American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers
  • JIMO Home
  • This Issue
  • Next Article
    A novel risk ranking method based on the single valued neutrosophic set
doi: 10.3934/jimo.2020104

A dynamical system method for solving the split convex feasibility problem

Zeng-Zhen Tan 1, , Rong Hu 2, , Ming Zhu 1, and  Ya-Ping Fang 3,,

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan, China
2. 

Department of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
3. 

Department of Mathematics, Sichuan University, Chengdu 610065, China

* Corresponding author: Ya-Ping Fang

Received  August 2019 Revised  March 2020 Published  June 2020

Fund Project: This work was partially supported by the National Science Foundation of China (No. 11471230) and the Scientific Research Foundation of the Education Department of Sichuan Province (No. 16ZA0213)

In this paper a dynamical system model is proposed for solving the split convex feasibility problem. Under mild conditions, it is shown that the proposed dynamical system globally converges to a solution of the split convex feasibility problem. An exponential convergence is obtained provided that the bounded linear regularity property is satisfied. The validity and transient behavior of the dynamical system is demonstrated by several numerical examples. The method proposed in this paper can be regarded as not only a continuous version but also an interior version of the known $ CQ $-method for solving the split convex feasibility problem.

Keywords: Split convex feasibility problem, dynamical system, CQ-algorithm, stability and convergence rate.
Mathematics Subject Classification: Primary: 65K05, 65K10, 90C25; Secondary: 47H09, 65L09.
Citation: Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020104
References:
[1]

B. Abbas and H. Attouch, Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator, Optimization, 64 (2015), 2223-2252.  doi: 10.1080/02331934.2014.971412.  Google Scholar

[2]

B. Abdellah and A. N. Muhammad, On descent-projection method for solving the split convex feasibility problems, J. Global Optim., 54 (2012), 627-639.   Google Scholar

[3]

H. Attouch, J. Bolte, P. Redont and M. Teboulle, Singular Riemannian barrier methods and gradient-projection dynamical systems for constrained optimization, Optimization, 53 (2004), 435-454.  doi: 10.1080/02331930412331327184.  Google Scholar

[4]

H. Attouch, Z. Chbani, J. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math. Program., 168 (2018), 123-175.  doi: 10.1007/s10107-016-0992-8.  Google Scholar

[5]

H. Attouch and B. F. Svaiter, A continuous dynamical Newton-like approach to solving monotone inclusions, SIAM J. Control Optim., 49 (2011), 574-598.  doi: 10.1137/100784114.  Google Scholar

[6]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 2nd Edn. 1988.  Google Scholar

[7]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.  Google Scholar

[8]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[9]

J. Bolte and M. Teboulle, Barrier operators and associated gradient-like dynamical systems for constrained minimization problems, SIAM J. Control Optim., 42 (2003), 1266-1292.  doi: 10.1137/S0363012902410861.  Google Scholar

[10]

B. I. Bot, E. R. Csetnek and S. C. Laszlo, A primal-dual dynamical approach to structured convex minimization problems, arXiv: 1905.08290, 2019. Google Scholar

[11]

B. I. Bot and E. R. Csetnek, A dynamical system associated with the fixed points set of a nonexpansive operator, J. Dynam. Differential Equations, 29 (2017), 155-168.  doi: 10.1007/s10884-015-9438-x.  Google Scholar

[12]

J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimization, Math. Program., 71 (1995), 179-194.  doi: 10.1007/BF01585997.  Google Scholar

[13]

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.  Google Scholar

[14]

C. Byrne, Iterative oblique projection onto convex sets and the split convex feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[15]

Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.  Google Scholar

[16]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[17]

Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.   Google Scholar

[18]

Y. Z. Dang, Z. H. Xue, Y. Gao and J. X. Li, Fast self-adaptive regularization iterative algorithm for solving split feasibility problem, J. Ind. Manag. Optim., 2019. doi: 10.3934/jimo.2019017.  Google Scholar

[19]

Y. Z. Dang, J. Sun and S. Zhang, Double projection algorithms for solving the split feasibility problems, J. Ind. Manag. Optim., 15 (2019), 2023-2034.  doi: 10.3934/jimo.2018135.  Google Scholar

[20]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer, New York, 2009. doi: 10.1007/978-0-387-87821-8.  Google Scholar

[21]

S. Effati, A. Ghomashi and A. R. Nazemi, Application of projection neural network in solving convex programming problems, Appl. Math. Comput., 188 (2007), 1103-1114.  doi: 10.1016/j.amc.2006.10.088.  Google Scholar

[22]

H. Federer, Geometric Measure Theory, Springer-Verlag Berlin Heidelberg, 1969.  Google Scholar

[23]

G. Franca, D. P. Robinson and R. Vidal, Admm and accelerated admm as continuous dynamical systems, Proceedings of the 35th International Conference on Machine Learning, PMLR, Stockholm Sweden, 80 (2018), 1559-1567.   Google Scholar

[24]

T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin and S. Ganjlizadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems, Oper. Res., 42 (1994), 1120-1136.  doi: 10.1287/opre.42.6.1120.  Google Scholar

[25]

A. Gibali, D. T Mai and N. T. Vinh, A new relaxed $CQ$ algorithm for solving split feasibility problems in Hilbert spaces and its applications, J. Ind. Manag. Optim., 15 (2019), 963-984.  doi: 10.3934/jimo.2018080.  Google Scholar

[26]

N. T. T. Ha, J. J. Strodiot and P. T. Vuong, On the global exponential stability of a projected dynamical system for strongly pseudomonotone variational inequalities, Optim. Lett., 12 (2018), 1625-1638.  doi: 10.1007/s11590-018-1230-5.  Google Scholar

[27]

A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems: Classical Methods and Recent Advances, Springer, Cham, 2015. doi: 10.1007/978-3-319-23407-6.  Google Scholar

[28]

H. J. He, C. Ling and H. K. Xu, An implementable splitting algorithm for the $l_1$-norm regularized split feasibility problem, J. Sci. Comput., 67 (2016), 281-298.  doi: 10.1007/s10915-015-0078-4.  Google Scholar

[29]

A. J. Hoffman, On approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Standards, 49 (1952), 263-265.  doi: 10.6028/jres.049.027.  Google Scholar

[30]

Y. H. Hu, C. Li and X. Q. Yang, On convergence rates of linearized proximal algorithms for convex composite optimization with applications, SIAM J. Optim., 26 (2016), 1207-1235.  doi: 10.1137/140993090.  Google Scholar

[31]

L. Landweber, An iterative formula for Fredholm integral equations of the first kind, Amer. J. Math., 73 (1951), 615-624.  doi: 10.2307/2372313.  Google Scholar

[32]

Q. S. Liu and J. Wang, $L_1$-minimization algorithms for sparse signal reconstruction based on a projection neural network, IEEE Trans. Neural Netw. Learn. Syst., 27 (2016), 698-707.   Google Scholar

[33]

Q. S. Liu and J. Wang, A projection neural network for constrained quadratic minimax optimization, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 2891-2900.  doi: 10.1109/TNNLS.2015.2425301.  Google Scholar

[34]

Q. S. Liu, J. D. Cao and Y. S. Xia, A delayed neural network for solving linear projection equations and its analysis, IEEE Trans. Neural Networks, 16 (2005), 834-843.  doi: 10.1109/TNN.2005.849834.  Google Scholar

[35]

D. A. Lorenz, F. Schöpfer and S. Wenger, The linearized Bregman method via split feasibility problems: Analysis and generalizations, SIAM J. Imaging Sci., 7 (2014), 1237-1262.  doi: 10.1137/130936269.  Google Scholar

[36]

I. B. Pyne, Linear programming on an electronic analogue computer, Trans. Amer. Inst. Elec. Engrs., 75 (1956), 139-143.  doi: 10.1109/TCE.1956.6372503.  Google Scholar

[37]

B. Qu, C. Y. Wang and N. H. Xiu, Analysis on Newton projection method for the split feasibility problem, Comput. Optim. Appl., 67 (2017), 175-199.  doi: 10.1007/s10589-016-9884-3.  Google Scholar

[38]

B. Qu and N. H. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665.  doi: 10.1088/0266-5611/21/5/009.  Google Scholar

[39]

S. M. Robinson, An application of error bounds for convex programming in a linear space, J. SIAM Control Ser. A, 13 (1975), 271-273.  doi: 10.1137/0313015.  Google Scholar

[40]

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Inc., New Jersey, 1991. Google Scholar

[41]

S. Suantai, N. Pholasa and P. Cholamjiak, The modified inertial relaxed $CQ$ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., 14 (2018), 1595-1615.  doi: 10.3934/jimo.2018023.  Google Scholar

[42]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 2012. doi: 10.1090/gsm/140.  Google Scholar

[43]

R. Tibshirani, Regression shrinkage and selection Via the lasso, J. Roy. Statist Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.  Google Scholar

[44]

J. H. Wang, Y. H. Hu, C. Li and J. C. Yao, Linear convergence of $CQ$ algorithms and applications in gene regulatory network inference, Inverse Problems, 33 (2017), 055017(25 pp). doi: 10.1088/1361-6420/aa6699.  Google Scholar

[45]

X. L. Wang, J. Zhao and D. F. Hou, Modified relaxed $CQ$ iterative algorithms for the split feasibility problem, Mathematics, 7 (2019), 119. Google Scholar

[46]

Y. S. Xia, H. Leung and J. Wang, A projection neural network and its application to constrained optimization problems, IEEE Trans. Circuits Syst. I. Regul. Pap., 49 (2002), 447-458.   Google Scholar

[47]

Y. S. Xia and J. Wang, A recurrent neural network for solving linear projected equations, Neural Network, 13 (2000), 337-350.   Google Scholar

[48]

Y. S. Xia and J. Wang, On the stability of globally projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150.  doi: 10.1023/A:1004611224835.  Google Scholar

[49]

Y. S. Xia and J. Wang, A bi-projection neural network for solving constrained quadratic optimization problems, IEEE Trans. Neural Netw. Learn. Syst., 27 (2016), 214-224.  doi: 10.1109/TNNLS.2015.2500618.  Google Scholar

[50]

J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser Boston, 1992.  Google Scholar

[51]

X. J. Zou, D. W. Gong, L. P. Wang and Z. Y. Chen, A novel method to solve inverse variational inequality problems based on neuralnetworks, Neurocomputing, 173 (2016), 1163-1168.   Google Scholar

show all references

References:
[1]

B. Abbas and H. Attouch, Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator, Optimization, 64 (2015), 2223-2252.  doi: 10.1080/02331934.2014.971412.  Google Scholar

[2]

B. Abdellah and A. N. Muhammad, On descent-projection method for solving the split convex feasibility problems, J. Global Optim., 54 (2012), 627-639.   Google Scholar

[3]

H. Attouch, J. Bolte, P. Redont and M. Teboulle, Singular Riemannian barrier methods and gradient-projection dynamical systems for constrained optimization, Optimization, 53 (2004), 435-454.  doi: 10.1080/02331930412331327184.  Google Scholar

[4]

H. Attouch, Z. Chbani, J. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math. Program., 168 (2018), 123-175.  doi: 10.1007/s10107-016-0992-8.  Google Scholar

[5]

H. Attouch and B. F. Svaiter, A continuous dynamical Newton-like approach to solving monotone inclusions, SIAM J. Control Optim., 49 (2011), 574-598.  doi: 10.1137/100784114.  Google Scholar

[6]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 2nd Edn. 1988.  Google Scholar

[7]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.  Google Scholar

[8]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[9]

J. Bolte and M. Teboulle, Barrier operators and associated gradient-like dynamical systems for constrained minimization problems, SIAM J. Control Optim., 42 (2003), 1266-1292.  doi: 10.1137/S0363012902410861.  Google Scholar

[10]

B. I. Bot, E. R. Csetnek and S. C. Laszlo, A primal-dual dynamical approach to structured convex minimization problems, arXiv: 1905.08290, 2019. Google Scholar

[11]

B. I. Bot and E. R. Csetnek, A dynamical system associated with the fixed points set of a nonexpansive operator, J. Dynam. Differential Equations, 29 (2017), 155-168.  doi: 10.1007/s10884-015-9438-x.  Google Scholar

[12]

J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimization, Math. Program., 71 (1995), 179-194.  doi: 10.1007/BF01585997.  Google Scholar

[13]

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.  Google Scholar

[14]

C. Byrne, Iterative oblique projection onto convex sets and the split convex feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.  Google Scholar

[15]

Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.  doi: 10.1088/0031-9155/51/10/001.  Google Scholar

[16]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[17]

Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.   Google Scholar

[18]

Y. Z. Dang, Z. H. Xue, Y. Gao and J. X. Li, Fast self-adaptive regularization iterative algorithm for solving split feasibility problem, J. Ind. Manag. Optim., 2019. doi: 10.3934/jimo.2019017.  Google Scholar

[19]

Y. Z. Dang, J. Sun and S. Zhang, Double projection algorithms for solving the split feasibility problems, J. Ind. Manag. Optim., 15 (2019), 2023-2034.  doi: 10.3934/jimo.2018135.  Google Scholar

[20]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer, New York, 2009. doi: 10.1007/978-0-387-87821-8.  Google Scholar

[21]

S. Effati, A. Ghomashi and A. R. Nazemi, Application of projection neural network in solving convex programming problems, Appl. Math. Comput., 188 (2007), 1103-1114.  doi: 10.1016/j.amc.2006.10.088.  Google Scholar

[22]

H. Federer, Geometric Measure Theory, Springer-Verlag Berlin Heidelberg, 1969.  Google Scholar

[23]

G. Franca, D. P. Robinson and R. Vidal, Admm and accelerated admm as continuous dynamical systems, Proceedings of the 35th International Conference on Machine Learning, PMLR, Stockholm Sweden, 80 (2018), 1559-1567.   Google Scholar

[24]

T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin and S. Ganjlizadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems, Oper. Res., 42 (1994), 1120-1136.  doi: 10.1287/opre.42.6.1120.  Google Scholar

[25]

A. Gibali, D. T Mai and N. T. Vinh, A new relaxed $CQ$ algorithm for solving split feasibility problems in Hilbert spaces and its applications, J. Ind. Manag. Optim., 15 (2019), 963-984.  doi: 10.3934/jimo.2018080.  Google Scholar

[26]

N. T. T. Ha, J. J. Strodiot and P. T. Vuong, On the global exponential stability of a projected dynamical system for strongly pseudomonotone variational inequalities, Optim. Lett., 12 (2018), 1625-1638.  doi: 10.1007/s11590-018-1230-5.  Google Scholar

[27]

A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems: Classical Methods and Recent Advances, Springer, Cham, 2015. doi: 10.1007/978-3-319-23407-6.  Google Scholar

[28]

H. J. He, C. Ling and H. K. Xu, An implementable splitting algorithm for the $l_1$-norm regularized split feasibility problem, J. Sci. Comput., 67 (2016), 281-298.  doi: 10.1007/s10915-015-0078-4.  Google Scholar

[29]

A. J. Hoffman, On approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Standards, 49 (1952), 263-265.  doi: 10.6028/jres.049.027.  Google Scholar

[30]

Y. H. Hu, C. Li and X. Q. Yang, On convergence rates of linearized proximal algorithms for convex composite optimization with applications, SIAM J. Optim., 26 (2016), 1207-1235.  doi: 10.1137/140993090.  Google Scholar

[31]

L. Landweber, An iterative formula for Fredholm integral equations of the first kind, Amer. J. Math., 73 (1951), 615-624.  doi: 10.2307/2372313.  Google Scholar

[32]

Q. S. Liu and J. Wang, $L_1$-minimization algorithms for sparse signal reconstruction based on a projection neural network, IEEE Trans. Neural Netw. Learn. Syst., 27 (2016), 698-707.   Google Scholar

[33]

Q. S. Liu and J. Wang, A projection neural network for constrained quadratic minimax optimization, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 2891-2900.  doi: 10.1109/TNNLS.2015.2425301.  Google Scholar

[34]

Q. S. Liu, J. D. Cao and Y. S. Xia, A delayed neural network for solving linear projection equations and its analysis, IEEE Trans. Neural Networks, 16 (2005), 834-843.  doi: 10.1109/TNN.2005.849834.  Google Scholar

[35]

D. A. Lorenz, F. Schöpfer and S. Wenger, The linearized Bregman method via split feasibility problems: Analysis and generalizations, SIAM J. Imaging Sci., 7 (2014), 1237-1262.  doi: 10.1137/130936269.  Google Scholar

[36]

I. B. Pyne, Linear programming on an electronic analogue computer, Trans. Amer. Inst. Elec. Engrs., 75 (1956), 139-143.  doi: 10.1109/TCE.1956.6372503.  Google Scholar

[37]

B. Qu, C. Y. Wang and N. H. Xiu, Analysis on Newton projection method for the split feasibility problem, Comput. Optim. Appl., 67 (2017), 175-199.  doi: 10.1007/s10589-016-9884-3.  Google Scholar

[38]

B. Qu and N. H. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665.  doi: 10.1088/0266-5611/21/5/009.  Google Scholar

[39]

S. M. Robinson, An application of error bounds for convex programming in a linear space, J. SIAM Control Ser. A, 13 (1975), 271-273.  doi: 10.1137/0313015.  Google Scholar

[40]

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Inc., New Jersey, 1991. Google Scholar

[41]

S. Suantai, N. Pholasa and P. Cholamjiak, The modified inertial relaxed $CQ$ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim., 14 (2018), 1595-1615.  doi: 10.3934/jimo.2018023.  Google Scholar

[42]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 2012. doi: 10.1090/gsm/140.  Google Scholar

[43]

R. Tibshirani, Regression shrinkage and selection Via the lasso, J. Roy. Statist Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.  Google Scholar

[44]

J. H. Wang, Y. H. Hu, C. Li and J. C. Yao, Linear convergence of $CQ$ algorithms and applications in gene regulatory network inference, Inverse Problems, 33 (2017), 055017(25 pp). doi: 10.1088/1361-6420/aa6699.  Google Scholar

[45]

X. L. Wang, J. Zhao and D. F. Hou, Modified relaxed $CQ$ iterative algorithms for the split feasibility problem, Mathematics, 7 (2019), 119. Google Scholar

[46]

Y. S. Xia, H. Leung and J. Wang, A projection neural network and its application to constrained optimization problems, IEEE Trans. Circuits Syst. I. Regul. Pap., 49 (2002), 447-458.   Google Scholar

[47]

Y. S. Xia and J. Wang, A recurrent neural network for solving linear projected equations, Neural Network, 13 (2000), 337-350.   Google Scholar

[48]

Y. S. Xia and J. Wang, On the stability of globally projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150.  doi: 10.1023/A:1004611224835.  Google Scholar

[49]

Y. S. Xia and J. Wang, A bi-projection neural network for solving constrained quadratic optimization problems, IEEE Trans. Neural Netw. Learn. Syst., 27 (2016), 214-224.  doi: 10.1109/TNNLS.2015.2500618.  Google Scholar

[50]

J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser Boston, 1992.  Google Scholar

[51]

X. J. Zou, D. W. Gong, L. P. Wang and Z. Y. Chen, A novel method to solve inverse variational inequality problems based on neuralnetworks, Neurocomputing, 173 (2016), 1163-1168.   Google Scholar

Figure 1.  The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [1, 1, 1]^T $ in Example $ 1 $ via ode45
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [1, 2, 3]^T $ in Example $ 1 $ via the central difference method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The transient behavior of dynamical system $ (4) $ with different $ \lambda $ in Example $ 1 $ via the explicit difference method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [5, -4]^T $ in Example $ 2 $ via the explicit difference method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [-10, 4]^T $ in Example $ 2 $ via the explicit difference method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The transient behavior of dynamical system $ (29) $ with 20 random initial points in Example $ 3 $ via the explicit difference method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The transient behavior of dynamical system $ (29) $ with $ x_0 = [-3, 5, 2]^T $ in Example $ 3 $ via the finite element method method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  The transient behavior of dynamical system $ (29) $ with $ x_0 = [-3, 5, 2]^T $ in Example $ 3 $ via the Piccard algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  The transient behavior of dynamical system $ (4) $ with the initial point that is generated randomly in Example $ 4 $ via ode45
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  The recovered sparse signal versus the true $ 50- $sparse signal in Example $ 4 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  The objective function value against the time for the $ LASSO $ problem solved through the dynamical system $ (4) $ with different choices of $ \lambda $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021011
[2]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[3]

Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053
[4]

Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063
[5]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040
[6]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[7]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210
[8]

Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066
[9]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056
[10]

Chenchen Wu, Wei Lv, Yujie Wang, Dachuan Xu. Approximation Algorithm for Spherical $ k $-Means Problem with Penalty. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021067
[11]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[12]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055
[13]

Yongjian Liu, Qiujian Huang, Zhouchao Wei. Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3357-3380. doi: 10.3934/dcdsb.2020235
[14]

Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021049
[15]

Tong Li, Nitesh Mathur. Riemann problem for a non-strictly hyperbolic system in chemotaxis. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021128
[16]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400
[17]

Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021064
[18]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065
[19]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003
[20]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (73)
  • HTML views (347)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences