American Institute of Mathematical Sciences

Advanced
November  2021, 17(6): 2989-3011. 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  November 2021 Early access  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 and Management Optimization, 2021, 17 (6) : 2989-3011. doi: 10.3934/jimo.2020104
References:
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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.

[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.

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

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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.

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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.

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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.

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

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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.

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[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.

[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.

[27]

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[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.

[29]

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[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[42]

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

[43]

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[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.

[45]

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

[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. 

[47]

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

[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.

[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.

[50]

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

[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. 

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.

[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. 

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[42]

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

[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.

[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.

[45]

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

[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. 

[47]

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

[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.

[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.

[50]

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

[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. 

Figure 1.  The transient behavior of dynamical system $ (4) $ with initial points $ x_0 = [1, 1, 1]^T $ in Example $ 1 $ via ode45
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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
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Figure 3.  The transient behavior of dynamical system $ (4) $ with different $ \lambda $ in Example $ 1 $ via the explicit difference method
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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
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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
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Figure 6.  The transient behavior of dynamical system $ (29) $ with 20 random initial points in Example $ 3 $ via the explicit difference method
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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
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Figure 8.  The transient behavior of dynamical system $ (29) $ with $ x_0 = [-3, 5, 2]^T $ in Example $ 3 $ via the Piccard algorithm
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Figure 9.  The transient behavior of dynamical system $ (4) $ with the initial point that is generated randomly in Example $ 4 $ via ode45
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Figure 10.  The recovered sparse signal versus the true $ 50- $sparse signal in Example $ 4 $
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Figure 11.  The objective function value against the time for the $ LASSO $ problem solved through the dynamical system $ (4) $ with different choices of $ \lambda $
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Suthep Suantai, Nattawut Pholasa, Prasit Cholamjiak. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1595-1615. doi: 10.3934/jimo.2018023
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