The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Mihai Postolache - One of the best experts on this subject based on the ideXlab platform.
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forward backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators
Arabian Journal of Mathematics, 2020Co-Authors: Vahid Dadashi, Mihai PostolacheAbstract:In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an $$\alpha $$-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a Nonexpansive Mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a Nonexpansive Mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming.
Hideaki Iiduka - One of the best experts on this subject based on the ideXlab platform.
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acceleration of the halpern algorithm to search for a fixed point of a Nonexpansive Mapping
Fixed Point Theory and Applications, 2014Co-Authors: Kaito Sakurai, Hideaki IidukaAbstract:This paper presents an algorithm to accelerate the Halpern fixed point algorithm in a real Hilbert space. To this goal, we first apply the Halpern algorithm to the smooth convex minimization problem, which is an example of a fixed point problem for a Nonexpansive Mapping, and indicate that the Halpern algorithm is based on the steepest descent method for solving the minimization problem. Next, we formulate a novel fixed point algorithm using the ideas of conjugate gradient methods that can accelerate the steepest descent method. We show that, under certain assumptions, our algorithm strongly converges to a fixed point of a Nonexpansive Mapping. We numerically compare our algorithm with the Halpern algorithm and show that it dramatically reduces the running time and iterations needed to find a fixed point compared with that algorithm. MSC:47H10, 65K05, 90C25.
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Acceleration method for convex optimization over the fixed point set of a Nonexpansive Mapping
Mathematical Programming, 2014Co-Authors: Hideaki IidukaAbstract:The existing algorithms for solving the convex minimization problem over the fixed point set of a Nonexpansive Mapping on a Hilbert space are based on algorithmic methods, such as the steepest descent method and conjugate gradient methods, for finding a minimizer of the objective function over the whole space, and attach importance to minimizing the objective function as quickly as possible. Meanwhile, it is of practical importance to devise algorithms which converge in the fixed point set quickly because the fixed point set is the set with the constraint conditions that must be satisfied in the problem. This paper proposes an algorithm which not only minimizes the objective function quickly but also converges in the fixed point set much faster than the existing algorithms and proves that the algorithm with diminishing step-size sequences strongly converges to the solution to the convex minimization problem. We also analyze the proposed algorithm with each of the Fletcher---Reeves, Polak---Ribiere---Polyak, Hestenes---Stiefel, and Dai---Yuan formulas used in the conventional conjugate gradient methods, and show that there is an inconvenient possibility that their algorithms may not converge to the solution to the convex minimization problem. We numerically compare the proposed algorithm with the existing algorithms and show its effectiveness and fast convergence.
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Approximate solutions to variational inequality over the fixed point set of a strongly Nonexpansive Mapping
Fixed Point Theory and Applications, 2014Co-Authors: Shigeru Iemoto, Kazuhiro Hishinuma, Hideaki IidukaAbstract:Variational inequality problems over fixed point sets of Nonexpansive Mappings include many practical problems in engineering and applied mathematics, and a number of iterative methods have been presented to solve them. In this paper, we discuss a variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a strongly Nonexpansive Mapping on a real Hilbert space. We then present an iterative algorithm, which uses the strongly Nonexpansive Mapping at each iteration, for solving the problem. We show that the algorithm potentially converges in the fixed point set faster than algorithms using firmly Nonexpansive Mappings. We also prove that, under certain assumptions, the algorithm with slowly diminishing step-size sequences converges to a solution to the problem in the sense of the weak topology of a Hilbert space. Numerical results demonstrate that the algorithm converges to a solution to a concrete variational inequality problem faster than the previous algorithm.
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three term conjugate gradient method for the convex optimization problem over the fixed point set of a Nonexpansive Mapping
Applied Mathematics and Computation, 2011Co-Authors: Hideaki IidukaAbstract:Many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain Nonexpansive Mapping, and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set. This paper presents a novel gradient method with a three-term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems. It is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions. Numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm.
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iterative algorithm for solving triple hierarchical constrained optimization problem
Journal of Optimization Theory and Applications, 2011Co-Authors: Hideaki IidukaAbstract:Many practical problems such as signal processing and network resource allocation are formulated as the monotone variational inequality over the fixed point set of a Nonexpansive Mapping, and iterative algorithms to solve these problems have been proposed. This paper discusses a monotone variational inequality with variational inequality constraint over the fixed point set of a Nonexpansive Mapping, which is called the triple-hierarchical constrained optimization problem, and presents an iterative algorithm for solving it. Strong convergence of the algorithm to the unique solution of the problem is guaranteed under certain assumptions.
Poom Kumam - One of the best experts on this subject based on the ideXlab platform.
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algorithms for solving the variational inequality problem over the triple hierarchical problem
Abstract and Applied Analysis, 2012Co-Authors: Thanyarat Jitpeera, Poom KumamAbstract:This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a Nonexpansive Mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions.
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a modified halpern typeiterative method of a system of equilibrium problems and a fixed point for a totally quasi asymptotically Nonexpansive Mapping in a banach space
Journal of Applied Mathematics, 2012Co-Authors: Preedaporn Kanjanasamranwong, Poom Kumam, Siwaporn SaewanAbstract:The purpose of this paper is to introduce the modified Halpern-type iterative method by the generalized f-projection operator for finding a common solution of fixed-point problem of a totally quasi-
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a new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a Nonexpansive Mapping
Journal of Applied Mathematics and Computing, 2009Co-Authors: Poom KumamAbstract:In this paper, we introduce an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a Nonexpansive Mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone Mappings in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under some parametric controlling conditions by the new hybrid method which is introduced by Takahashi et al. (J. Math. Anal. Appl., doi: 10.1016/j.jmaa.2007.09.062, 2007). The results are connected with Tada and Takahashi’s result [A. Tada and W. Takahashi, Weak and strong convergence theorems for a Nonexpansive Mappings and an equilibrium problem, J. Optim. Theory Appl. 133, 359–370, 2007]. Moreover, our result is applicable to a wide class of Mappings.
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a hybrid approximation method for equilibrium and fixed point problems for a monotone Mapping and a Nonexpansive Mapping
Nonlinear Analysis: Hybrid Systems, 2008Co-Authors: Poom KumamAbstract:The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the set of fixed points of a Nonexpansive Mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone Mappings in the framework of a Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under appropriate conditions. Additionally, the idea of our results are applied to find a zero of a maximal monotone operator and a strictly pseudocontractive Mapping in a real Hilbert space.
Meijuan Shang - One of the best experts on this subject based on the ideXlab platform.
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Three-step iterations for Nonexpansive Mappings and inverse-strongly monotone Mappings
Journal of Systems Science and Complexity, 2009Co-Authors: Meijuan Shang, Xiaolong QinAbstract:This paper introduces a three-step iteration for finding a common element of the set of fixed points of a Nonexpansive Mapping and the set of solutions of the variational inequality for an inverse-strongly monotone Mapping by viscosity approximation methods in a Hilbert space. The authors show that the iterative sequence converges strongly to a common element of the two sets, which solves some variational inequality. Subsequently, the authors consider the problem of finding a common fixed point of a Nonexpansive Mapping and a strictly pseudo-contractive Mapping and the problem of finding a common element of the set of fixed points of a Nonexpansive Mapping and the set of zeros of an inverse-strongly monotone Mapping. The results obtained in this paper extend and improve the corresponding results announced by Nakajo, Takahashi, and Toyoda.
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A general iterative scheme for Nonexpansive Mappings and inverse-strongly monotone Mappings
Journal of Applied Mathematics and Computing, 2008Co-Authors: Meijuan Shang, Xiaolong QinAbstract:In this paper, we introduce an general iterative scheme for finding a common element of the set of fixed points of a Nonexpansive Mapping and the set of solutions of the variational inequality for an inverse-strongly monotone Mapping in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the two sets. Using this results, we consider the problem of finding a common fixed point of a Nonexpansive Mapping and a strictly pseudocontractive Mapping and the problem of finding a common element of the set of fixed points of a Nonexpansive Mapping and the set of zeros of an inverse-strongly monotone Mapping. The results of this paper extended and improved the results of Iiduka and Takahashi (Nonlinear Anal. 61:341–350, 2005).
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strong convergence of monotone hybrid algorithm for hemi relatively Nonexpansive Mappings
Fixed Point Theory and Applications, 2008Co-Authors: Dongxing Wang, Meijuan ShangAbstract:The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively Nonexpansive Mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively Nonexpansive Mapping, but it cannot be used for hemi-relatively Nonexpansive Mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others.
Wojciech M Kozlowski - One of the best experts on this subject based on the ideXlab platform.
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pointwise lipschitzian Mappings in uniformly convex and uniformly smooth banach spaces
Nonlinear Analysis-theory Methods & Applications, 2013Co-Authors: Wojciech M KozlowskiAbstract:Abstract Let X be a uniformly convex and uniformly smooth Banach space. Let T : C → C be an asymptotic pointwise Nonexpansive Mapping, where C is a bounded, closed and convex subset of X . In this paper we investigate conditions sufficient for the weak convergence of the generalized Mann and Ishikawa processes to a fixed point of T .
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fixed point iteration processes for asymptotic pointwise Nonexpansive Mapping in modular function spaces
Fixed Point Theory and Applications, 2012Co-Authors: Buthinah Bin A Dehaish, Wojciech M KozlowskiAbstract:Let Lρ be a uniformly convex modular function space with a strong Opial property. Let T : C → C be an asymptotic pointwise Nonexpansive Mapping, where C is a ρ-a.e. compact convex subset of Lρ. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point. MSC: Primary 47H09; Secondary 47H10
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fixed point iteration processes for asymptotic pointwise Nonexpansive Mapping in modular function spaces
Fixed Point Theory and Applications, 2012Co-Authors: Buthinah Bin A Dehaish, Wojciech M KozlowskiAbstract:Let L ρ Open image in new window be a uniformly convex modular function space with a strong Opial property. Let T : C → C Open image in new window be an asymptotic pointwise Nonexpansive Mapping, where C is a ρ-a.e. compact convex subset of L ρ Open image in new window. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.
