The Experts below are selected from a list of 15033 Experts worldwide ranked by ideXlab platform
Ming Tian - One of the best experts on this subject based on the ideXlab platform.
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regularized gradient projection methods for finding the minimum norm solution of the constrained Convex Minimization Problem
Journal of Inequalities and Applications, 2017Co-Authors: Ming Tian, Huifang ZhangAbstract:Let H be a real Hilbert space and C be a nonempty closed Convex subset of H. Assume that g is a real-valued Convex function and the gradient ∇g is $\frac{1}{L}$ -ism with $L>0$ . Let $0<\lambda <\frac{2}{L+2}$ , $0<\beta_{n}<1$ . We prove that the sequence $\{x_{n}\} $ generated by the iterative algorithm $x_{n+1}=P_{C}(I-\lambda(\nabla g+\beta_{n}I))x_{n}$ , $\forall n\geq0$ converges strongly to $q\in U$ , where $q=P_{U}(0)$ is the minimum-norm solution of the constrained Convex Minimization Problem, which also solves the variational inequality $\langle-q, p-q\rangle\leq0$ , $\forall p\in U$ . Under suitable conditions, we obtain some strong convergence theorems. As an application, we apply our algorithm to solving the split feasibility Problem in Hilbert spaces.
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weak convergence theorem for variational inequality Problems with monotone mapping in hilbert space
Journal of Inequalities and Applications, 2016Co-Authors: Ming Tian, Bingnan JiangAbstract:We know that variational inequality Problem is very important in the nonlinear analysis. The main purpose of this paper is to propose an iterative method for finding an element of the set of solutions of a variational inequality Problem with a monotone and Lipschitz continuous mapping in Hilbert space. This iterative method is based on the extragradient method. We get a weak convergence theorem. Using this result, we obtain three weak convergence theorems for the equilibrium Problem, the constrained Convex Minimization Problem, and the split feasibility Problem.
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regularized gradient projection methods for the constrained Convex Minimization Problem and the zero points of maximal monotone operator
Fixed Point Theory and Applications, 2015Co-Authors: Ming Tian, Siwen JiaoAbstract:In this paper, based on the viscosity approximation method and the regularized gradient-projection algorithm, we find a common element of the solution set of a constrained Convex Minimization Problem and the set of zero points of the maximal monotone operator Problem. In particular, the set of zero points of the maximal monotone operator Problem can be transformed into the equilibrium Problem. Under suitable conditions, new strong convergence theorems are obtained, which are useful in nonlinear analysis and optimization. As an application, we apply our algorithm to solving the split feasibility Problem and the constrained Convex Minimization Problem in Hilbert spaces.
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strong convergence of modified algorithms based on the regularization for the constrained Convex Minimization Problem
Abstract and Applied Analysis, 2014Co-Authors: Ming Tian, Junying GongAbstract:As is known, the regularization method plays an important role in solving constrained Convex Minimization Problems. Based on the idea of regularization, implicit and explicit iterative algorithms are proposed in this paper and the sequences generated by the algorithms can converge strongly to a solution of the constrained Convex Minimization Problem, which also solves a certain variational inequality. As an application, we also apply the algorithm to solve the split feasibility Problem.
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general iterative methods for equilibrium and constrained Convex Minimization Problem
Optimization, 2014Co-Authors: Ming Tian, Lei LiuAbstract:The gradient-projection algorithm (GPA) plays an important role in solving constrained Convex Minimization Problems. Based on Marino and Xu's method [G. Marino and H.-K. Xu, A general method for nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 318 (2006), pp. 43–52], we combine GPA and averaged mapping approach to propose implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium and a constrained Convex Minimization Problem for the first time in this article. Under suitable conditions, strong convergence theorems are obtained.
Fumiaki Kohsaka - One of the best experts on this subject based on the ideXlab platform.
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an implicitly defined iterative sequence for monotone operators in banach spaces
Journal of Inequalities and Applications, 2014Co-Authors: Fumiaki KohsakaAbstract:Given a monotone operator in a Banach space, we show that an iterative sequence, which is implicitly defined by a fixed point theorem for mappings of firmly nonexpansive type, converges strongly to a minimum norm zero point of the given operator. Applications to a Convex Minimization Problem and a variational inequality Problem are also included.
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weak and strong convergence theorems for maximal monotone operators in a banach space
Set-valued Analysis, 2004Co-Authors: Shoji Kamimura, Fumiaki Kohsaka, Wataru TakahashiAbstract:In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly Convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim.14 (1976), 877–898] and Kamimura and Takahashi [J. Approx. Theory106 (2000), 226–240]. Finally, we apply our convergence theorem to the Convex Minimization Problem and the variational inequality Problem.
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strong convergence of an iterative sequence for maximal monotone operators in a banach space
Abstract and Applied Analysis, 2004Co-Authors: Fumiaki KohsakaAbstract:We first introduce a modified proximal point algorithm for maximal monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of maximal monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the Convex Minimization Problem and the variational inequality Problem in a Banach space.
Siwen Jiao - One of the best experts on this subject based on the ideXlab platform.
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regularized gradient projection methods for the constrained Convex Minimization Problem and the zero points of maximal monotone operator
Fixed Point Theory and Applications, 2015Co-Authors: Ming Tian, Siwen JiaoAbstract:In this paper, based on the viscosity approximation method and the regularized gradient-projection algorithm, we find a common element of the solution set of a constrained Convex Minimization Problem and the set of zero points of the maximal monotone operator Problem. In particular, the set of zero points of the maximal monotone operator Problem can be transformed into the equilibrium Problem. Under suitable conditions, new strong convergence theorems are obtained, which are useful in nonlinear analysis and optimization. As an application, we apply our algorithm to solving the split feasibility Problem and the constrained Convex Minimization Problem in Hilbert spaces.
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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weak convergence of a projection algorithm for variational inequalities in a banach space
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Hideaki Iiduka, Wataru TakahashiAbstract:Abstract Let C be a nonempty, closed Convex subset of a Banach space E. In this paper, motivated by Alber [Ya.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, in: Lecture Notes Pure Appl. Math., vol. 178, Dekker, New York, 1996, pp. 15–50], we introduce the following iterative scheme for finding a solution of the variational inequality Problem for an inverse-strongly-monotone operator A in a Banach space: x 1 = x ∈ C and x n + 1 = Π C J −1 ( J x n − λ n A x n ) for every n = 1 , 2 , … , where Π C is the generalized projection from E onto C, J is the duality mapping from E into E ∗ and { λ n } is a sequence of positive real numbers. Then we show a weak convergence theorem (Theorem 3.1). Finally, using this result, we consider the Convex Minimization Problem, the complementarity Problem, and the Problem of finding a point u ∈ E satisfying 0 = A u .
Yekini Shehu - One of the best experts on this subject based on the ideXlab platform.
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a convergence analysis result for constrained Convex Minimization Problem
Optimization, 2015Co-Authors: Yekini ShehuAbstract:AbstractOur purpose in this paper is to obtain strong convergence result for approximation of solution to constrained Convex Minimization Problem using a new iterative scheme in a real Hilbert space. Furthermore, we give numerical analysis of our iterative scheme.
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Approximation of Solutions to Constrained Convex Minimization Problem in Hilbert Spaces
Vietnam Journal of Mathematics, 2015Co-Authors: Yekini ShehuAbstract:The idea of this paper is to perturb Mann iteration scheme and obtain a strong convergence result for approximation of solutions to constrained Convex Minimization Problem in a real Hilbert space. Furthermore, we give computational analysis of our iterative scheme.
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Iterative approximation of solutions for constrained Convex Minimization Problem
Arabian Journal of Mathematics, 2013Co-Authors: Yekini Shehu, Olaniyi S. Iyiola, Cyril Dennis EnyiAbstract:In this paper, we propose a new iterative scheme for finding a minimizer of a constrained Convex Minimization Problem and prove that the sequence generated by our new scheme converges strongly to a solution of the constrained Convex Minimization Problem in a real Hilbert space.