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Projection methods for solving split equilibrium problems

  • * Corresponding author: dangvanhieu@tdtu.edu.vn

    * Corresponding author: dangvanhieu@tdtu.edu.vn
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  • 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.

    Mathematics Subject Classification: Primary: 65K10, 65K15; Secondary: 90C33.


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  • 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] 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. StarkImage 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.
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