In this paper, we present a logarithmic-quadratic proximal (LQP) type prediction-correction methods for solving constrained variational inequalities VI(S,f), where S is a convex set with linear constraints. The comput...
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In this paper, we present a logarithmic-quadratic proximal (LQP) type prediction-correction methods for solving constrained variational inequalities VI(S,f), where S is a convex set with linear constraints. The computational load in each iteration is quite tiny. However, the number of iterations is significantly dependent on a parameter which balances the primal and dual variables. We then propose a self-adaptive prediction-correction method that adjusts the scalar parameter automatically. Under certain conditions, the global convergence of the proposed method is established. In order to demonstrate the efficiency of the proposed method, we provide numerical results for a convex nonlinear programming and traffic equilibrium problems. (c) 2006 Elsevier Inc. All rights reserved.
In the alternating directions method, the relaxation factor gamma is an element of (0, root 5+1/2) by Glowinski is useful in practical computations for structured variational inequalities. This paper points out that t...
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In the alternating directions method, the relaxation factor gamma is an element of (0, root 5+1/2) by Glowinski is useful in practical computations for structured variational inequalities. This paper points out that the same restriction region of the relaxation factor is also valid in the proximal alternating directions method.
We study stability properties of a proximal point algorithm for solving the inclusion 0 is an element of T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergenc...
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We study stability properties of a proximal point algorithm for solving the inclusion 0 is an element of T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of our algorithm is uniform, in the sense that it is stable under small perturbations whenever the set-valued mapping T is metrically regular at a given solution. We present also an inexact proximalpoint method for strongly metrically subregular mappings and show that it is super-linearly convergent to a solution to the inclusion 0 is an element of T(x). (C) 2007 Elsevier Inc. All rights reserved.
We introduce the concept of hypomonotone point-to-set operators in Banach spaces, with respect to a regularizing function. This notion coincides with the one given by Rockafellar and Wets in Hilbertian spaces, when th...
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We introduce the concept of hypomonotone point-to-set operators in Banach spaces, with respect to a regularizing function. This notion coincides with the one given by Rockafellar and Wets in Hilbertian spaces, when the regularizing function is the square of the norm. We study the associated proximal mapping, which leads to a hybrid proximal-extragradient and proximal-projection methods for nonmonotone operators in reflexive Banach spaces. These methods allow for inexact solution of the proximal subproblems with relative error criteria. We then consider the notion of local hypomonotonicity and propose localized versions of the algorithms, which are locally convergent. (C) 2006 Elsevier Inc. All rights reserved.
In this paper, we first prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method, which is a generalization of the results of Reich [J. Math. Ana...
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In this paper, we first prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method, which is a generalization of the results of Reich [J. Math. Anal. Appl. 75 (1980), 287-292], and Takahashi and Ueda [J. Math. Anal. Appl. 104 (1984), 546-553]. Further using this result, we consider the proximal point algorithm in a Banach space by the viscosity approximation method, and obtain a strong convergence theorem which is a generalization of the result of Kamimura and Takahashi [Set-Valued Anal. 8 (2000), 361-374].
Since proximal point algorithms (abbreviated as PPA) are attractive for solving monotone variational inequality problems, various approximate versions of PPA (APPA) are developed for practical applications. In this pa...
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Since proximal point algorithms (abbreviated as PPA) are attractive for solving monotone variational inequality problems, various approximate versions of PPA (APPA) are developed for practical applications. In this paper, we make a comparison between two different versions of APPAs (APPA I and APPA II) in the literature which share some common properties. Both of the algorithms use the same inexactness restriction and the same step length. The only difference is that they use different search directions. Through theoretical analysis and numerical experiment, we can see that APPA II usually performs better than APPA I. (c) 2006 Elsevier Ltd. All rights reserved.
In this paper we present some algorithms for minimization of DC function (difference of two convex functions). They are descent methods of the proximal-type which use the convex properties of the two convex functions ...
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In this paper we present some algorithms for minimization of DC function (difference of two convex functions). They are descent methods of the proximal-type which use the convex properties of the two convex functions separately. We also consider an approximate proximal point algorithm. Some properties of the ε-subdifferential and the ε-directional derivative are discussed. The convergence properties of the algorithms are established in both exact and approximate forms. Finally, we give some applications to the concave programming and maximum eigenvalue problems.
In this paper, we first introduce an iterative sequence of Mann's type and Halpern's type for finding a zero point of an m-accretive operator in a real Banach space. Then we obtain the strong and weak converge...
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In this paper, we first introduce an iterative sequence of Mann's type and Halpern's type for finding a zero point of an m-accretive operator in a real Banach space. Then we obtain the strong and weak convergence by changing control conditions of the sequence. The result improves and extends a strong convergence theorem and a weak convergence theorem obtained by Kamimura and Takahashi [9], simultaneously.
In this paper, we introduce a new class of equilibrium problems known as the multivalued regularized equilibrium problems. We use the auxiliary principle technique to suggest some iterative methods for solving multiva...
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In this paper, we introduce a new class of equilibrium problems known as the multivalued regularized equilibrium problems. We use the auxiliary principle technique to suggest some iterative methods for solving multivalued regularized equilibrium problems. The convergence of the proposed methods is studied under some mild conditions. As special cases, we obtain a number of known and new results for solving various classes of regularized equilibrium problems and related optimization problems.
Some algorithms in signal and image processing may be formulated in the Krasnoselski-Mann ( KM) iteration form and the KM theorem asserts the convergence of this iteration under certain assumptions. We give more gener...
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Some algorithms in signal and image processing may be formulated in the Krasnoselski-Mann ( KM) iteration form and the KM theorem asserts the convergence of this iteration under certain assumptions. We give more general iterative schemes which include the KM iteration as a special case and establish the convergence of extended iterations. Based on the generalized KM theorems, some algorithms with a broader scope are analysed and treated in new settings.
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