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

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

* Corresponding author: MIN LI

Received  June 2016 Revised  December 2016 Published  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.

Citation: Su-Hong Jiang, Min Li. A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming. Journal of Industrial & Management Optimization, 2018, 14 (1) : 397-412. doi: 10.3934/jimo.2017052
References:
[1]

S. BoydN. ParikhE. ChuB. 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.   Google Scholar

[2]

X. J. CaiD. 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.  Google Scholar

[3]

C. H. ChenB. S. HeY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Y. H. DaiD. R. HanX. 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.  Google Scholar

[7]

W. DengM.-J. LaiZ. 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.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

[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). Google Scholar

[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.   Google Scholar

[12]

D. R. HanX. M YuanW. 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.  Google Scholar

[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. Google Scholar

[14]

B. S. HeH. LiuZ. 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.  Google Scholar

[15]

B. S. HeM. 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.  Google Scholar

[16]

B. S. HeM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

X. D. LiD. 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.  Google Scholar

[21]

T. Y. LinS. 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.  Google Scholar

[22]

T. Y. LinS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Y. G. PengA. GaneshJ. WrightW. 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.  Google Scholar

[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.  Google Scholar

[30]

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

show all references

References:
[1]

S. BoydN. ParikhE. ChuB. 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.   Google Scholar

[2]

X. J. CaiD. 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.  Google Scholar

[3]

C. H. ChenB. S. HeY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

Y. H. DaiD. R. HanX. 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.  Google Scholar

[7]

W. DengM.-J. LaiZ. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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). Google Scholar

[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.   Google Scholar

[12]

D. R. HanX. M YuanW. 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.  Google Scholar

[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. Google Scholar

[14]

B. S. HeH. LiuZ. 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.  Google Scholar

[15]

B. S. HeM. 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.  Google Scholar

[16]

B. S. HeM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

X. D. LiD. 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.  Google Scholar

[21]

T. Y. LinS. 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.  Google Scholar

[22]

T. Y. LinS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Y. G. PengA. GaneshJ. WrightW. 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.  Google Scholar

[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.  Google Scholar

[30]

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

Figure 1.  Evolutions of objective function values w.r.t. CPU 20:20:30
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. $
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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