American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs
  • JIMO Home
  • This Issue
  • Next Article
    An economic order quantity for deteriorating items with allowable rework of deteriorated products
October  2019, 15(4): 1881-1896. doi: 10.3934/jimo.2018127

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

Kangkang Deng , Zheng Peng , and  Jianli Chen

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.

Keywords: Sparse probabilistic Boolean network, $\ell_{\frac{1}{2}}$-regularization, separable optimization problem, method of multipliers, proximal point algorithm.
Mathematics Subject Classification: Primary: 90B15, 90C52; Secondary: 90C90.
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:
[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

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

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

Jun Wang, Xing Tao Wang. Sparse signal reconstruction via the approximations of $ \ell_{0} $ quasinorm. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019035
[2]

Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems & Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024
[3]

Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057
[4]

Victor Churchill, Rick Archibald, Anne Gelb. Edge-adaptive $ \ell_2 $ regularization image reconstruction from non-uniform Fourier data. Inverse Problems & Imaging, 2019, 13 (5) : 931-958. doi: 10.3934/ipi.2019042
[5]

Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $ \ell_p $ penalty. Journal of Industrial & Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006
[6]

Sugata Gangopadhyay, Constanza Riera, Pantelimon Stăniă. Gowers $ U_2 $ norm as a measure of nonlinearity for Boolean functions and their generalizations. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020056
[7]

James Montaldi, Amna Shaddad. Generalized point vortex dynamics on $ \mathbb{CP} ^2 $. Journal of Geometric Mechanics, 2019, 11 (4) : 601-619. doi: 10.3934/jgm.2019030
[8]

Junlin Xiong, Wenjie Liu. $ H_{\infty} $ observer-based control for large-scale systems with sparse observer communication network. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2020005
[9]

Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109
[10]

Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial & Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949
[11]

Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $ L_1 $ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037
[12]

Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $ l_1 $ norm measure. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019061
[13]

Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094
[14]

Hao Li, Hai Bi, Yidu Yang. The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1775-1789. doi: 10.3934/dcdsb.2020002
[15]

Genni Fragnelli, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Operators of order 2$ n $ with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020128
[16]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-19. doi: 10.3934/dcds.2019228
[17]

Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020015
[18]

Valeria Banica, Luis Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1317-1329. doi: 10.3934/cpaa.2018064
[19]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[20]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (77)
  • HTML views (1092)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences