An inertial-viscosity algorithm for solving split generalized equilibrium problem and a system of demimetric mappings in Hilbert spaces
Tóm tắt
In this paper, we introduce a new inertial-viscosity approximation method for solving a split generalized equilibrium problem and common fixed point problem in real Hilbert spaces. The algorithm is designed such that its convergence does not require the norm of the bounded linear operator underlying the split equilibrium problem. Moreover, a strong convergence result is proved under mild conditions in real Hilbert spaces. Furthermore, we give some numerical examples to show the efficiency and accuracy of the proposed method and we also compare the performance of our algorithm with other related methods in the literature.
Tài liệu tham khảo
Abass, H. A., Aremu, K. O., Jolaoso, L. O., Mewomo, O. T.: An Inertial forward-backward splitting method for approximating solutions of certain optimization of certain optimization problems. J. Nonlin. Funct. Anal., Article ID 6 (2020), https://doi.org/10.23952/jnfa.2020.6
Ali, B., Haruna, L.Y.: Fixed point approximations of noncommutative generic 2-generalized Bregman nonspreading mappings with equilibriums. J. Nonlinear Sci. Appl. 13(6), 303–316 (2020)
Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal. 9, 3–11 (2001)
Ansari, Q.H., Rehan, A.: Split feasibility and fixed point problems. In: Ansari, Q.H. (ed.) it Nonlinear Analysis: Approximation Theory, Optimization and Application, pp. 281–322. Springer, New York (2014)
Aremu, K.O., Izuchuwkwu, C., Ugwunnadi, G.C., Mewomo, O.T.: On the proximal point algorithm and demimetric mappings in CAT(0) spaces. Demonstr. Math. 51, 277–294 (2018)
Bryne, C.: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18(2), 441–453 (2002)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8(2), 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Prob. 21, 2071–2084 (2005)
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., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Cheawchan, K., Kangtunyakarn, A.: The modified split generalized equilibrium problem for quasi-nonexpansive mappings and applications. J. Inequal. Appl. 2018, 122 (2018). https://doi.org/10.1186/s13660-018-1716-9
Cholamjiak, P., Shehu, Y.: Inertial forward-backward splitting method in Banach spaces with application to compressed sensing. Appl. Math. 64, 409–435 (2019)
Cholamjiak, P., Thong, D.V., Cho, Y.J.: A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl. Math. (2019). https://doi.org/10.1007/s10440-019-00297-7
Cholamjiak, P., Suantai, S., Sunthrayuth, P.: Strong convergence of a general viscosity explicit rule for the sum of two monotone operators in Hilbert spaces. J. Appl. Anal. Comput. 9, 2137–2155 (2019)
Cholamjiak, P., Suantai, S., Sunthrayuth, P.: An explicit parallel algorithm for solving variational inclusion problem and fixed point problem in Banach spaces. Banach J. Math. Anal. 14, 20–40 (2020)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6(1), 117–136 (2005)
Eslamian, M., Saadati, R., Vahidi, J.: Viscosity iterative process for demicontractive mappings and multivalued mappings and equilibrium problems. Comp Appl Math. 36, 1239–1253 (2017)
Farid, M., Kazmi, K. R.: A new mapping for finding a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem. Korean J. Math., 27(2), 297–327 (2019)
Gibali, A., Shehu, Y.: An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces. Optimization 68(1), 13–32 (2019)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Rehman, H., Pakkaranang, N., Hussain, A., Wairojjana, N.: A modified extra-gradient method for a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces. J. Math. Comput SCI-JM 22(1), 38–48 (2020)
Jantakarn, K., Kaewcharoen, A.: Strong convergence theorems for mixed equilibrium problems and Bregman relatively nonexpansive mappings in reflexive Banach spaces. J. Nonlinear Sci. Appl. 14(2), 63–79 (2021)
Kazmi, K.R., Raouf, A.: A class of operator equilibrium problems. J. Math. Anal. Appl. 308, 554–566 (2005)
Kazmi, K.R., Khan, S.A.: Existence of solutions to a generalized system. J. Optim. Theory Appl. 142, 355–361 (2009)
Kazmi, K. R., Rizvi, S. H.: Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup. Math. Sci. 7(1) (2013). https://doi.org/10.1186/2251-7456-7-1
Khan, M.A.A., Cholamjiak, P.: A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces. J. Fixed Point Theory Appl. 22, 62 (2020). https://doi.org/10.1007/s11784-020-00796-3
Jolaoso, L.O., Lukumon, G.A., Aphane, M.: Convergence theorem for system of pseudomonotone equilibrium and split common fixed point problems in Hilbert spaces. Boll Unione Mat Ital (2021). https://doi.org/10.1007/s40574-020-00271-4
Jolaoso, L.O., Aphane, M.: Strong convergence inertial projection and contraction method with self adaptive stepsize for pseudomonotone variational inequalities and fixed point problems. J. Inequal. Appl. (2020). Article: 261. https://doi.org/10.1186/s13660-020-02536-0
Jolaoso, L.O., Taiwo, A., Alakoya, A., Mewomo, O.T.: A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem. Comput. Appl. Math. (2019). https://doi.org/10.1007/s40314-019-1014-2
Maingè, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal. 16, 899–912 (2008)
Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)
Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)
Moudafi, A., Thakur, B.S.: Solving proximal split feasibility problems without prior knowledge of operator norms. Opt. Lett. 8, 2099–2110 (2014)
Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18, 1159–1166 (1992)
Polyak, B. T.: Some methods of speeding up the convergence of iteration methods. U. S. S. R. Comput. Math, Math, Phy., 4(5), 1–17 (1964)
Shehu, Y., Li, X.-H., Dong, Q.-L.: An efficient projection-type method for monotone variational inequalities in Hilbert spaces. Numer. Algorithms 84, 365–388 (2020)
Shehu, Y., Gang Cai, Iyiola, O. S.: Iterative approximation of solutions for proximal split feasibility problems. Fixed Point Theory Appl., 2015, 123 (2015)
Suantai, S., Pholasa, N., Cholamjiak, P.: Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems. RACSAM 113, 1081–1099 (2019)
Suantai, S., Shehu, Y., Cholamjiak, P., Iyiola, O.S.: Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces. J. Fixed Point Theory Appl. 20, 68 (2018). https://doi.org/10.1007/s11784-018-0549-y
Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Indust. Manag. Optim. 14, 1595–1615 (2018)
Taiwo, A., Mewomo, O. T.: Inertial-viscosity-type algorithms for solving generalized equilibrium and fixed point problems in Hilbert spaces. Viet. J. Mathem. (2020)
Takahashi, W., Wen, C.-F., Yao, J.-C.: The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory 19(1), 407–420 (2018)
Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Xu, H. K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl., 26(10) (2010). Article ID 105018
Zhao, J.: Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optimization 64, 2619–2630 (2015)
Zhao, J., Zhang, H.: Solving split common fixed-point problem of firmly quasi-nonexpansive mappings without prior knowledge of operators norms. Abstr. Appl. Analy. (2014). Article ID 389689