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

• * Corresponding author: Zheng Peng
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.

Mathematics Subject Classification: Primary: 90B15, 90C52; Secondary: 90C90.

 Citation: • • Figure 1.  The probability distribution $x$ for the case $j = 2$ and $l = 2$

Figure 2.  The probability distribution $x$ for the case $j = 2$ and $l = 3$

Figure 3.  The probability distribution $x$ for the case $j = 3$ and 1024 BNs

Figure 4.  The probability distribution $x$ for the case $j = 3$ and 2048 BNs

Figure 5.  The probability distribution $x$ for the case $j = 4$ and 4096 BNs

Table 1.  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
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