Abstract
In this paper, using a hybrid extragradient method, we introduce a new iterative process for finding a common element of the solution set of the Split Equality Common Equilibrium Problem for a finite family of pseudomonotone bifunctions and the solution set of the Split Equality Common Null Point Problem for a finite family of monotone operators in certain Banach spaces. We establish strong convergence of the proposed algorithm. This paper concludes with certain applications where we utilize our results to study the determination of a solution of the Split Equality Common Variational Inequality Problem and a solution of the Split Equality Common Null Point Problem. A numerical example to support our main theorem will be exhibited. The theorems proved improve and complement a host of important recent results.
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References
Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Applied Mathematics, vol. 178, pp. 15–50. Dekker, New York (1996)
Alsulami, S.M., Hussain, N., Takahashi, W.: Weakly convergent iterative method for the splite commone null point problem in Banach spaces. J. Nonlinear Convex Anal. 11, 2333–2342 (2015)
Anh, P.N.: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 154, 303–320 (2012)
Antipin, A.S.: Equilibrium programming problems: prox-regularization and prox-methods. In: Gritzmann, P., Horst, R., Sachs, E., Tichatschke, R. (eds.) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 452, pp. 1–18. Springer, Heidelberg (1997)
Antipin, A.S.: Extraproximal approach to calculating equilibriums in pure exchange models. Comput. Math. Math. Phys. 46, 1687–1998 (2006)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 34, 367–426 (1996)
Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibriume problems. J. Optim. Theory Appl. 90, 31–43 (1996)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Byrne, C.L.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, Heidelberg (2012)
Cegielski, A.: General method for solving the split common fixed point problem. J. Optim. Theory. Appl. 165, 385–404 (2015)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Cegielski, A.: Projection methods: an annotated bibliography of books and reviews. Optimization 64, 2343–2358 (2015)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Chidume, C.E., Romanus, O.M., Nnyaba, U.V.: An iterative algorithm for solving split equality fixed point problems for a class of nonexpansive-type mappings in Banach spaces. Numer. Algorithms. https://doi.org/10.1007/s11075-018-0638-4
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2011)
Eskandani, G.Z., Raeisi, M., Rassias, ThM: A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance. J. Fixed Point Theory Appl. 20, 132 (2018)
Eslamian, M., Eskandani, G.Z., Raeisi, M.: Split common null point and common fixed point problems between Banach spaces and Hilbert spaces. Mediterr. J. Math. 14, 119 (2017)
Gibali, A.: A new split inverse problem and application to least intensity feasible solutions. Pure Appl. Funct. Anal. 2, 243–258 (2017)
Gibali, A., Küfer, K.-H., Süss, P.: Successive linear programing approach for solving the nonlinear split feasibility problem. J. Nonlinear Convex Anal. 15, 345–353 (2014)
Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudmonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)
Iusem, A.N., Nasri, M.: Inexact proximal point methods for equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim. 28, 1279–1308 (2007)
Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21, 1319–1344 (2011)
Kohsaka, F., Takahashi, W.: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr. Appl. Anal. 3, 239–249 (2004)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976). (In Russian)
Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optimiz. Theory App. 119, 317–333 (2003)
Latif, A., Eslamian, M.: Strong convergence of split equality Ky Fan inequality problem. RACSAM (2017). https://doi.org/10.1007/s13398-017-0407-6
Li, Z., Han, D., Zhang, W.: A self-adaptive projection-type method for nonlinear multiple-sets split feasibility problem. Inverse Probl. Sci. Eng. iFirst 21, 1–16 (2012)
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)
Maingé, P.E.: A viscosity method with no spectral radius requirements for the split common fixed point problem. Eur. J. Oper. Res. 235, 17–27 (2014)
Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-valued Anal. 16, 899–912 (2008)
Mastroeni, G.: On Auxiliary Principle for Equilibrium Problems, vol. 3, pp. 1244–1258. Publicatione del Dipartimento di Mathematica dell, Universita di Pisa, Pisa (2000)
Mastroeni, G.: Gap function for equilibrium problems. J. Glob. Optim. 27, 411–426 (2003)
Moudafi, A.: A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal. 79, 117–121 (2013)
Moudafi, A.: Proximal point algorithm extended to equilibrum problem. J. Nat. Geometry 15, 91–100 (1999)
Moudafi, A.: Alternating CQ-algorithm for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15(4), 809–818 (2014)
Moudafi, A., Al-Shemas, E.: Simultaneous iterative methods for split equality problems and application. Trans. Math. Program. Appl. 1, 1–11 (2013)
Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2013)
Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Sijthoff & Nordhoff International Publishers, Alphen aan den Rijn (1987)
Penfold, S., Zalas, R., Casiraghi, M., Brooke, M., Censor, Y., Schulte, R.: Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy. Phys. Med. Biol. 62, 3599–3618 (2017)
Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)
Raeisi, M., Eskandani, G.Z., Eslamian, M.: A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings. Optimization 68, 309–327 (2018)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Sabach, S.: Product of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces. SIAM J Optim. 21, 1289–1308 (2011)
Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)
Schöpfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Probl. 24, 0550088 (2008)
Semenov, V.V.: A strongly convergent splitting method for systems of operator inclusions with monotone operators. J. Autom. Inf. Sci. 46, 45–56 (2014)
Semenov, V.V.: Hybrid splitting methods for the system of operator inclusions with monotone operators. Cybern. Syst. Anal. 50, 741–749 (2014)
Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Glob. Optim. 56, 373–397 (2013)
Takahashi, W.: Nonlinear Functional Analysis. Kindaikagaku (Japanese), Tokyo (1988)
Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009)
Takahashi, W.: The split common null point problem in Banach spaces. Archiv der Mathematik 104, 357–365 (2015)
Takahashi, W.: The split common null point problem and the shrinking projection method in Banach spaces. Optimization 65, 281–287 (2016)
Takahashi, W., Yao, J.C.: Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces. Fixed Point Theory Appl. 2015, 87 (2015)
Takahashi, W.: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 15, 1349–1355 (2014)
Tiel, J.V.: Convex Analysis: An Introductory Text. Wiley, Hoboken (1984)
Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000)
Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155, 605–627 (2012)
Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65, 109–113 (2002)
Yao, J.C.: Variational inequalities with generalized monotone operators. Math. Oper. Res. 19, 691–705 (1994)
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Raeisi, M., Eskandani, G.Z. A hybrid extragradient method for a general split equality problem involving resolvents and pseudomonotone bifunctions in Banach spaces. Calcolo 56, 43 (2019). https://doi.org/10.1007/s10092-019-0341-4
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DOI: https://doi.org/10.1007/s10092-019-0341-4
Keywords
- Split equality equilibrium problem
- Pseudomonotone bifunction
- \(\phi \)-Lipschitz-type continuity
- Variational inequality