• Previous Article
    A new concave reformulation and its application in solving DC programming globally under uncertain environment
  • JIMO Home
  • This Issue
  • Next Article
    Differential equation approximation and enhancing control method for finding the PID gain of a quarter-car suspension model with state-dependent ODE
September  2020, 16(5): 2331-2349. doi: 10.3934/jimo.2019056

Projection methods for solving split equilibrium problems

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author: dangvanhieu@tdtu.edu.vn

Received  July 2018 Revised  January 2019 Published  September 2020 Early access  May 2019

The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others.

Citation: Dang Van Hieu. Projection methods for solving split equilibrium problems. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2331-2349. doi: 10.3934/jimo.2019056
References:
[1]

P. N. Anh and L. D. Muu, A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems, Optim. Lett., 8 (2014), 727-738.  doi: 10.1007/s11590-013-0612-y.

[2]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Program., 63 (1994), 123-145. 

[3]

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

[4]

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

[5]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensitymodulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. 

[6]

Y. Censor and T. Elving, A multiprojections algorithm using Bregman projections in a product spaces, Numer. Algor., 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[7]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algor., 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.

[8]

Y. Censor and A. Segalh, Iterative projection methods in biomedical inverse problems. In: Censor Y, Jiang M, Louis AK (eds) Mathematical methods in biomedical imaging and intensity-modulated therapy, IMRT, Edizioni della Norale, Pisa, (2008), 65–96.

[9]

S. ChangL. WangX. R. Wang and G. Wang, General split equality equilibrium problems with application to split optimization problems, J. Optim. Theory Appl., 166 (2015), 377-390.  doi: 10.1007/s10957-015-0739-3.

[10]

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. 

[11]

J. ContrerasM. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, EEE Trans. Power. Syst., 19 (2004), 195-206.  doi: 10.1109/TPWRS.2003.820692.

[12]

J. DeephoJ. Martnez-MorenoK. Sitthithakerngkiet and P. Kumam, Convergence analysis of hybrid projection with Cesaro mean method for the split equilibrium and general system of finite variational inequalities, J. Comput. Appl. Math., 318 (2017), 658-673.  doi: 10.1016/j.cam.2015.10.006.

[13]

J. DeephoW. Kumam and P. Kumam, A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems, J. Math. Model. Algor., 13 (2014), 405-423.  doi: 10.1007/s10852-014-9261-0.

[14]

B. V. DinhD. X. Son and T. V. Anh, Extragradient-proximal methods for split equilibrium and fixed point problems in Hilbert spaces, Vietnam J. Math., 45 (2017), 651-668.  doi: 10.1007/s10013-016-0237-4.

[15]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.

[16]

S. D. Flam and A. S. Antipin, Equilibrium programming and proximal-like algorithms, Math. Program., 78 (1997), 29-41.  doi: 10.1016/S0025-5610(96)00071-8.

[17]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.

[18]

Z. He, The split equilibrium problems and its convergence algorithms, J. Inequal. Appl., 2012 (2012), 15pp. doi: 10.1186/1029-242X-2012-162.

[19]

D. V. Hieu, Projected subgradient algorithms on systems of equilibrium problems, Optim. Lett., 12 (2018), 551-566.  doi: 10.1007/s11590-017-1127-8.

[20]

D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21 (2016), 478-501.  doi: 10.3846/13926292.2016.1183527.

[21]

D. V. Hieu, Two hybrid algorithms for solving split equilibrium problems, Inter. J. Comput. Math., 95 (2018), 561-583.  doi: 10.1080/00207160.2017.1291934.

[22]

D. V. Hieu and A. Moudafi, A barycentric projected-subgradient algorithm for equilibrium problems, J. Nonlinear Var. Anal., 1 (2017), 43-59. 

[23]

D. V. Hieu and J. J. Strodiot, Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl., 20 (2018), Art. 131, 32 pp. doi: 10.1007/s11784-018-0608-4.

[24]

D. V. Hieu, An inertial-like proximal algorithm for equilibrium problems, Math. Meth. Oper. Res., 88 (2018), 399-415.  doi: 10.1007/s00186-018-0640-6.

[25]

D. V. HieuY. J. Cho and Y.-B. Xiao, Modified extragradient algorithms for solving equilibrium problems, Optimization, 67 (2018), 2003-2029.  doi: 10.1080/02331934.2018.1505886.

[26]

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction, Kluwer Academic, Dordrecht, The Netherlands, 1989. doi: 10.1007/978-94-010-9608-9.

[27]

A. N. Iusem and W. Sosa, Iterative algorithms for equilibrium problems, Optimization, 52 (2003), 301-316.  doi: 10.1080/0233193031000120039.

[28]

K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egyptian Math. Society, 21 (2013), 44-51.  doi: 10.1016/j.joems.2012.10.009.

[29]

A. Moudafi, Proximal point algorithm extended to equilibrum problem, J. Nat. Geometry, 15 (1999), 91-100. 

[30]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.

[31]

A. Moudafi and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Lett., 8 (2014), 2099-2110.  doi: 10.1007/s11590-013-0708-4.

[32]

A. Moudafi and E. Al-Shemas, Simultaneously iterative methods for split equality problem, Trans. Math. Program. Appl., 1 (2013), 1-11. 

[33]

A. Moudafi, A relaxed alternating CQ algorithm for convex feasibility problems, Nonlinear Anal. TMA, 79 (2013), 117-121.  doi: 10.1016/j.na.2012.11.013.

[34]

L. D. Muu and W. Oettli, Convergence of an adative penalty scheme for finding constrained equilibria, Nonlinear Anal. TMA, 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C.

[35]

T. D. QuocL. D. Muu and N. V. Hien, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749-776.  doi: 10.1080/02331930601122876.

[36]

P. Santos and S. Scheimberg, An inexact subgradient algorithm for equilibrium problems, Comput. Appl. Math., 30 (2011), 91-107. 

[37] H. Stark, Image Recovery: Theory and Applications, Academic Press, Orlando, FL, 1987. 
[38]

P. T. VuongJ. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2012), 605-627.  doi: 10.1007/s10957-012-0085-7.

[39]

H. K. Xu, Viscosity approximation methods for nonexpansive mappings, Math. Anal. Appl., 298 (2004), 279-291.  doi: 10.1016/j.jmaa.2004.04.059.

[40]

L. H. YenL. D. Muu and N. T. T. Huyen, An algorithm for a class of split feasibility problems: Application to a model in electricity production, Math. Meth. Oper. Res., 84 (2016), 549-565.  doi: 10.1007/s00186-016-0553-1.

show all references

References:
[1]

P. N. Anh and L. D. Muu, A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems, Optim. Lett., 8 (2014), 727-738.  doi: 10.1007/s11590-013-0612-y.

[2]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Program., 63 (1994), 123-145. 

[3]

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

[4]

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

[5]

Y. CensorT. BortfeldB. Martin and A. Trofimov, A unified approach for inversion problems in intensitymodulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365. 

[6]

Y. Censor and T. Elving, A multiprojections algorithm using Bregman projections in a product spaces, Numer. Algor., 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[7]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algor., 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.

[8]

Y. Censor and A. Segalh, Iterative projection methods in biomedical inverse problems. In: Censor Y, Jiang M, Louis AK (eds) Mathematical methods in biomedical imaging and intensity-modulated therapy, IMRT, Edizioni della Norale, Pisa, (2008), 65–96.

[9]

S. ChangL. WangX. R. Wang and G. Wang, General split equality equilibrium problems with application to split optimization problems, J. Optim. Theory Appl., 166 (2015), 377-390.  doi: 10.1007/s10957-015-0739-3.

[10]

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. 

[11]

J. ContrerasM. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, EEE Trans. Power. Syst., 19 (2004), 195-206.  doi: 10.1109/TPWRS.2003.820692.

[12]

J. DeephoJ. Martnez-MorenoK. Sitthithakerngkiet and P. Kumam, Convergence analysis of hybrid projection with Cesaro mean method for the split equilibrium and general system of finite variational inequalities, J. Comput. Appl. Math., 318 (2017), 658-673.  doi: 10.1016/j.cam.2015.10.006.

[13]

J. DeephoW. Kumam and P. Kumam, A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems, J. Math. Model. Algor., 13 (2014), 405-423.  doi: 10.1007/s10852-014-9261-0.

[14]

B. V. DinhD. X. Son and T. V. Anh, Extragradient-proximal methods for split equilibrium and fixed point problems in Hilbert spaces, Vietnam J. Math., 45 (2017), 651-668.  doi: 10.1007/s10013-016-0237-4.

[15]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.

[16]

S. D. Flam and A. S. Antipin, Equilibrium programming and proximal-like algorithms, Math. Program., 78 (1997), 29-41.  doi: 10.1016/S0025-5610(96)00071-8.

[17]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.

[18]

Z. He, The split equilibrium problems and its convergence algorithms, J. Inequal. Appl., 2012 (2012), 15pp. doi: 10.1186/1029-242X-2012-162.

[19]

D. V. Hieu, Projected subgradient algorithms on systems of equilibrium problems, Optim. Lett., 12 (2018), 551-566.  doi: 10.1007/s11590-017-1127-8.

[20]

D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21 (2016), 478-501.  doi: 10.3846/13926292.2016.1183527.

[21]

D. V. Hieu, Two hybrid algorithms for solving split equilibrium problems, Inter. J. Comput. Math., 95 (2018), 561-583.  doi: 10.1080/00207160.2017.1291934.

[22]

D. V. Hieu and A. Moudafi, A barycentric projected-subgradient algorithm for equilibrium problems, J. Nonlinear Var. Anal., 1 (2017), 43-59. 

[23]

D. V. Hieu and J. J. Strodiot, Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl., 20 (2018), Art. 131, 32 pp. doi: 10.1007/s11784-018-0608-4.

[24]

D. V. Hieu, An inertial-like proximal algorithm for equilibrium problems, Math. Meth. Oper. Res., 88 (2018), 399-415.  doi: 10.1007/s00186-018-0640-6.

[25]

D. V. HieuY. J. Cho and Y.-B. Xiao, Modified extragradient algorithms for solving equilibrium problems, Optimization, 67 (2018), 2003-2029.  doi: 10.1080/02331934.2018.1505886.

[26]

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction, Kluwer Academic, Dordrecht, The Netherlands, 1989. doi: 10.1007/978-94-010-9608-9.

[27]

A. N. Iusem and W. Sosa, Iterative algorithms for equilibrium problems, Optimization, 52 (2003), 301-316.  doi: 10.1080/0233193031000120039.

[28]

K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egyptian Math. Society, 21 (2013), 44-51.  doi: 10.1016/j.joems.2012.10.009.

[29]

A. Moudafi, Proximal point algorithm extended to equilibrum problem, J. Nat. Geometry, 15 (1999), 91-100. 

[30]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.

[31]

A. Moudafi and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Lett., 8 (2014), 2099-2110.  doi: 10.1007/s11590-013-0708-4.

[32]

A. Moudafi and E. Al-Shemas, Simultaneously iterative methods for split equality problem, Trans. Math. Program. Appl., 1 (2013), 1-11. 

[33]

A. Moudafi, A relaxed alternating CQ algorithm for convex feasibility problems, Nonlinear Anal. TMA, 79 (2013), 117-121.  doi: 10.1016/j.na.2012.11.013.

[34]

L. D. Muu and W. Oettli, Convergence of an adative penalty scheme for finding constrained equilibria, Nonlinear Anal. TMA, 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C.

[35]

T. D. QuocL. D. Muu and N. V. Hien, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749-776.  doi: 10.1080/02331930601122876.

[36]

P. Santos and S. Scheimberg, An inexact subgradient algorithm for equilibrium problems, Comput. Appl. Math., 30 (2011), 91-107. 

[37] H. Stark, Image Recovery: Theory and Applications, Academic Press, Orlando, FL, 1987. 
[38]

P. T. VuongJ. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2012), 605-627.  doi: 10.1007/s10957-012-0085-7.

[39]

H. K. Xu, Viscosity approximation methods for nonexpansive mappings, Math. Anal. Appl., 298 (2004), 279-291.  doi: 10.1016/j.jmaa.2004.04.059.

[40]

L. H. YenL. D. Muu and N. T. T. Huyen, An algorithm for a class of split feasibility problems: Application to a model in electricity production, Math. Meth. Oper. Res., 84 (2016), 549-565.  doi: 10.1007/s00186-016-0553-1.

Figure 1.  Algorithm 1 for $ (m, k) = (30, 20) $ and different sequences of $ \beta_n $. The number of iterations is 360,353,339,360,355,376, respectively
Figure 2.  Algorithm 1 for (m; k) = (60; 40) and different sequences of βn. The number of iterations is 258,333,336,326,291,293, respectively
Figure 3.  Algorithm 1 for (m; k) = (100; 50) and different sequences of βn. The number of iterations is 215,236,283,280,321,290, respectively
Figure 4.  Algorithm 1 for $ (m, k) = (150,100) $ and different sequences of $ \beta_n $. The number of iterations is 161,188,219,209,245,264, respectively
Figure 5.  Experiment for the algorithms with $ (m, k) = (30, 20) $. The number of iterations is 334,240,379,168,130, respectively
Figure 6.  Experiment for the algorithms with (m; k) = (60; 40). The number of iterations is 326,221,292,129,108, respectively
Figure 7.  Experiment for the algorithms with (m; k) = (100; 50). The number of iterations is 308,250,356,114, 89, respectively
Figure 8.  Experiment for the algorithms with $ (m, k) = (150,100) $. The number of iterations is 254,192,271, 87, 69, respectively
[1]

Jamilu Abubakar, Poom Kumam, Abor Isa Garba, Muhammad Sirajo Abdullahi, Abdulkarim Hassan Ibrahim, Wachirapong Jirakitpuwapat. An efficient iterative method for solving split variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021160

[2]

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

[3]

Yazheng Dang, Marcus Ang, Jie Sun. An inertial triple-projection algorithm for solving the split feasibility problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022019

[4]

Emeka Chigaemezu Godwin, Adeolu Taiwo, Oluwatosin Temitope Mewomo. Iterative method for solving split common fixed point problem of asymptotically demicontractive mappings in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022005

[5]

Jacob Ashiwere Abuchu, Godwin Chidi Ugwunnadi, Ojen Kumar Narain. Inertial Mann-Type iterative method for solving split monotone variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022075

[6]

Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2557-2572. doi: 10.3934/jimo.2020082

[7]

Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021046

[8]

Yazheng Dang, Jie Sun, Honglei Xu. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1383-1394. doi: 10.3934/jimo.2016078

[9]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[10]

Preeyanuch Chuasuk, Ferdinard Ogbuisi, Yekini Shehu, Prasit Cholamjiak. New inertial method for generalized split variational inclusion problems. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3357-3371. doi: 10.3934/jimo.2020123

[11]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021037

[12]

Ya-zheng Dang, Jie Sun, Su Zhang. Double projection algorithms for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2023-2034. doi: 10.3934/jimo.2018135

[13]

Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial and Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749

[14]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205

[15]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[16]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial and Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101

[17]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123

[18]

Guirong Pan, Bing Xue, Hongchun Sun. An optimization model and method for supply chain equilibrium management problem. Mathematical Foundations of Computing, 2022, 5 (2) : 145-156. doi: 10.3934/mfc.2022001

[19]

Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203

[20]

Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial and Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (393)
  • HTML views (918)
  • Cited by (3)

Other articles
by authors

[Back to Top]