Some fundamental properties of resolvents of monotone operators in Banach spaces are investigated. Using them, we study the asymptotic behavior of the sequences generated by two modifications of the proximalpoint alg...
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Some fundamental properties of resolvents of monotone operators in Banach spaces are investigated. Using them, we study the asymptotic behavior of the sequences generated by two modifications of the proximal point algorithm for monotone operators satisfying a range condition defined in Banach spaces.
In the paper, we introduce two iterative sequences for finding a point in the intersection of the zero set of a inverse strongly monotone or inverse-monotone operator and the zero set of a maximal monotone operator in...
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In the paper, we introduce two iterative sequences for finding a point in the intersection of the zero set of a inverse strongly monotone or inverse-monotone operator and the zero set of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. We prove weak convergence theorems under appropriate conditions, respectively. (C) 2006 Elsevier Ltd. All rights reserved.
In this paper, we introduce two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applicatio...
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In this paper, we introduce two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we consider a problem of finding a minimizer of a convex function.
The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35-61] has revealed that the sequence generated by the method of alternating projections converges...
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The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35-61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, A and B, that can be used to approximate common zeros of A and B. In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals. (C) 2011 Elsevier Ltd. All rights reserved.
We propose two algorithms for finding (common) zeros of finitely many maximal monotone mappings in reflexive Banach spaces. These algorithms are based on the Bregman distance related to a well-chosen convex function a...
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We propose two algorithms for finding (common) zeros of finitely many maximal monotone mappings in reflexive Banach spaces. These algorithms are based on the Bregman distance related to a well-chosen convex function and improve previous results. Finally, we mention two applications of our algorithms for solving equilibrium problems and convex feasibility problems.
Due to its significant efficiency, the alternating direction method (ADM) has attracted a lot of attention in solving linearly constrained structured convex optimization. In this paper, in order to make implementation...
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Due to its significant efficiency, the alternating direction method (ADM) has attracted a lot of attention in solving linearly constrained structured convex optimization. In this paper, in order to make implementation of ADM relatively easy, some linearized proximal ADMs are proposed and the associated convergence results of the proposed linearized proximal ADMs are given. Additionally, theoretical analysis shows that the relaxation factor for the linearized proximal ADMs can have the same restriction region as that for the general ADM.
Given a Hilbert space H and a closed convex function Phi : H -> R boolean OR {+infinity}, we consider the inertial proximalalgorithm x(n+1) - x(n) - alpha(n)(x(n) - x(n-1)) + beta(n)partial derivative Phi(x(n+1)) ...
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Given a Hilbert space H and a closed convex function Phi : H -> R boolean OR {+infinity}, we consider the inertial proximalalgorithm x(n+1) - x(n) - alpha(n)(x(n) - x(n-1)) + beta(n)partial derivative Phi(x(n+1)) (sic) 0, (A) where (alpha(n)) and (beta(n)) are nonnegative sequences. The notation partial derivative Phi stands for the subdifferential of Phi in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to inf Phi as n -> +infinity. When Phi has a unique minimum, the question of the convergence of (x(n)) is solved. In the case of multiple minima, it is proved that if (Pi k=1n alpha k) is not an element of l(1) and if a suitable geometric condition on the set argmin Phi is fulfilled, then non stationary sequences of (A) cannot converge.
In this article, we introduce composite iterative schemes for finding a zero point of a finite family of maximal monotone operators in a reflexive Banach space. Then, we prove strong convergence theorems by using a sh...
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In this article, we introduce composite iterative schemes for finding a zero point of a finite family of maximal monotone operators in a reflexive Banach space. Then, we prove strong convergence theorems by using a shrinking projection method. Moreover, we also apply our results to a system of convex minimization problems in reflexive Banach spaces.
A regularization method for the proximal point algorithm of finding a zero for a maximal monotone operator in a Hilbert space is proposed. Strong convergence of this algorithm is proved.
A regularization method for the proximal point algorithm of finding a zero for a maximal monotone operator in a Hilbert space is proposed. Strong convergence of this algorithm is proved.
Two strong convergence theorems for a proximal method for finding common zeroes of maximal monotone operators in reflexive Banach spaces are established. Both theorems take into account possible computational errors.
Two strong convergence theorems for a proximal method for finding common zeroes of maximal monotone operators in reflexive Banach spaces are established. Both theorems take into account possible computational errors.
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