American Institute of Mathematical Sciences

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January  2018, 14(1): 397-412. doi: 10.3934/jimo.2017052

A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming

Su-Hong Jiang and  Min Li ,

School of Management and Engineering, Nanjing University, Nanjing 210093, China

* Corresponding author: MIN LI

Received  June 2016 Revised  December 2016 Published  January 2018 Early access  June 2017

Fund Project: This research was supported by National Science Foundation of China (Grant No. 11401300,71602083,71671085 and 11001053) and Program for New Century Excellent Talents in University (Grant No. NCET-12-0111).

We propose a modified splitting method for a linearly constrained minimization model whose objective function is the sum of three convex functions without coupled variables. Our work is mainly inspired by the recently proposed strictly contractive Peaceman-Rachford splitting method (SC-PRSM) for a two-block separable convex minimization model. For the new method, we prove its convergence and estimate its convergence rates measured by iteration complexity in the nonergodic sense. We also test the SC-PRSM on the continuous resource allocation problem, and the numerical results show that our method has a competitive performance with the direct extension of ADMM which usually works well in practice but may fail to converge in theory.

Keywords: Convex programming, multi-block, Peaceman-Rachford splitting method, convergence rate, resource allocation.
Mathematics Subject Classification: Primary: 90C25; Secondary: 90B50.
Citation: Su-Hong Jiang, Min Li. A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming. Journal of Industrial and Management Optimization, 2018, 14 (1) : 397-412. doi: 10.3934/jimo.2017052
References:
[1]

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

[2]

X. J. Cai, D. R. Han and X. M. Yuan, On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function, Computational Optimization and Applications, 66 (2017), 39-73.  doi: 10.1007/s10589-016-9860-y.

[3]

C. H. Chen, B. S. He, Y. Y. Ye and X. M. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Mathematical Programming Ser. A, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.

[4]

C. H. Chen, Y. Shen and Y. F. You, On the convergence analysis of the alternating direction method of multipliers with three blocks. Abstract and Applied Analysis, (2013), Article ID 183961, 7 pages.

[5]

E. Corman and X. M. Yuan, A generalized proximal point algorithm and its convergence rate, SIAM Journal on Optimization, 24 (2014), 1614-1638.  doi: 10.1137/130940402.

[6]

Y. H. Dai, D. R. Han, X. M. Yuan and W. X. Zhang, A sequential updating scheme of the Lagrange multiplier for separable convex programming, Mathematics of Computation, 86 (2017), 315-343.  doi: 10.1090/mcom/3104.

[7]

W. Deng, M.-J. Lai, Z. M. Peng and W. T. Yin, Parallel multi-block ADMM with o(1/k) convergence, Journal of Scientific Computing, 71 (2017), 712-736.  doi: 10.1007/s10915-016-0318-2.

[8]

J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in $2$ and $3$ space variables, Transactions of the American Mathematical Society, 82 (1956), 421-439.  doi: 10.1090/S0002-9947-1956-0084194-4.

[9]

D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrange Methods: Applications to the Solution of Boundary-valued Problems (eds. M. Fortin and R. Glowinski), North Holland, Amsterdam, The Netherlands, (1983), 299-331.

[10]

R. Glowinski, T. Kärkkäinen and K. Majava, On the convergence of operator-splitting methods, in Numerical Methods for Scientific Computing, Variational Problems and Applications (eds. Y. Kuznetsov, P. Neittanmaki and O. Pironneau), Barcelona, (2003).

[11]

R. Glowinski and A. Marrocco, Approximation par $\acute{e}$l$\acute{e}$ments finis d'ordre un et r$\acute{e}$solution par p$\acute{e}$nalisation-dualit$\acute{e}$ d'une classe de probl$\grave{e}$mes non lin$\acute{e}$aires, R.A.I.R.O., 9 (1975), 41-76. 

[12]

D. R. Han, X. M Yuan, W. X. Zhang and X. J. Cai, An ADM-based splitting method for separable convex programming, Computational Optimization and Applications, 54 (2013), 343-369.  doi: 10.1007/s10589-012-9510-y.

[13]

B. S. He, H. Liu, J. W. Lu and X. M. Yuan, Application of the strictly contractive PeacemanRachford splitting method to multi-block seperable convex programming, manuscript, in Splitting Methods in Communication and Imaging, Science, and Engineering (eds. R. Glowinski, S. Osher and W. Yin), Springer, Switzerland, (2016), 195-235.

[14]

B. S. He, H. Liu, Z. R. Wang and X. M. Yuan, A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), 1101-1140.  doi: 10.1137/13090849X.

[15]

B. S. He, M. Tao and X. M. Yuan, A splitting method for separable convex programming, IMA Journal of Numerical Analysis, 35 (2015), 394-426.  doi: 10.1093/imanum/drt060.

[16]

B. S. He, M. Tao and X. M. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming, SIAM Journal on Optimization, 22 (2012), 313-340.  doi: 10.1137/110822347.

[17]

B. S. He and X. M. Yuan, On the O(1/n) convergence rate of Douglas-Rachford alternating direction method, SIAM Journal on Numerical Analysis, 50 (2012), 700-709.  doi: 10.1137/110836936.

[18]

B. S. He and X. M. Yuan, On nonergodic convergence rate of Douglas-Rachford alternating direction method of multipliers, Numerische Mathematik, 30 (2015), 567-577.  doi: 10.1007/s00211-014-0673-6.

[19]

M. Li, D. F. Sun and K. -C. Toh, A convergent 3-block semi-proximal ADMM for convex minimization problems with one strongly convex block, Asia Pacific Journal of Operational Research, 32 (2015), 1550024, 19 pp. doi: 10.1142/S0217595915500244.

[20]

X. D. Li, D. F. Sun and K.-C. Toh, A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Mathematical Programming Ser. A, 155 (2016), 333-373.  doi: 10.1007/s10107-014-0850-5.

[21]

T. Y. Lin, S. Q. Ma and S. Z. Zhang, On the global linear convergence of the ADMM with multi-block variables, SIAM Journal on Optimization, 25 (2015), 1478-1497.  doi: 10.1137/140971178.

[22]

T. Y. Lin, S. Q. Ma and S. Z. Zhang, On the sublinear convergence rate of multi-block {ADMM}, Journal of the Operations Research Society of China, 3 (2015), 251-274.  doi: 10.1007/s40305-015-0092-0.

[23]

P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), 964-979.  doi: 10.1137/0716071.

[24]

S. Q. Ma, Alternating proximal gradient method for convex minimization, Journal of Scientific Computing, 68 (2016), 546-572.  doi: 10.1007/s10915-015-0150-0.

[25]

Y. E. Nesterov, Gradient methods for minimizing composite objective function, Mathematical Programming Ser. B, 140 (2013), 125-161.  doi: 10.1007/s10107-012-0629-5.

[26]

M. Patriksson, A survey on the continuous nonlinear resource allocation Problem, European Journal of Operations Research, 185 (2008), 1-46.  doi: 10.1016/j.ejor.2006.12.006.

[27]

D. H. Peaceman and H. H. Rachford, The numerical solution of parabolic elliptic differential equations, SIAM Journal on Applied Mathematics, 3 (1955), 28-41.  doi: 10.1137/0103003.

[28]

Y. G. Peng, A. Ganesh, J. Wright, W. L. Xu and Y. Ma, Robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 2233-2246.  doi: 10.1109/CVPR.2010.5540138.

[29]

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

[30]

H. Uzawa, Market mechanisms and mathematical programming, Econometrica, 28 (1960), 872-881.  doi: 10.2307/1907569.

show all references

References:
[1]

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

[2]

X. J. Cai, D. R. Han and X. M. Yuan, On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function, Computational Optimization and Applications, 66 (2017), 39-73.  doi: 10.1007/s10589-016-9860-y.

[3]

C. H. Chen, B. S. He, Y. Y. Ye and X. M. Yuan, The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Mathematical Programming Ser. A, 155 (2016), 57-79.  doi: 10.1007/s10107-014-0826-5.

[4]

C. H. Chen, Y. Shen and Y. F. You, On the convergence analysis of the alternating direction method of multipliers with three blocks. Abstract and Applied Analysis, (2013), Article ID 183961, 7 pages.

[5]

E. Corman and X. M. Yuan, A generalized proximal point algorithm and its convergence rate, SIAM Journal on Optimization, 24 (2014), 1614-1638.  doi: 10.1137/130940402.

[6]

Y. H. Dai, D. R. Han, X. M. Yuan and W. X. Zhang, A sequential updating scheme of the Lagrange multiplier for separable convex programming, Mathematics of Computation, 86 (2017), 315-343.  doi: 10.1090/mcom/3104.

[7]

W. Deng, M.-J. Lai, Z. M. Peng and W. T. Yin, Parallel multi-block ADMM with o(1/k) convergence, Journal of Scientific Computing, 71 (2017), 712-736.  doi: 10.1007/s10915-016-0318-2.

[8]

J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in $2$ and $3$ space variables, Transactions of the American Mathematical Society, 82 (1956), 421-439.  doi: 10.1090/S0002-9947-1956-0084194-4.

[9]

D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrange Methods: Applications to the Solution of Boundary-valued Problems (eds. M. Fortin and R. Glowinski), North Holland, Amsterdam, The Netherlands, (1983), 299-331.

[10]

R. Glowinski, T. Kärkkäinen and K. Majava, On the convergence of operator-splitting methods, in Numerical Methods for Scientific Computing, Variational Problems and Applications (eds. Y. Kuznetsov, P. Neittanmaki and O. Pironneau), Barcelona, (2003).

[11]

R. Glowinski and A. Marrocco, Approximation par $\acute{e}$l$\acute{e}$ments finis d'ordre un et r$\acute{e}$solution par p$\acute{e}$nalisation-dualit$\acute{e}$ d'une classe de probl$\grave{e}$mes non lin$\acute{e}$aires, R.A.I.R.O., 9 (1975), 41-76. 

[12]

D. R. Han, X. M Yuan, W. X. Zhang and X. J. Cai, An ADM-based splitting method for separable convex programming, Computational Optimization and Applications, 54 (2013), 343-369.  doi: 10.1007/s10589-012-9510-y.

[13]

B. S. He, H. Liu, J. W. Lu and X. M. Yuan, Application of the strictly contractive PeacemanRachford splitting method to multi-block seperable convex programming, manuscript, in Splitting Methods in Communication and Imaging, Science, and Engineering (eds. R. Glowinski, S. Osher and W. Yin), Springer, Switzerland, (2016), 195-235.

[14]

B. S. He, H. Liu, Z. R. Wang and X. M. Yuan, A strictly contractive Peaceman-Rachford splitting method for convex programming, SIAM Journal on Optimization, 24 (2014), 1101-1140.  doi: 10.1137/13090849X.

[15]

B. S. He, M. Tao and X. M. Yuan, A splitting method for separable convex programming, IMA Journal of Numerical Analysis, 35 (2015), 394-426.  doi: 10.1093/imanum/drt060.

[16]

B. S. He, M. Tao and X. M. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming, SIAM Journal on Optimization, 22 (2012), 313-340.  doi: 10.1137/110822347.

[17]

B. S. He and X. M. Yuan, On the O(1/n) convergence rate of Douglas-Rachford alternating direction method, SIAM Journal on Numerical Analysis, 50 (2012), 700-709.  doi: 10.1137/110836936.

[18]

B. S. He and X. M. Yuan, On nonergodic convergence rate of Douglas-Rachford alternating direction method of multipliers, Numerische Mathematik, 30 (2015), 567-577.  doi: 10.1007/s00211-014-0673-6.

[19]

M. Li, D. F. Sun and K. -C. Toh, A convergent 3-block semi-proximal ADMM for convex minimization problems with one strongly convex block, Asia Pacific Journal of Operational Research, 32 (2015), 1550024, 19 pp. doi: 10.1142/S0217595915500244.

[20]

X. D. Li, D. F. Sun and K.-C. Toh, A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions, Mathematical Programming Ser. A, 155 (2016), 333-373.  doi: 10.1007/s10107-014-0850-5.

[21]

T. Y. Lin, S. Q. Ma and S. Z. Zhang, On the global linear convergence of the ADMM with multi-block variables, SIAM Journal on Optimization, 25 (2015), 1478-1497.  doi: 10.1137/140971178.

[22]

T. Y. Lin, S. Q. Ma and S. Z. Zhang, On the sublinear convergence rate of multi-block {ADMM}, Journal of the Operations Research Society of China, 3 (2015), 251-274.  doi: 10.1007/s40305-015-0092-0.

[23]

P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16 (1979), 964-979.  doi: 10.1137/0716071.

[24]

S. Q. Ma, Alternating proximal gradient method for convex minimization, Journal of Scientific Computing, 68 (2016), 546-572.  doi: 10.1007/s10915-015-0150-0.

[25]

Y. E. Nesterov, Gradient methods for minimizing composite objective function, Mathematical Programming Ser. B, 140 (2013), 125-161.  doi: 10.1007/s10107-012-0629-5.

[26]

M. Patriksson, A survey on the continuous nonlinear resource allocation Problem, European Journal of Operations Research, 185 (2008), 1-46.  doi: 10.1016/j.ejor.2006.12.006.

[27]

D. H. Peaceman and H. H. Rachford, The numerical solution of parabolic elliptic differential equations, SIAM Journal on Applied Mathematics, 3 (1955), 28-41.  doi: 10.1137/0103003.

[28]

Y. G. Peng, A. Ganesh, J. Wright, W. L. Xu and Y. Ma, Robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 2233-2246.  doi: 10.1109/CVPR.2010.5540138.

[29]

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

[30]

H. Uzawa, Market mechanisms and mathematical programming, Econometrica, 28 (1960), 872-881.  doi: 10.2307/1907569.

Figure 1.  Evolutions of objective function values w.r.t. CPU 20:20:30
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Table 1.  The function $\phi(s)$ for generating the cost function
Name $\phi(s_i):\Re_+\rightarrow [0, \, \infty)$ Parameters
Linear cost $w_i s_i$ $w_i \in U(1, \, 5)\\ k_i \in U(1, \, 5)$
Power cost $k_i s_i^{q_i}$
Piecewise quadratic cost $\displaystyle \left\{ \begin{array}{ll} k_is_i^2, ~~~~~~~~~~~~~~~~~~~{\rm {if}}\, s_i \leq w_i/\sqrt{2k_i}\\ w_i\sqrt{2k}s- {w_i^2/2}, ~~~~{\rm{otherwise}}. \end{array} \right. $
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Name $\phi(s_i):\Re_+\rightarrow [0, \, \infty)$ Parameters
Linear cost $w_i s_i$ $w_i \in U(1, \, 5)\\ k_i \in U(1, \, 5)$
Power cost $k_i s_i^{q_i}$
Piecewise quadratic cost $\displaystyle \left\{ \begin{array}{ll} k_is_i^2, ~~~~~~~~~~~~~~~~~~~{\rm {if}}\, s_i \leq w_i/\sqrt{2k_i}\\ w_i\sqrt{2k}s- {w_i^2/2}, ~~~~{\rm{otherwise}}. \end{array} \right. $
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