A modified method with inertial-type for solving fixed point and variational inequalities problems in reflexive Banach spaces

Bashir Ali, Ajio Terlemun Jude, Godwin Chidi Ugwunnadi

Abstract


In this paper, using Bregman distance technique, we introduce aN inertial type algorithm with self - adaptive step size for approximating a common element of the set of solutions of pseudomonotone variational inequality problem and the set of common fixed point of a finite family of generic generalized Bregman nonspreading mapping in a real reflexive Banach space. Furthermore, we prove a strong convergence theorem of our algorithm without prior knowledge of the Lipschitz constant of the operator under some mild assumptions. We also give a numerical example to illustrate the performance of our algorithm. Our result generalize and improve many existing results in the literature.


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Y.I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, In: (A.G. Kartsatos (Ed.)) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes Pure Appl. Math., Dekker, New York, NY, USA, 178 (1996), 15-50.

B. Ali, G.C. Ugwunnadi, M.S. Lawan, A.R. Khan, Modified inertial subgradient extragradient method in reflexive Banach spaces, Bol. Soc. Mat. Mex. 27, (2021), 30.

B. Ali, L.Y. Haruna, M.H. Harbau, Common Attractive Point approximations for family of generic generalized Bregman nonspreading mappings in Banach spaces, Azerbaijan Journal of Mathematics 13 (2023), no. 2, 2218-6816. https://doi.org/10.59849/2218-6816.2023.2.27.

B. Ali, J.T. Ajio, Convergence theorem for equilibrium, split null point and common fixed point problems in Banach spaces, J. Adv. Math. Stud. 16 (2023), no. 2, 167-189.

B. Ali, J.T. Ajio, An inertial type subgradient extragradient method for common fixed point and variational inequalities problems in real Banach space, Journal of Mathematical Extension, Submitted.

E. Asplund, R.T. Rockafellar, Gradient of convex function, Trans. Am. Math. Soc. 228 (1969), 443-467.

H.H. Bauschke, J.M. Borwein, Legendre function and the method of Bregman projections, J. Convex Anal. 4(1997), 27-67.

H.H. Bauschke, P.L. Combettes, J.M. Borwein, Essential smoothness, essential strict convexity, and Legendre functions in Banach Spaces, Commun. Contemp. Math. 3 (2001), 615-647.

Y. A. Belay, H. Zegeye, O.A. Boikanyo, Approximation methods for solving split equality of variational inequality and f,g - fixed point problems in reflexive Banach spaces, Nonlinear Functional Anal. and Appl. 28 (2023), no. 1, 135-173.

J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York 2000.

L.M. Bregman, The realization method for finding the common point of convex set and its application to solution of convex programming, USSR Comput. Math. Phys. 7 (1967), 200-217.

D. Butnariu, A.N. Iusem, Totally convex functions for fixed points computation and infinite dimensional optimization, Applied Optimization 40, Kluwer Academic Publishers, Dordrecht, 2000.

D. Butnariu, E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. Article 2006 (2006), ID 84919, 1-39.

L. C. Ceng, A. Petrusel, X. Qin, J. C. Yao, Pseudomonotone variational inequalities and fixed point, Fixed Point Theory 22 (2021), 543-558.

Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim Meth Softw. 26 (2011), 827-845.

Y. Censor, A. Lent, An iterative row-action method interval convex programming, J. Optim. Theory Appl. 34 (1981), 321-353.

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

C.E. Chidume, A. Adamu, L.C. Okereke, A Krasnoselskii-type algorithm for approximating solutions of variational inequality problems and convex feasibility problem, J. Nonlinear Var. Anal. 2 (2018), 203-218.

T.H. Cuong, J.C. Yao, N.D. Yen, Qualitative properties of the minimum sum of squares clustering problem, Optimization 69 (2020), no. 9, 2131-2154.

A. Gibali, D.V. Hieu, A new inertial double-projection method for solving variational inequalities, J. of Fixed Point Theory and Applications 21 (2019), no. 4.

A.A. Goldstein, Convex programming in Hilbert space, Bull. Am. Math. Soc. 70 (1964), 709-710.

P. Hartman, G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966), 271-310.

H. He, C. Ling, H.K. Xu, A relaxed projection method for split variational inequalities, J. Optim. Theory Appl. 166 (2015), 213-233.

B.S. He, A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim. 35 (1997), 69-76.

L.O. Jolaoso, A. Taiwo, T.O. Alakoya, O.T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods, Journal of Optimization Theory and Applications 185 (2020), 744-766.

A.A. Khan, S. Migorski, M. Sama, Inverse problems for multivalued quasi variational inequalities and noncoercive variational in equalities with noisy data, Optimization 68 (2019), no. 10, 1897-1931.

F. Kohsaka, W. Takahashi, Proximal point algorithm with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6 (2005), 505-523.

G.M. Korpelevich, An extragradient method for solving saddle points and for other problems, Ekon. Mat. Metody 12 (1976), 747-756.

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

E.S. Levitin, B.T. Polyak, Constrained minimization problems, USSR Comput. Math. Math. Phys. 6 (1966), 1-50.

L.J. Lin, M.F. Yang, Q.H. Ansari, G. Kassay, Existence results for Stampacchia and Minty type implicit variational inequalities with multivalued maps, Nonlinear Anal. Theory Methods Appl. 61 (2005), 1-19.

J.L. Lions, G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-517.

P.E. Mainge, Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints, Eur. J. Oper. Res. 205 (2010), 501-506.

P.E Mainge, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim. 47 (2008), 1499-1515.

P.E. Mainge, Strong convergence of projected subgradient methods for non-smooth and nonstrictly convex minimization, Set-valued Anal. 16 (2008), 899-912.

R. Maluleka, G.C. Ugwunnadi, M. Aphane, Inertial method for solving pseudomonotone variational inequality and fixed Point problems in Banach spaces, Axioms 12 (2023), no. 10, Article ID 960.

F. Ma, A subgradient extragradient algorithm for solving monotone variational inequalities in Banach spaces, J. Inequal. Appl. 26 (2020), 2020.

J. Mashregi, M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with applications in game theory, Nonlinear Anal. 72 (2010), 2086-2099.

E. Naraghirad, J.C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2013 (2013), Article ID 141.

R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd edition, Lecture Notes in Mathematics 1364, Springer, Berlin, Germany, 1993.

B.T. Polyak, Some methods of speeding up the convergence of the iteration methods, USSR Comput. Math. Phys. 4 (1964), 1-17.

S. Reich, S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), 471-485.

S. Reich, S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), no. 13, 22-44.

Y. Shehu, A. Gibali, New inertial relaxed method for solving split feasibilities, Optimization Letters 15 (2021), 2109-2126.

M.V. Solodov, B.F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim. 37 (1999), 765-776.

W. Takahashi, N.C. Wong, J.C. Yao, Attractive point and mean convergence theorem for new generalized nonspreading mappings in Banach spaces, Contemporary Math. 636 (2015), 225-245.

B. Tan, X. Qin, Self adaptive viscosity-type inertial extragradient algorithms for Solving variational inequalities with applications, Mathematical Modelling and Analysis 27 (2022), no. 1, 41-58.

D.V. Thong, Y. Shehu, O.S. Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numer. Algor. 84 (2019), 795-823.

P. Tseng, A modfied Forward-Backward splitting method for maximal monotone mappings, SIAM J. Control Optim. 38 (2000), 431-446.

H.K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66 (2002), 240-256.

C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002.




DOI: https://doi.org/10.52846/ami.v52i2.1980