Abstract
In this paper, we introduce a new general alternative regularization algorithm for solving split equilibrium and fixed point problems in real Hilbert spaces. The proposed method does not require a prior estimate of the norm of the bounded linear operator nor a fixed stepsize for its convergence. Instead, we employ a line search technique and prove a strong convergence result for the sequence generated by the algorithm. A numerical experiment is given to show that the proposed method converges faster in terms of number of iteration and CPU time of computation than some existing methods in the literature.
Similar content being viewed by others
References
Abbas M, Al Sharani M, Ansari QH, Iyiola OS, Shehu Y (2018) Iterative methods for solving proximal split minimization problem. Numer Algorithm 78(1):193–215
Ansari QH, Rehan A (2014) Split feasibility and fixed point problems. In: Ansari QH (ed) Nonlinear analysis: approximation theory, optimization and application. Springer, New York, pp 281–322
Bigi G, Passacantando M (2015) Descent and penalization techniques for equilibrium problems with nonlinear constraints. J Optim Theory Appl 164:804–818
Bigi G, Castellani M, Pappalardo M (2009) A new solution method for equilibrium problems. Optim Methods Softw 24:895–911
Bigi G, Castellani M, Pappalardo M, Passacantando M (2019) Nonlinear programming technique for equilibria. Spinger Nature, Switzerland
Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Stud 63:123–145
Browder FE (1968) Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull Am Math Soc 74:660–665
Bryne C (2002) Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl 18(2):441–453
Censor Y, Elfving T, Kopf N, Bortfeld T (2005) The multiple-sets split feasibility problem and its applications for inverse problmes. Inverse Probl 21:2071–2084
Censor Y, Bortfeld T, Martin B, Trofimov A (2006) A unified approach for inversion problems in intensity-modulated radiation therapy. Phys Med Biol 51:2353–2365
Cholamjiak P, Suantai S (2012) Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions. J Glob Optim 54:185–197
Cholamjiak W, Cholamjiak P, Suantai S (2015) Convergence of iterative schemes for solving fixed point of multi-valued nonself mappings and equilibrium problems. J Nonlinear Sci Appl 8:1245–1256
Combettes PL, Hirstoaga SA (2005) Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 6(1):117–136
He Z (2012) The split equilibrium problem and its convergence algorithms. J Inequal Appl 2012:162. https://doi.org/10.1186/1029-242X-2012-162
Hieu DV (2015) A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space. J Korean Math Soc 52(2):373–388
Hieu DV (2016) Common solutions to pseudomonotone equilibrium problems. Bull Iran Math Soc 42(5):1207–1219
Hieu DV (2017) Halpern subgradient extragradient method extended to equilibrium problems. Rev R Acad Cienc Exactas F’i,s Nat Ser A Math RACSAM 111:823–840
Iiduka H (2010) A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping. Optimization 59:873–885
Iiduka H, Yamada I (2009) A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J Optim 19:1881–1893
Iiduka H, Yamada I (2009) A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58:251–261
Jolaoso LO, Abass HA (2019) A viscosity-proximal gradient method with inertial extrapolation for solving certain minimization problems in Hilbert space. Arch Math 55(3):167–194
Jolaoso LO, Ogbuisi FU, Mewomo OT (2018a) An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces. Adv Pure Appl Math 9(3):167–184
Jolaoso LO, Oyewole OK, Okeke CC, Mewomo OT (2018b) A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space. Demonstr Math 51:211–232
Jolaoso LO, Taiwo A, Alakoya TO, Mewomo OT (2019) A strong convergence theorem for solving variational inequalities using an inertial viscosity subgradient extragradient algorithm with self adaptive stepsize. Demonstr Math 52:183–203
Kazmi KR, Rizvi SH (2013) Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J Egypt Math Soc 21:44–51
Kesornprom S, Cholamjiak P (2019) Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications. Optimization 68:2365–2391
Kesornprom S, Pholasa N, Cholamjiak P (2019) On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem. Numer Algorithm. https://doi.org/10.1007/s11075-019-00790-y
Konnov IV (2007) Equilibrium models and variational inequalities. Elsevier, Amsterdam
Konnov IV, Ali MSS (2006) Descent methods for monotone equilibrium problems in Banach spaces. J Comput Appl Math 188:165–179
Konnov IV, Pinyagina OV (2003) D-gap functions and descent methods for a class of monotone equilibrium problems. Lobachevskii J Math 13:57–65
Maingé PE (2008) A hybrid extragradient viscosity method for monotone operators and fixed point problems. SIAM J Control Optim 47(3):1499–1515
Maingé PE (2008) Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal 16:899–912
Maingé (2010) Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. Eur J Oper Res 205:501–506
Moudafi A (2000) Viscosity approximation methods for fixed-point problems. J Math Anal Appl 241:46–55
Moudafi A (2011) Split monotone variational inclusions. J Optim Theory Appl 150:275–283
Moudafi A, Thakur BS (2014) Solving proximal split feasibility problems without prior knowledge of operator norms. Optim Lett 8(7):2099–2110
Onjai-uea N, Phuengrattana W (2017) On solving split mixed equilibrium problems and fixed point problems of hybrid-type multivalued mappings in Hilbert spaces. J Inequal Appl 2017:137. https://doi.org/10.1186/s13660-017-1416-x
Rahaman M, Liou YC, Ahmad R, Ahmad I (2015) Convergence theorems for split equality generalized mixed equilibrium problems for demicontractive mappings. J Inequal Appl 2015:418
Shehu Y, Iyiola OS (2017) Strong convergence result for proximal split feasibility problem in Hilbert spaces. Optimization 66(12):2275–2250
Shehu Y, Iyiola OS (2018) Nonlinear iterative method for proximal split feasibility problem. Math Methods Appl Sci 41(2):781–802
Shehu Y, Ogbuisi FU (2015) Convergence analysis for proximal split feasibility problem and fixed point problems. J Appl Math Comput 48(1–2):221–239
Suntai S, Cholamjiak P, Cho YJ, Cholamjiak W (2016) On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert space. Fixed Point Theory Appl 2016:35. https://doi.org/10.1186/s13663-016-0509-4 16 pp
Taiwo A, Jolaoso LO, Mewomo OT (2019) A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces. Comput Appl Math 38(2):77
Takahashi W, Xu HK, Yao JC (2015) Iterative methods for generalized split feasibility problem in Hilbert spaces. Set Valued Var Anal 23:205–221
Tseng P (2000) A modified forward–backward splitting method for maximal monotone mappings. SIAM J Control Optim 38:431–446
Xu HK (2002) Another control condition in an iterative method for nonexpansive mappings. Bull Aust Math Soc 65(1):109–113
Yamada I (2001) The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Inherently parallel algorithm for feasibility and optimization and their applications. Stud Comput Math 8:473–504
Yang C, He S (2014) General alternative regularization method for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2014:203
Acknowledgements
L. O. Jolaoso is supported by the Postdoctoral research grant from the Sefako Makgatho Health Sciences University, South Africa. He acknowledge with thanks the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University for making their facilities available for the research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Carlos Conca.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Jolaoso, L.O., Karahan, I. A general alternative regularization method with line search technique for solving split equilibrium and fixed point problems in Hilbert spaces. Comp. Appl. Math. 39, 150 (2020). https://doi.org/10.1007/s40314-020-01178-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01178-8