Abstract
In this paper, we propose a new algorithm for solving a split equilibrium problem involving nonmonotone and monotone equilibrium bifunctions in real Hilbert spaces by using a shrinking projection method with a general Armijo line search rule on the ε-subdifferential. We obtain a strong convergence theorem for the new algorithm.
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References
Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. European J. Oper. Res. 227, 1–11 (2013)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127–149 (1994)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18, 441–453 (2002)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 18, 103–120 (2004)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Problems 21, 2071–2084 (2005)
Censor, Y., Motova, X. A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)
Combettes, P. L., Hirstoaga, A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)
Dinh, B. V., Kim, D. S.: Projection algolithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. Math. 302, 106–117 (2016)
Dinh, B.V., Son, D.X., Jiao, L., Kim, D.S.: Linesearch algorithms for split equilibrium problems and nonexpansive map**s. Fixed Point Theory Appl. 2016, 27 (2016). https://doi.org/10.1186/s13663-016-0518-3
Dinh, B. V., Kim, D. S.: Extragradient algorithms for equilibrium problems and symmetric generalized hybrid map**s. Optim Lett. 11, 537–553 (2017)
He, Z.: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. 2012, 162 (2012). https://doi.org/10.1186/1029-242X-2012-162
Iusem, A. N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. 52, 621–635 (2003)
Takahashi, W.: Nonlinear variational inequalities and fixed point theorems. J. Math. Soc. Japan 28, 168–181 (1976)
Xu, H.K.: Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Problems 2010, 26 (2010). https://doi.org/10.1088/0266-5611/26/10/105018
Acknowledgments
The first author would like to thank the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program and Naresuan University for supporting his research via Grant No. PHD/0032/2555.
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Dedicated to Professor Hoang Tuy
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Yuying, T., Plubtieng, S. A New Linesearch Algorithm for Split Equilibrium Problems. Acta Math Vietnam 45, 397–409 (2020). https://doi.org/10.1007/s40306-020-00361-7
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DOI: https://doi.org/10.1007/s40306-020-00361-7