Acceleration of the Modified \(S\)-Algorithm to Search for a Fixed Point of a Nonexpansive Mapping

D. Kitkuan, J. Zhao, H. Zong, W. Kumam


The purpose of this paper is to present accelerations of the \(S\)-algorithm. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerated the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the \(S\)-algorithm. Under certain assumptions, our algorithm strongly converges to a fixed point with the S-algorithm and show that it dramatically reduces the running time and iteration needed to find a fixed point compared with that algorithm.


Demicontractive mappings; Common fixed point

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R. P. Agarwal, D. O’Regan and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8(1) (2007), 61 – 79, DOI: 10.1016/0022-247X(91)90245-U.

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim. 14 (2004), 773 – 782, DOI: 10.1137/S1052623403427859.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2011), DOI: 10.1007/978-3-319-48311-5.

R. I. Bo¸t, E. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput. 256 (2015), 472 – 487, DOI: 10.1016/j.amc.2015.01.017.

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Dekker, New York (1984), DOI: 10.1112/blms/17.3.293.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990), DOI: 10.1017/CBO9780511526152.

H. Iiduka, Iterative algorithm for solving triple-hierarchical constrained optimization problem, J. Optim. Theory Appl. 148 (2011), 580 – 592, DOI: 10.1007/s10957-010-9769-z.

M. A. Krasnoselski, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk. 10 (1955), 123 – 127, DOI: 10.1016/S0076-5392(08)61895-0.

P. Kumam, Strong convergence theorems by an extra gradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turkish Journal of Mathematics 33(1) (2009), 85 – 98, DOI: 10.3906/mat-0804-11.

W. R. Mann, Mean value methods in iteration, Bull. Amer. Math. Soc. 4 (1953), 506–510, DOI: 10.2307/2032162.

A. Mouad and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Lett. 8(7) (2014), 2099–2110, DOI: 10.1007/s11590-013-0708-4.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, Berlin (2006).

Z. Opial,Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bulletin of the American Mathematical Society 73 (1967), 591 – 597, DOI: 10.1090/S0002-9904-1967-11761-0.

S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009), 614 – 621, DOI: 10.1016/

K. Sakurai and H. Liduka, Acceleration of the Halpern algorithm to search for a fixed point of a nonexpansive mapping, Fixed Point Theory Appl. 2014 (2014), 202, DOI: 10.1186/1687-1812-2014-202.

Y. Shehu, G. Cai and O. S. Iyiola, Iterative approximation of solutions for proximal split feasibility problems, Fixed Point Theory Appl. 2015 (2015), 123, DOI: 10.1186/s13663-015-0375-5.

T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305(1) (2005), 227 – 239, DOI: 10.1016/j.jmaa.2004.11.017.

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

K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301 – 308, DOI: 10.1006/jmaa.1993.1309.

H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), 279 – 291, DOI: 10.1016/j.jmaa.2004.04.059.

Y. Yao, G. Marino and L. Muglia, A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality, Optimization 63(4) (2012), 1 – 11, DOI: 10.1080/02331934.2012.674947.

Z. Yao, S. Y. Cho, S. M. Kang and L.-J. Zhu, A regularized algorithm for the proximal split feasibility problem, Abstract Appl. Anal. 2014 (2014), Article ID 894272, 6 pages, DOI: 10.1155/2014/894272.

Y. Yao, Z. Yao, A. A. Abdou and Y. J. Cho, Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis, Fixed Point Theory Appl. 2015(2015), 205, DOI: 10.1186/s13663-015-0462-7.


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