A strong convergence algorithm for approximating a common solution of variational inequality and fixed point problems in real Hilbert space

Olawale K. Oyewole, Akindele A. Mebawondu, Oluwatosin T. Mewomo

Abstract


In this paper, we propose an iterative algorithm for approximating a common solution of a variational inequality and fixed point problem. The algorithm combines the subgradient extragradient technique, inertial method and a modified viscosity approach. Using this algorithm, we state and prove a strong convergence algorithm for obtaining a common solution of a pseudomonotone variational inequality problem and fixed point of an \(\eta\)-demimetric mapping in a real Hilbert space. We give an application of this result to some theoretical optimization problems. Furthermore, we report some numerical examples to show the efficiency of our method by comparing with previous methods in the literature. Our result extend, improve and unify many other results in this direction in the literature.


Keywords


variational inequality; pseudomonotone operator; strong convergence; fixed point; extragradient algorithm, linesearch rule, Hilbert space.

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References


Alakoya, T.O., Jolaoso, L.O., Mewomo, O.T., Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems, Demonstr. Math., 53(2020), no. 1, 208-224.

Alakoya, T.O., Jolaoso, L.O., Mewomo, O.T., Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces, Afr. Mat., (2021), DOI:10.1007/s13370-020-00869-z.

Alakoya, T.O., Jolaoso, L.O., Taiwo, A., Mewomo, O.T., Inertial algorithm with self-adaptive stepsize for split common null point and common fixed point problems for multivalued mappings in Banach spaces, Optimization, (2021),

DOI:10.1080/02331934.2021.1895154.

Alakoya, T.O., Taiwo, A., Mewomo, O.T., Cho, Y.J., An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 67(2021), no. 1, 1-31.

Anh, P.K., Hieu, D.V., Parallel and sequential hybrid methods for a finite family of asymptotically quasi- -nonexpensive mappings, J. Appl. Math. Comput., 48(2015), 241-263.

Anh, P.N., Phuong, N.X., A parallel extragradient-like projection method for unrelated variational inequalities and fixed point problems, J. Fixed Point Theory Appl., 20(2018), Art. 17.

Antipin, A.S., On a method for convex program using a symmetrical modification of the Lagrange function, Ekonomika i Matematicheskie Metody., 12(1976), 1164-1173.

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

Browder, F.E., Petryshyn, W.V., Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20(1967), 19-228.

Byrne, C., Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl., 18(2002), 441-453.

Cai, G., Dong, Q.L., Peng, Y., Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz operators, J. Optim. Theory Appl., 188 (2021), no. 2, 447-472.

Ceng, L.C., Ansari, Q.H., Yao, J.C., An extragradient method for solving split feasibility and xed point problems, Comput. Math. Appl., 64(2012), 633-642.

Censor, Y., Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8(1994), no. 2, 221-239.

Censor, Y., Gibali, A., Reich, S., The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148(2011), 318-335.

Censor, Y., Gibali, A., Reich, S., Strong convergence of subgradient extragradient for the variational inequality problem in Hilbert space, Optim. Method Soft., 6(2011), 827-845.

Gibali, A., Jolaoso, L.O., Mewomo, O.T., Taiwo, A., Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces,

Results Math., 75(2020), Art. No. 179, 36 pp.

Godwin, E.C., Izuchukwu, C., Mewomo, O.T., An inertial extrapolation method for solving generalized split feasibility problems in real Hilbert spaces, Boll. Unione Mat. Ital., 14(2021), no. 2, 379-401.

Hao, Y., Some results of variational inclusion problems and fixed point problems with applications, Appl. Math. Mech. (English Ed.), 30(2009), 1589-1596.

Harker, P.T., Pang, J.S., A damped-Newton method for the linear complementarity problem, In: Allgower, G., Georg, K. (eds.) Computational Solution of Nonlinear Systems of Equations. Lectures in Applied Mathematics, 26(1990), AMS, Providence, 265-284.

He, Z., Chen, C., Gu, F., Viscosity approximation method for nonexpansive nonself nonexpansive mappings and variational inequality, J. Nonlinear Sci. Appl., 1(2008), 169-178.

Hieu, D.V., Anh, P.K., Muu, L.D., Modi ed hybrid projection methods for nding common solutions to variational inequality problems, Comput Optim Appl., 66(2017), 75-96.

Hieu, D.V., Anh, P.K., Muu, L.D., Modi ed extragradient-like algorithms with new step-sizes for variational inequalities, Comput Optim Appl., 73(2019), 913-932.

Hieu, D.V., Thong, D.V., New extragradient-like algorithms for strongly pseudomonotone variational inequalities, J. Glob Optim., 70(2018), 385-399.

Hu, X., Wang, J., Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network IEEE Trans., Neural Netw., 17(2006), 1487-1499.

Iiduka, H., Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program, 133(2012), 227-242.

Iiduka, H., Takahashi, W., Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61(2005), 341-350.

Iiduka, H., Yamada, I., An ergodic algorithm for the power-control games for CDMA data networks, J. Math. Model. Algorithms, 8(2009), 1-18.

Izuchukwu, C., Mebawondu, A.A., Mewomo, O.T., A new method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl.,

(2020), no. 4, Art. No. 98, 23 pp.

Izuchukwu, C., Ogwo, G.N., Mewomo, O.T., An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, (2020), DOI:10.1080/02331934.2020.1808648.

Jolaoso, L.O., Oyewole, O.K., Aremu, K.O., Mewomo, O.T., A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions

of split generalized equilibrium problems in Hilbert spaces, Int. J. Comput. Math., (2020), DOI:10.1080/00207160.2020.1856823.

Khanh, P.D., A new extragradient method for strongly pseudomonotone variational inequalities, Numer Funct. Anal. Optim., 37(2016), 1131-1143.

Kopeck, E., Reich, S., A note on alternating projections in Hilbert space, J. Fixed Point Theory Appl., 12(2012), 41-47.

Korpelevich, G.M., The extragradient method for finding saddle points and other problems, Ekon. Mate. Metody., 12(1976), 747-756.

Kraikaew, R., Saejung, S., Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163(2014), 399-412.

Malitsky, Yu. V., Semenov, V.V., A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61(2015), 193-202.

Ogwo, G.N., Izuchukwu, C., Aremu, K.O., Mewomo, O.T., A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27(2020), 1-26.

Ogwo, G.N., Izuchukwu, C., Mewomo, O.T., Inertial methods for nding minimum-norm solutions of the split variational inequality problem beyond monotonicity, Numer. Algorithms, (2021), DOI:10.1007/s11075-021-01081-1.

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), 1-21.

Olona, M.A., Alakoya, T.O., Owolabi, A.O.E., Mewomo, O.T., Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an in nite family of strictly pseudocontractive mappings, J. Nonlinear Funct. Anal., 2021(2021), Art. ID

, 21 pp.

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., (2021),

DOI:10.3934/naco.2021004.

Oyewole, O.K., Abass, H.A., Mewomo, O.T., A strong convergence algorithm for a fixed point constraint null point problem, Rend. Circ. Mat. Palermo II, 70(2021), 389-408.

Oyewole, O.K., Mewomo, O.T., Jolaoso, L.O., Khan, S.H., An extragradient algorithm for split generalized equilibrium problem and the set of xed points of quasi- -nonexpansive mappings in Banach spaces, Turkish J. Math., 44(4)(2020), 1146-1170.

Slavakis, K., Yamada, I., Robust wideband beamforming by the hybrid steepest descent method, IEEE Trans. Signal Process., 55(2007), 4511-4522.

Su, M., Xu, H.K., Remarks on the Gradient-Projection algorithm, J. Nonlinear Anal. Optim., 1(2010), 35-43.

Taiwo, A., Alakoya, T.O., Mewomo, O.T., Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces,

Numer. Algorithms, 86(2021), no. 4, 1359-1389.

Taiwo, A., Alakoya, T.O., Mewomo, O.T., Strong convergence theorem for solving equilibrium problem and fixed point of relatively nonexpansive multi-valued mappings in a Banach space with AG2applications, Asian-Eur. J. Math., (2020), DOI:10.1142/S1793557121501370.

Taiwo, A., Jolaoso, L.O., Mewomo, O.T., Inertial-type algorithm for solving split common fixed-point problem in Banach spaces, J. Sci. Comput., 86(2021), no. 1, Paper No. 12, 30 pp.

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., (2020), DOI:10.1007/s13370-020-00832-y.

Takahashi, W., Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.

Thong, D.V., Hieu, D.V., Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems, Numer. Algorithms, 82(2019), 761-789.

Thong, D.V., Hieu, D.V., Rassias, T.M., Self adaptive inertial subgradient extragradient algorithms for solving psedomonotone variational inequality problems, Optim. Lett.,

(2020), 115-144.

Yang, J., Liu, H., A modified projected gradient method for monotone variational inequalities, J. Optim Theory Appl., 179(2018), 197-211.

Zhang, Y., Yuan, Q., Iterative common soultions of fixed point and variational inequality problems, J. Nonlinear Sci. Appl., 9(2016), 1882-1890.




DOI: http://dx.doi.org/10.24193/subbmath.2024.1.12

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