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October  2019, 15(4): 1881-1896. doi: 10.3934/jimo.2018127

## Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method

 College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China

* Corresponding author: Zheng Peng

Received  January 2018 Revised  April 2018 Published  August 2018

Fund Project: The work is supported by NSFC grants 11571074, 11726505, and the Natural Science Foundation of FuJian Province grant 2015J01010

The sparse probabilistic Boolean network (SPBN) model has been applied in various fields of industrial engineering and management. The goal of this model is to find a sparse probability distribution based on a given transition-probability matrix and a set of Boolean networks (BNs). In this paper, a partial proximal-type operator splitting method is proposed to solve a separable minimization problem arising from the study of the SPBN model. All the subproblem-solvers of the proposed method do not involve matrix multiplication, and consequently the proposed method can be used to deal with large-scale problems. The global convergence to a critical point of the proposed method is proved under some mild conditions. Numerical experiments on some real probabilistic Boolean network problems show that the proposed method is effective and efficient compared with some existing methods.

Citation: Kangkang Deng, Zheng Peng, Jianli Chen. Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1881-1896. doi: 10.3934/jimo.2018127
##### References:
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Swaleh, On fixed points of asymptotically regular mappings, Journal of the Australian Mathematical Society, 43 (1987), 328-346. doi: 10.1017/S1446788700029621. Google Scholar [23] I. Shmulevich, E.-R. Dougherty, S. Kim and W. Zhang, Probabilistic Boolean networks: A rule- based uncertainty model for gene regulatory networks, Bioinformatics, 18 (2002), 261-274. doi: 10.1093/bioinformatics/18.2.261. Google Scholar [24] I. Shmulevich, E-R. Dougherty and W. Zhang, From Boolean to probabilistic Boolean networks as models of genetic regulatory networks, Proceedings of the IEEE, 90 (2002), 1778-1792. doi: 10.1109/JPROC.2002.804686. Google Scholar [25] I. Shmulevich, E.-R. Dougherty and W. Zhang, Gene perturbation and intervention in probabilistic Boolean networks, Bioinformatics, 18 (2002), 1319-1331. doi: 10.1093/bioinformatics/18.10.1319. Google Scholar [26] B. Tian, X.-Q. Yang and K.-W. Meng, An interior-point $\ell_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization, Journal of Industrial & Management Optimization, 12 (2016), 949-973. doi: 10.3934/jimo.2016.12.949. Google Scholar [27] X.-F. Wang and G. Chen, Complex networks: Small-world, scale-free and beyond, IEEE Circuits and Systems Magazine, 3 (2003), 6-20. Google Scholar [28] Z.-M. Wu, X.-J. Cai and D.-R. Han, Linearized block-wise alternating direction method of multipliers for multiple-block convex programming, Journal of Industrial & Management Optimization, 14 (2018), 833-855. doi: 10.3934/jimo.2017078. Google Scholar [29] M.-H. Xu, Proximal alternating directions method for structured variational inequalities, Journal of Optimization Theory and Applications, 134 (2007), 107-117. doi: 10.1007/s10957-007-9192-2. Google Scholar [30] Z.-B. Xu, H. Guo, Y. Wang and H. Zhang, The representation of $\ell_{\frac{1}{2}}$ regularizer among $\ell_q (0 < q < 1)$ regularizer: an experimental study based on phase diagram, Acta Automatica Sinica, 38 (2012), 1225-1228. doi: 10.3724/SP.J.1004.2012.01225. Google Scholar [31] Z.-B. Xu, X.-Y. Chang, F.-M. Xu and H. Zhang, $\ell_{\frac{1}{2}}$ regularization: a thresholding representation theory and a fast slover, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1013-1027. Google Scholar [32] F.-M. Xu, Y.-H. Dai, Z.-H. Zhao and Z.-B. Xu, Efficient projected gradient methods for a class of $\ell_0$ constrained optimization problems, Mathematical Programming, to appear.Google Scholar [33] J. Yang, Y.-Q. Dai, Z. Peng, J.-P. Zhuang and W.-X. Zhu, A homotopy alternating direction method of multipliers for linearly constrained separable convex optimization, Journal of the Operations Research Society of China, 5 (2017), 271-290. doi: 10.1007/s40305-017-0170-6. Google Scholar [34] J. Zeng, S. Lin, Y. Wang and Z.-B. Xu, $\ell_\frac{1}{2}$ Regularization: convergence of iterative half thresholding algorithm, IEEE Transactions on Signal Processing, 62 (2014), 2317-2329. doi: 10.1109/TSP.2014.2309076. Google Scholar [35] K.-Z. Zhang and L.-Z. Zhang, Controllability of probabilistic Boolean control networks with time-variant delays in states, Science China: Information Sciences, 59(9)(2016), 092204, 10pp. doi: 10.1007/s11432-015-5423-6. Google Scholar

show all references

##### References:
 [1] Y.-Q. Bai and K.-J. Shen, Alternating direction method of multipliers for $\ell1-\ell 2$ regularized Logistic regression model, Journal of the Operations Research Society of China, 4 (2016), 243-253. doi: 10.1007/s40305-015-0090-2. Google Scholar [2] 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. Google Scholar [3] M. Caetano and T. Yoneyama, An autocatalytic network model for stock markets, Physica A, 419 (2015), 122-127. doi: 10.1016/j.physa.2014.10.052. Google Scholar [4] X. Chen, W.-K. Ching and X.-S. Chen, Construction of probabilistic boolean networks from a prescribed transition probability matrix: A maximum entropy rate approach, East Asian Journal on Applied Mathematics, 1 (2011), 132-154. doi: 10.4208/eajam.080310.200910a. Google Scholar [5] X. Chen, H. Jiang and W.-K. Ching, Construction of sparse probabilistic boolean networks, East Asian Journal of Applied Mathematics, 2 (2012), 1-18. doi: 10.4208/eajam.030511.060911a. Google Scholar [6] Y.-H. Dai, D.-R. Han, X.-M. Yuan and W.-X. Zhang, A sequential updating scheme of Lagrange multiplier for separable convex programming, Mathematics of Computation, 86 (2017), 315-343. doi: 10.1090/mcom/3104. Google Scholar [7] J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in Large Scale Optimization: State of the Art Springer US, (1994), 115-134. Google Scholar [8] M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Computational Optimization and Applications, 1 (1992), 93-111. doi: 10.1007/BF00247655. Google Scholar [9] J.-W. Gu, W.-K. Ching, T.-K. Siu and H. Zheng, On modeling credit defaults: A probabilistic Boolean network approach, Risk and Decision Analysis, 4 (2013), 119-129. Google Scholar [10] D.-R. Han, X.-M. Yuan and W.-X. Zhang, An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing, Mathematics of Computation, 83 (2014), 2263-2291. doi: 10.1090/S0025-5718-2014-02829-9. Google Scholar [11] B.-S. He and X.-M. Yuan, Alternating direction method of multipliers for linear programming, Journal of the Operations Research Society of China, 4 (2016), 425-436. doi: 10.1007/s40305-016-0136-0. Google Scholar [12] 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. Google Scholar [13] I. Ivanov, R. Pal and E.-R. Dougherty, Dynamics preserving size reduction mappings for probabilistic Boolean networks, IEEE Transactions on Signal Processing, 55 (2007), 2310-2322. doi: 10.1109/TSP.2006.890929. Google Scholar [14] K. Kobayashi and K. Hiraishi, An integer programming approach to optimal control problems in context-sensitive probabilistic Boolean networks, Automatica, 47 (2011), 1260-1264. doi: 10.1016/j.automatica.2011.01.035. Google Scholar [15] K. Kobayashi and K. Hiraishi, A probabilistic approach to control of complex systems and its application to real-time pricing, Mathematical Problems in Engineering, Volume (2014), Art. ID 906717, 8 pp. doi: 10.1155/2014/906717. Google Scholar [16] K. Kobayashi and K. Hiraishi, Verification of real-time pricing systems based on probabilistic Boolean networks, Applied Mathematics, 7 (2016), Article ID: 70627, 14 pages. doi: 10.4236/am.2016.715146. Google Scholar [17] J. Li, A. Ritter and D. Jurafsky, Inferring user preferences by probabilistic logical reasoning over social networks, preprint, arXiv: 1411.2679.Google Scholar [18] R. Liang, Y. Qiu and W.-K. Ching, Construction of probabilistic Boolean network for credit default data, Computational Sciences and Optimization (CSO), 2014 Seventh International Joint Conference on. IEEE, (2014), 11-15. Google Scholar [19] 2003. Available from: http://code.google.com/p/pbn-matlab-toolbox.Google Scholar [20] B.-K. Natraajan, Sparse approximation to linear systems, SIAM Journal on Computing, 24 (1995), 227-234. doi: 10.1137/S0097539792240406. Google Scholar [21] Z. Peng and D.-H. Wu, A partial parallel splitting augmented Lagrangian method for solving constrained matrix optimization problems, Computers and Mathematics with Applications, 60 (2010), 1515-1524. doi: 10.1016/j.camwa.2010.06.035. Google Scholar [22] B.-E. Rhoades, S. Sessa, M.-S. Khan and M. Swaleh, On fixed points of asymptotically regular mappings, Journal of the Australian Mathematical Society, 43 (1987), 328-346. doi: 10.1017/S1446788700029621. Google Scholar [23] I. Shmulevich, E.-R. Dougherty, S. Kim and W. Zhang, Probabilistic Boolean networks: A rule- based uncertainty model for gene regulatory networks, Bioinformatics, 18 (2002), 261-274. doi: 10.1093/bioinformatics/18.2.261. Google Scholar [24] I. Shmulevich, E-R. Dougherty and W. Zhang, From Boolean to probabilistic Boolean networks as models of genetic regulatory networks, Proceedings of the IEEE, 90 (2002), 1778-1792. doi: 10.1109/JPROC.2002.804686. Google Scholar [25] I. Shmulevich, E.-R. Dougherty and W. Zhang, Gene perturbation and intervention in probabilistic Boolean networks, Bioinformatics, 18 (2002), 1319-1331. doi: 10.1093/bioinformatics/18.10.1319. Google Scholar [26] B. Tian, X.-Q. Yang and K.-W. Meng, An interior-point $\ell_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization, Journal of Industrial & Management Optimization, 12 (2016), 949-973. doi: 10.3934/jimo.2016.12.949. Google Scholar [27] X.-F. Wang and G. Chen, Complex networks: Small-world, scale-free and beyond, IEEE Circuits and Systems Magazine, 3 (2003), 6-20. Google Scholar [28] Z.-M. Wu, X.-J. Cai and D.-R. Han, Linearized block-wise alternating direction method of multipliers for multiple-block convex programming, Journal of Industrial & Management Optimization, 14 (2018), 833-855. doi: 10.3934/jimo.2017078. Google Scholar [29] M.-H. Xu, Proximal alternating directions method for structured variational inequalities, Journal of Optimization Theory and Applications, 134 (2007), 107-117. doi: 10.1007/s10957-007-9192-2. Google Scholar [30] Z.-B. Xu, H. Guo, Y. Wang and H. Zhang, The representation of $\ell_{\frac{1}{2}}$ regularizer among $\ell_q (0 < q < 1)$ regularizer: an experimental study based on phase diagram, Acta Automatica Sinica, 38 (2012), 1225-1228. doi: 10.3724/SP.J.1004.2012.01225. Google Scholar [31] Z.-B. Xu, X.-Y. Chang, F.-M. Xu and H. Zhang, $\ell_{\frac{1}{2}}$ regularization: a thresholding representation theory and a fast slover, IEEE Transactions on Neural Networks and Learning Systems, 23 (2012), 1013-1027. Google Scholar [32] F.-M. Xu, Y.-H. Dai, Z.-H. Zhao and Z.-B. Xu, Efficient projected gradient methods for a class of $\ell_0$ constrained optimization problems, Mathematical Programming, to appear.Google Scholar [33] J. Yang, Y.-Q. Dai, Z. Peng, J.-P. Zhuang and W.-X. Zhu, A homotopy alternating direction method of multipliers for linearly constrained separable convex optimization, Journal of the Operations Research Society of China, 5 (2017), 271-290. doi: 10.1007/s40305-017-0170-6. Google Scholar [34] J. Zeng, S. Lin, Y. Wang and Z.-B. Xu, $\ell_\frac{1}{2}$ Regularization: convergence of iterative half thresholding algorithm, IEEE Transactions on Signal Processing, 62 (2014), 2317-2329. doi: 10.1109/TSP.2014.2309076. Google Scholar [35] K.-Z. Zhang and L.-Z. Zhang, Controllability of probabilistic Boolean control networks with time-variant delays in states, Science China: Information Sciences, 59(9)(2016), 092204, 10pp. doi: 10.1007/s11432-015-5423-6. Google Scholar
The probability distribution $x$ for the case $j = 2$ and $l = 2$
The probability distribution $x$ for the case $j = 2$ and $l = 3$
The probability distribution $x$ for the case $j = 3$ and 1024 BNs
The probability distribution $x$ for the case $j = 3$ and 2048 BNs
The probability distribution $x$ for the case $j = 4$ and 4096 BNs
The computational results of Algorithm 1 with different stopping error
 Stopping error $\varepsilon$ $10^{-2}$ $10^{-3}$ $5\times 10^{-4}$ $10^{-4}$ $5\times 10^{-5}$ $10^{-5}$ Total iteration number $k$ 97 260 267 520 562 722 Identified major BNs 104 118 189 118 118 118 118 118 358 360 360 360 360 360 360 395 395 395 395 395 376 594 594 594 594 594 395 836 836 836 836 836 594 911 911 911 911 911 836 939 939 939 939 939 911 939
 Stopping error $\varepsilon$ $10^{-2}$ $10^{-3}$ $5\times 10^{-4}$ $10^{-4}$ $5\times 10^{-5}$ $10^{-5}$ Total iteration number $k$ 97 260 267 520 562 722 Identified major BNs 104 118 189 118 118 118 118 118 358 360 360 360 360 360 360 395 395 395 395 395 376 594 594 594 594 594 395 836 836 836 836 836 594 911 911 911 911 911 836 939 939 939 939 939 911 939
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