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A dynamical system method for solving the split convex feasibility problem
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 |
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.
References:
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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. |
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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 |
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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. |
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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. |
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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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On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.
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H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
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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. |
[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 |
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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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A Gauss-Newton method for convex composite optimization, Math. Program., 71 (1995), 179-194.
doi: 10.1007/BF01585997. |
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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. |
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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.
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A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.
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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. |
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Double projection algorithms for solving the split feasibility problems, J. Ind. Manag. Optim., 15 (2019), 2023-2034.
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Application of projection neural network in solving convex programming problems, Appl. Math. Comput., 188 (2007), 1103-1114.
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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. 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. |
[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. |
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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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. Google Scholar |










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