A modified inertial shrinking projection algorithm with adaptive step size for solving split generalized equilibrium, monotone inclusion and fixed point problems

Abd-Semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Oluwatosin Temitope Mewomo

Abstract


In this paper, we study the common solution problem of split generalized equilibrium problem, monotone inclusion problem and common fixed point problem for a countable family of strict pseudo-contractive multivalued mappings. We propose a modified shrinking projection algorithm of inertial form with self-adaptive step sizes for finding a common solution of the aforementioned problem. The self-adaptive step size eliminates the difficulty of computing the operator norm while the inertial term accelerates the rate of convergence of the proposed algorithm. Moreover, unlike several of the existing results in the literature, the monotone inclusion problem considered is a more general problem involving the sum of Lipschitz continuous monotone operators and maximal monotone operators, and knowledge of the Lipschitz constant is not required to implement our algorithm. Under some mild conditions, we establish strong convergence result for the proposed method. Finally, we present some applications and numerical experiments to illustrate the usefulness and applicability of our algorithm as well as comparing it with some related methods. Our results improve and extend corresponding results in the literature.


Keywords


Split generalized equilibrium problem; monotone inclusion problem,; inertial method; fixed point problem, strict pseudo-contractions, multivalued mappings.

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Alakoya, T.O., Mewomo, O.T., Viscosity S-iteration method with inertial technique and self-adaptive step size for split variational inclusion, equilibrium and fixed point problems, Comput. Appl. Math., 41(1)(2022), Paper No. 39, 31 pp.

Alakoya, T.O., Mewomo, O.T., S-Iteration inertial subgradient extragradient method for variational inequality and xed point problems, Optimization, (2023), DOI: 10.1080/02331934.2023.2168482.

Alakoya, T.O., Uzor, V.A., Mewomo, O.T., A new projection and contraction method for solving split monotone variational inclusion, pseudomonotone variational inequality, and common fixed point problems, Comput. Appl. Math., 42(2023), Art. No. 33 pp.

Alakoya, T.O., Uzor, V.A., Mewomo, O.T., Yao, J.-C., On a system of monotone variational inclusion problems with xed-point constraint, J. Inequ. Appl., 2022(2022), Paper No. 47, 33 pp.

Alvarez, F., Attouch, H., An inertial proximal method for monotone operators via discretization of a nonlinera oscillator with damping, Set Valued Anal., 9(2001), 3-11.

Attouch, H., Peypouquet, J., Redont, P., Backward-forward algorithm for structured monotone inclusions in Hilbert spaces, J. Math. Anal. Appl., 457 (2018), 1095-1117.

Bauschke, H.H., Combettes, P.L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, vol. 408, 2011.

Blum, E., Oettli, W., From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63(1-4)(1994), 123-145.

Bot, R.I., Csetnek, E.R., An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems, Numer. Algorithms, 71(2016), 519-540.

Brezis, H., Operateurs Maximaux Monotones, Chapitre II, North-Holland Math. Stud., 5(1973), 19-51.

Ceng, L.C., Yao, J.C., A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214(2008), 186-201.

Censor, Y., Gibali, A., Reich, S., Algorithms for the split variational inequality problem, Numer. Algorithms, 59(2012), 301-323.

Chen, R., Yao, Y., Strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Appl. Math. Comput., 32(2010), 69-82.

Cholamjiak, W., Cholamjiak, P., Suantai, S., An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces, J. Fixed Point Theory Appl., 20(1)(2018), 1-17.

Combettes, P.L., Wajs, V.R., Signal recovery by proximal forward-backward splitting, Multiscale Model Simul., 4(2005), 1168-1200.

Deepho, J., Martinez-Moreno, J., Kumam, P., A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems, J. Nonlinear Sci. Appl., 9(4)(2016), 1475-1496.

Ecksten, J., Svaiter, B.F., A family of projective splitting splitting methods for the sum of two maximal monotone operators, Math. Progr. Ser B, 111(2008), 173-199.

Ecksten, J., Svaiter, B.F., General projective splitting methods for sum of maximal monotone operators, SIAM J. Control Optim., 48(2009), 787-811.

Fichera, G., Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 34(8)(1963), 138-142.

Gibali, A., Thong, D.V., Tseng type methods for solving inclusion problems and its applications, Calcolo, 55(2018), 1-22.

Godwin, E.C., Alakoya, T.O., Mewomo, O.T., Yao, J.-C., Relaxed inertial Tseng extragradient method for variational inequality and fixed point problems, Appl. Anal., (2022), DOI:10.1080/00036811.2022.2107913.

Godwin, E.C., Izuchukwu, C., Mewomo, O.T., Image restoration using a modi ed relaxed inertial method for generalized split feasibility problems, Math. Methods Appl. Sci., (2022), DOI:10.1002/mma.8849.

Kazmi, K.R., Rizvi, S.H., Iterative approximation of a common solution of a split generalized equilibrium problem and a xed point problem for nonexpansive semigroup, Math. Sci., 7(2013), no. 1, Art. 1, 10 pp.

Kim, T.H., Xu, H.H., Strong convergence of modi ed Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 64(2006), 1140-1152.

Lohawech, P., Kaewcharoen, A., Farajzadeh, A., Algorithms for the common solution of the split variational inequality problems and fixed point problems with applications, J. Inequal. Appl., 2018(358)(2018).

Lorenz, D.A., Pock, T., An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51(2015), 311-325.

Ma, Z., Wang, L., Chang, S.S., Duan, W., Convergence theorems for split equality mixed equilibrium problems with applications, Fixed Point Theory Appl., 2015(2015), Art. 31.

Marino, G., Xu, H.H., Weak and strong convergence theorems for pseudo-contraction in Hilbert spaces, J. Math. Anal. Appl., 329(2007), 336-346.

Martinet, B., R egularisation, din equations variationelles par approximations succesives,

Rev. Francaise Informat., Recherche Operationelle 4, Ser. R-3, 154-159.

Mouda , A., Oliny, M., Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155(2003), 447-454.

Mouda , A., Thera, M., Finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl., 94(2)(1997), 425-448.

Nakajo, K., Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279(2003), 372-379.

Ogwo, G.N., Alakoya, T.O., Mewomo, O.T., Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point

problems, Optimization, (2021), DOI:10.1080/02331934.2021.1981897.

Ogwo, G.N., Alakoya, T.O., Mewomo, O.T., Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces, Demonstr. Math., 55(1)(2022), 193-216.

Ogwo, G.N., Alakoya, T.O., Mewomo, O.T., An inertial subgradient extragradient method with Armijo type step size for pseudomonotone variational inequalities with non-Lipschitz operators in Banach spaces, J. Ind. Manag. Optim., (2022), doi:10.3934/jimo.2022239.

Ogwo, G.N., Izuchukwu, C., Mewomo, O.T., Relaxed inertial methods for solving split variational inequality problems without product space formulation, Acta Math. Sci. Ser. B (Engl. Ed.), 42(5)(2022), 1701-1733.

Olona, M.A., Alakoya, T.O., Owolabi, A.O.-E., Mewomo, O.T., Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings, Demonstr. Math., 54(2021), 47-67.

Owolabi, A.O.-E., Alakoya, T.O., Taiwo, A., Mewomo, O.T., A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings, Numer. Algebra Control Optim., 12(2)(2022), 255-278.

Phuengrattana, W., Lerkchaiyaphum, K., On solving the split generalized equilibrium problem and the fixed point problem for a countable family of nonexpansive multivalued mappings, Fixed Point Theory Appl., 2018(2018), Art. 6.

Rockafellar, R.T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14(1976), 877-898.

Shahazad, N., Zegeye, H., Approximating of common point of fixed points of a pseudocontractive mapping and zeros of sum of monotone mappings, Fixed Point Theory Appl., 2014(2014), Art. 85.

Sitthithakerngkiet, K., Deepho, J., Martinez-Moreno, J., Kumam, P., An iterative approximation scheme for solving a split generalized equilibrium, variational inequalities and fixed point problems, Int. J. Comput. Math., 94(12)(2017), 2373-2395.

Stampacchia, G., Formes bilinearies coercitives sur les ensembles convexes, Acad. Sci. Paris, 258(1964), 4413-4416.

Suantai, S., Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311(2005), 506-517.

Taiwo, A., Owolabi, A. O.-E., Jolaoso, L.O., Mewomo, O.T., Gibali, A., A new approximation scheme for solving various split inverse problems, Afr. Mat., 32(3-4)(2021), 369-401.

Thong, D.V., Cholamjiak, P., Strong convergence of a forward-backward splitting method with a new step size for solving monotone inclusions, Comput. Appl. Math., 38(2)(2019), Paper No. 94, 16 pp.

Tseng, P., A modi ed forward-backward splitting method for maximal method for maximal monotone mappings, SIAM J. Control Optim., 38(2000), 431-446.

Uzor, V.A., Alakoya, T.O., Mewomo, O.T., Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems, Open Math., 20(2022), 234-257.

Uzor, V.A., Alakoya, T.O., Mewomo, O.T., On split monotone variational inclusion problem with multiple output sets with fixed point constraints, Comput. Methods Appl.

Math., (2022), DOI: 10.1515/cmam-2022-0199.

Yuying, T., Plubtieng, S., Strong convergence theorems by hybrid and shrinking projection methods for sums of two monotone operators, J. Ineq. Appl., 2017(2017), Art. 72.

Zhou, Y., Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 69(2008), 456-462.




DOI: http://dx.doi.org/10.24193/subbmath.2024.3.12

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