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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  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 & 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.  Google Scholar

[2]

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

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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

[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.  Google Scholar

[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.  Google Scholar

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
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