google Scholar

[14]

C. FengH. Xu and B. Li, An Alternating direction method approach to cloud traffic management, IEEE T. Parall. Distr., 28 (2017), 2145-2158.  doi: 10.1109/TPDS.2017.2658620.  Google Scholar

[15]

M. Fortin and R. Glowinski, Augmented Lagrangian methods: Applications to the numerical solution of boundary value problems, Stud. Math. Appl., 15 (1983), xix+340 pp.  Google Scholar

[16]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Optim. Appl., 2 (1976), 17-40.  doi: 10.1016/0898-1221(76)90003-1.  Google Scholar

[17]

X. Gao and S. Zhang, First-Order algorithms for convex optimization with nonseparable objective and coupled constraints, J Oper. Res. Soc. China., 5 (2017), 131-159.  doi: 10.1007/s40305-016-0131-5.  Google Scholar

[18]

R. Glowinski and A. Marroco, Sur l'approximation, par elements finis d'ordre un, et la resolution, par penalisation-dualite, d'une classe de problemes de dirichlet non lineares, J Equine. Vet. Sci., 9 (1975), 41-76.   Google Scholar

[19]

D. HanX. Yuan and W. Zhang, An augmented-Lagrangian-based parallel splitting method for separable convex minimization with applications to image processing, Math. Comput., 83 (2014), 2263-2291.  doi: 10.1090/S0025-5718-2014-02829-9.  Google Scholar

[20]

D. Han and X. Yuan, A note on the alternating direction method of multipliers, J. Optim. Theory Appl., 155 (2012), 227-238.  doi: 10.1007/s10957-012-0003-z.  Google Scholar

[21]

D. HanX. YuanW. Zhang and X. Cai, An ADM-based splitting method for separable convex programming, Comput. Optim. Appl., 54 (2013), 343-369.  doi: 10.1007/s10589-012-9510-y.  Google Scholar

[22]

B. He, Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities, Comput. Optim. Appl., 42 (2009), 195-212.  doi: 10.1007/s10589-007-9109-x.  Google Scholar

[23]

B. HeH. YangQ. Meng and D. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities, J. Optim. Theory Appl., 112 (2002), 129-143.  doi: 10.1023/A:1013048729944.  Google Scholar

[24]

B. HeM. Tao and X. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming, SIAM J. Optim., 22 (2012), 313-340.  doi: 10.1137/110822347.  Google Scholar

[25]

B. HeM. Tao and X. Yuan, A splitting method for separable convex programming, IMA J Numer. Anal., 35 (2015), 394-426.  doi: 10.1093/imanum/drt060.  Google Scholar

[26]

B. HeL. Hou and X. Yuan, On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, SIAM J. Optim., 25 (2015), 2274-2312.  doi: 10.1137/130922793.  Google Scholar

[27]

M. Hong, T. Chang, X. Wang, M. Razaviyayn, S. Ma and Z. Luo, A block successive upper bound minimization method of multipliers for linearly constrained convex optimization, Mathematics, 2014. Google Scholar

[28]

M. HongZ. Luo and M. Razaviyayn, Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM J. Optim., 26 (2016), 337-364.  doi: 10.1137/140990309.  Google Scholar

[29]

M. Hong and Z. Luo, On the linear convergence of the alternating direction method of multipliers, Math. Program., 162 (2017), 165-199.  doi: 10.1007/s10107-016-1034-2.  Google Scholar

[30]

G. James, C. Paulson and P. Rusmevichientong, The Constrained Lasso, Technical report, University of Southern California, 2013. Google Scholar

[31]

X. LiD. Sun and K. C. Toh, A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Math. Program., 155 (2016), 333-373.  doi: 10.1007/s10107-014-0850-5.  Google Scholar

[32]

T. LinS. Ma and S. Zhang, On the global linear convergence of the ADMM with multi-block variables, SIAM J. Optim., 25 (2015), 1478-1497.  doi: 10.1137/140971178.  Google Scholar

[33]

J. F. MotaJ. M. XavierP. M. Aguiar and M. Puschel, Distributed optimization with local domains: Application in MPF and network flows, IEEE T. Automat. Contr., 60 (2015), 2004-2009.  doi: 10.1109/TAC.2014.2365686.  Google Scholar

[34]

Y. PengA. GaneshJ. WrightW. Xu and Y. Ma, Robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE T. Pattern. Anal., 34 (2012), 2233-2246.  doi: 10.1109/CVPR.2010.5540138.  Google Scholar

[35]

R. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97-116.  doi: 10.1287/moor.1.2.97.  Google Scholar

[36]

D. SunK. C. Toh and L. Yang, A convergent 3-block semi-proximal alternating direction method of multipliers for conic programming with 4-block constraints, SIAM J. Optim., 25 (2015), 882-915.  doi: 10.1137/140964357.  Google Scholar

[37]

L. Xu and D. Han, A proximal alternating direction method for weakly coupled variational inequalities, Pacific J. Optim., 9 (2013), 155-166.   Google Scholar

[38]

J. Yang and Y. Zhang, Alternating direction algorithms for $ \ell_1$-Problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar

[39]

X. Yuan, An improved proximal alternating directions method for monotone variational inequalities with separable structure, Comput. Optim. Appl., 49 (2011), 17-29.  doi: 10.1007/s10589-009-9293-y.  Google Scholar

show all references

References:
[1]

A. AgarwalS. Negahban and M. J. Wainwright, Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions, Ann. Appl. Stat., 40 (2012), 1171-1197.  doi: 10.1214/12-AOS1000.  Google Scholar

[2]

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

[3]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, FnT Mach. Learn., 3 (2010), 1-122.  doi: 10.1561/2200000016.  Google Scholar

[4]

X. CaiD. Han and X. Yuan, On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function, Comput. Optim. Appl., 66 (2017), 39-73.  doi: 10.1007/s10589-016-9860-y.  Google Scholar

[5]

C. ChenB. HeX. Yuan and Y. Ye, The direct extension of ADMM for Muti-block convex minimization problems is not necessarily convergent, Math. Program., 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.  Google Scholar

[6]

C. ChenM. LiX. Liu and Y. Ye, Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: Convergence analysis and insights, Mathematics, 65 (2017), 1231-1249.  doi: 10.1287/opre.2017.1615.  Google Scholar

[7]

C. Chen, Y. Shen and Y. You, On the convergence analysis of the alternating direction method of multipliers with three blocks, Abstr. Appl. Anal., 2013 (2013), Art. ID 183961, 7 pp.  Google Scholar

[8]

Y. CuiX. LiD. Sun and K. C. Toh, On the convergence properties of a majorized alternating direction method of multipliers for linearly constrained convex optimization problems with coupled objective functions, J. Optim. Theory Appl., 169 (2016), 1013-1041.  doi: 10.1007/s10957-016-0877-2.  Google Scholar

[9]

D. Davis and W. Yin, Convergence rate analysis of several splitting schemes, UCLA CAM Report, (2014), 14-51.   Google Scholar

[10]

W. Deng and W. Yin, On the global and linear convergence of the generalized alternating direction method of multipliers, J Sci. Comput., 66 (2016), 889-916.  doi: 10.1007/s10915-015-0048-x.  Google Scholar

[11]

J. Eckstein and M. Fukushima, Some reformulation and applications of the alternating direction method of multipliers, Scale Optim., (1994), 115-134.   Google Scholar

[12]

J. Eckstein and D. 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

[13]

F. Facchinei and C. Kanzow, Penalty methods for the solution of generalized Nash equilibrium problems, SIAM J. Control. Optim., 20 (2010), 2228-2253.  doi: 10.1137/090749499.  Google Scholar

[14]

C. FengH. Xu and B. Li, An Alternating direction method approach to cloud traffic management, IEEE T. Parall. Distr., 28 (2017), 2145-2158.  doi: 10.1109/TPDS.2017.2658620.  Google Scholar

[15]

M. Fortin and R. Glowinski, Augmented Lagrangian methods: Applications to the numerical solution of boundary value problems, Stud. Math. Appl., 15 (1983), xix+340 pp.  Google Scholar

[16]

D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Optim. Appl., 2 (1976), 17-40.  doi: 10.1016/0898-1221(76)90003-1.  Google Scholar

[17]

X. Gao and S. Zhang, First-Order algorithms for convex optimization with nonseparable objective and coupled constraints, J Oper. Res. Soc. China., 5 (2017), 131-159.  doi: 10.1007/s40305-016-0131-5.  Google Scholar

[18]

R. Glowinski and A. Marroco, Sur l'approximation, par elements finis d'ordre un, et la resolution, par penalisation-dualite, d'une classe de problemes de dirichlet non lineares, J Equine. Vet. Sci., 9 (1975), 41-76.   Google Scholar

[19]

D. HanX. Yuan and W. Zhang, An augmented-Lagrangian-based parallel splitting method for separable convex minimization with applications to image processing, Math. Comput., 83 (2014), 2263-2291.  doi: 10.1090/S0025-5718-2014-02829-9.  Google Scholar

[20]

D. Han and X. Yuan, A note on the alternating direction method of multipliers, J. Optim. Theory Appl., 155 (2012), 227-238.  doi: 10.1007/s10957-012-0003-z.  Google Scholar

[21]

D. HanX. YuanW. Zhang and X. Cai, An ADM-based splitting method for separable convex programming, Comput. Optim. Appl., 54 (2013), 343-369.  doi: 10.1007/s10589-012-9510-y.  Google Scholar

[22]

B. He, Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities, Comput. Optim. Appl., 42 (2009), 195-212.  doi: 10.1007/s10589-007-9109-x.  Google Scholar

[23]

B. HeH. YangQ. Meng and D. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities, J. Optim. Theory Appl., 112 (2002), 129-143.  doi: 10.1023/A:1013048729944.  Google Scholar

[24]

B. HeM. Tao and X. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming, SIAM J. Optim., 22 (2012), 313-340.  doi: 10.1137/110822347.  Google Scholar

[25]

B. HeM. Tao and X. Yuan, A splitting method for separable convex programming, IMA J Numer. Anal., 35 (2015), 394-426.  doi: 10.1093/imanum/drt060.  Google Scholar

[26]

B. HeL. Hou and X. Yuan, On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, SIAM J. Optim., 25 (2015), 2274-2312.  doi: 10.1137/130922793.  Google Scholar

[27]

M. Hong, T. Chang, X. Wang, M. Razaviyayn, S. Ma and Z. Luo, A block successive upper bound minimization method of multipliers for linearly constrained convex optimization, Mathematics, 2014. Google Scholar

[28]

M. HongZ. Luo and M. Razaviyayn, Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM J. Optim., 26 (2016), 337-364.  doi: 10.1137/140990309.  Google Scholar

[29]

M. Hong and Z. Luo, On the linear convergence of the alternating direction method of multipliers, Math. Program., 162 (2017), 165-199.  doi: 10.1007/s10107-016-1034-2.  Google Scholar

[30]

G. James, C. Paulson and P. Rusmevichientong, The Constrained Lasso, Technical report, University of Southern California, 2013. Google Scholar

[31]

X. LiD. Sun and K. C. Toh, A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Math. Program., 155 (2016), 333-373.  doi: 10.1007/s10107-014-0850-5.  Google Scholar

[32]

T. LinS. Ma and S. Zhang, On the global linear convergence of the ADMM with multi-block variables, SIAM J. Optim., 25 (2015), 1478-1497.  doi: 10.1137/140971178.  Google Scholar

[33]

J. F. MotaJ. M. XavierP. M. Aguiar and M. Puschel, Distributed optimization with local domains: Application in MPF and network flows, IEEE T. Automat. Contr., 60 (2015), 2004-2009.  doi: 10.1109/TAC.2014.2365686.  Google Scholar

[34]

Y. PengA. GaneshJ. WrightW. Xu and Y. Ma, Robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE T. Pattern. Anal., 34 (2012), 2233-2246.  doi: 10.1109/CVPR.2010.5540138.  Google Scholar

[35]

R. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97-116.  doi: 10.1287/moor.1.2.97.  Google Scholar

[36]

D. SunK. C. Toh and L. Yang, A convergent 3-block semi-proximal alternating direction method of multipliers for conic programming with 4-block constraints, SIAM J. Optim., 25 (2015), 882-915.  doi: 10.1137/140964357.  Google Scholar

[37]

L. Xu and D. Han, A proximal alternating direction method for weakly coupled variational inequalities, Pacific J. Optim., 9 (2013), 155-166.   Google Scholar

[38]

J. Yang and Y. Zhang, Alternating direction algorithms for $ \ell_1$-Problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar

[39]

X. Yuan, An improved proximal alternating directions method for monotone variational inequalities with separable structure, Comput. Optim. Appl., 49 (2011), 17-29.  doi: 10.1007/s10589-009-9293-y.  Google Scholar

Figure 1.  Convergence precision of all algorithms, the error is given by $\|Ax-b\|.$.
Figure 2.  Solve GNEP with self-adaptive stepsize
Figure 3.  Solve GNEP with fixed stepsize $\alpha_k = 0.2$
Figure 4.  The Basis Pursuit Problem, $Q = 10, \beta = 0.08.$
Figure 5.  The Constrained LASSO with $\beta = 0.4,\lambda = 0.1,Q = 190$.
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