Hybrid proximal point algorithm for solving split equilibrium problems and its applications

Year 2022, , 932 - 957, 01.08.2022

Maryam Safari , Fridoun Moradlou , Ali Asghar Khalilzadeh

https://doi.org/10.15672/hujms.1023754

Cited By: 2

Abstract

This paper deals with split equilibrium problems in Banach spaces. The presented algorithm is based on the hybrid algorithm and the proximal point algorithm and has been used for finding the solution of split equilibrium problems. Under some standard assumptions on equilibrium bifunctions, it is proven that the generated sequences by the presented scheme are strongly convergent. Finally, the efficiency of the proposed method is demonstrated through some examples. Also, comparative results verify that the proposed method is more effective than the other existing methods in the literature. Furthermore, an application of the presented algorithm in Hilbert spaces and an application of our method to solve the $LASSO$ problem in the field of compressed sensing are given.

Keywords

split equilibrium problem, strong convergence, uniformly convex Banach space, uniformly smooth Banach space

References

• [1] S. Alizadeh and F. Moradlou, Strong convergence theorems for m-generalized hybrid mappings in Hilbert spaces, Topol. Methods Nonlinear Anal. 46, 315-328, 2015.
• [2] S. Alizadeh and F. Moradlou, A strong convergence theorem for equilibrium problems and generalized hybrid mappings, Mediterr. J. Math. 13, 379-390, 2016.
• [3] A.S. Antipin, The convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence, Comput. Math. Math. Phys. 35, 539-551, 1995.
• [4] E. Blum and W. Oettli, From Optimization and variational inequalities to equilibrium problems, Math. Stud. 63, 123-145, 1994.
• [5] A. Cegielski, Iterative methods for fixed point problems in Hilbert spaces, Lecture Notes in Mathematics, Vol. 2057, Springer, 2012.
• [6] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms 59, 301-323, 2012.
• [7] C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, in: Lecture Notes in Mathematics, vol. 1965, Springer, Berlin, 2009.
• [8] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.
• [9] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6, 117-136, 2005.
• [10] J. Contreras, M. Klusch and J.B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, EEE Trans. Power. Syst. 19, 195- 206, 2004.
• [11] J. Deepho, W. Kumm and P. Kumm, A new hybrid projection algorithm for solving the split generalized equilibrium problems and the system of variational inequality problems, J. Math. Model. Algorithms 13, 405-423, 2014.
• [12] B.V. Dinh, D.X. Son and T.V. Anh, Extragradient-Proximal Methods for Split Equilibrium and Fixed Point Problems in Hilbert Spaces, Vietnam J. Math. 45, 651-668, 2015.
• [13] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, Berlin, 2002.
• [14] Z. He, The split equilibrium problem and its convergence algorithms, J. Inequal. Appl. 162, 2012.
• [15] D.V. Hieu, Parallel Extragradient-Proximal Methods for Split Equilibrium Problems, Math. Model. Anal. 21, 478-501, 2016.
• [16] D.V. Hieu, Two hybrid algorithms for solving split equilibrium problems, Int. J. Comput. Math. 95, 561-583, 2018.
• [17] D.V. Hieu, Projection methods for solving split equilibrium problems, J. Ind. Manag. Optim. 16, 2331-2349, 2020.
• [18] D.V. Hieu, L.D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms 73, 197-217, 2016.
• [19] Z. Jouymandi and F. Moradlou, Extragradient methods for solving equilibrium problems, variational inequalities and fixed point problems, Numer. Funct. Anal. Optim. 38, 1391-1409, 2017.
• [20] Z. Jouymandi and F. Moradlou, Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems and fixed point problems in Banach spaces, Numer. Algorithms 78, 1153-1182, 2018.
• [21] Z. Jouymandi and F. Moradlou, Extragradient and linesearch algorithms for solving equilibrium problems, variational inequalities and fixed point problems in Banach spaces, Fixed Point Theory 20, 523-540, 2019.
• [22] G. Kassay, T.N. Hai and N.T. Vinh, Coupling Popov’s algorithm with subgradient extragradient method for solving equilibrium problems, J. Nonlinear Convex Anal. 19, 959-986, 2018.
• [23] D.S. Kim and B.V. Dinh, Parallel extragradient algorithms for multiple set split equilibrium problems in Hilbert spaces, Numer. Algorithms 77, 741-761, 2018.
• [24] D. Kinderlehrar and D. Stampacchia, An introduction to variational inequality and their application, Academic Press, New York, 1980.
• [25] S.I. Lyashko and V.V. Semenov, A new two-step proximal algorithm of solving the problem of equilibrium programming, in: Optimization and Applications in Control and Data Sciences 115, 315-326, Springer, 2016.
• [26] Y. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim. 25, 502-520, 2015.
• [27] B. Martinet, Régularisation d′inéquations variationnelles par approximations successives, Rev Française Informat Recherche Opérationnelle 4, 154-158, 1970.
• [28] S. Matsushita and L. Xu, On convergence of the proximal point algorithm in Banach spaces, Proc. Amer. Math. Soc. 139, 4087-4095, 2011.
• [29] S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2004 37-47, 2004.
• [30] S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134, 257-266, 2005.
• [31] A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems 26, (Article ID 055007), 2010 .
• [32] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150, 275- 283, 2011.
• [33] L.D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. 18, 1159-1166, 1992.
• [34] T.D. Quoc, P. N. Anh and L. D. Muu, Dual extragradient algorithms to equilibrium Problems, J. Glob. Optim. 52, 139-159, 2012.
• [35] T.D. Quoc, L.D. Muu and V.H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization 57, 749-776, 2008.
• [36] S. Reich, A weak convergence theorem for the alternating method with Bregman distance, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996.
• [37] S. Reich and S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10, 471-485, 2009.
• [38] R.T. Rockafellar, Maximal monotone operators and proximal point algorithm, SIAM J. Control Optim. 14, 877-898, 1976.
• [39] M. Safari and F. Moradlou, Shrinking hybrid method for multiple-sets split feasibility problems and variational inequality problems, Ric. Mat., accepted, doi:10.1007/s11587-021-00676-z.
• [40] D. Sahu, D. O’Regan and R. P. Agarwal, Fixed point theory for Lipschitzian-type mappings with Applications, Springer, New York, 2009.
• [41] F. Sch¨opfer, T. Schuster and A. K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24, (Article ID 055008), 2008.
• [42] J.J. Strodiot, P.T. Vuong and N.T.T. Van, A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces, J. Global Optim. 64, 159-178, 2016.
• [43] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69, 1025-1033, 2008.
• [44] R. Tibshirani, Regression shrinkage and selection via LASSO, J. R. Stat. Soc. Ser. B. Stat. Methodol. 58, 267-288, 1996.
• [45] S. Wang, X. Gong, A.N. Abdou and Y.J. Cho, Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications, Fixed Point Theory Appl. 2016, 1-22, 2016.