Until now, a few bundle methods for general maximal monotone operators exist and they were only employed with one polyhedral approximation of the -enlargement of the maximal monotone operator considered. However, we f...
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Until now, a few bundle methods for general maximal monotone operators exist and they were only employed with one polyhedral approximation of the -enlargement of the maximal monotone operator considered. However, we find in the literature several hybrid-proximal methods which could be adapted with a great deal of bundle techniques in order to find a zero of a maximal monotone operator;yet, we could also consider the use of two polyhedral approximations. The method developed in this study has used a double polyhedral approximation at each iteration. Besides, as an application, we give a bundle method for a forward-backward type algorithm.
In this note, a new algorithm is presented for finding a zero of difference of two maximal monotone operators T and S, i.e., T - S in finite dimensional real Hilbert space H in which operator S has local boundedness p...
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In this note, a new algorithm is presented for finding a zero of difference of two maximal monotone operators T and S, i.e., T - S in finite dimensional real Hilbert space H in which operator S has local boundedness property. This condition is weaker than Moudafi's condition on operator S in [13]. Moreover, applying some conditions on inertia term in new algorithm, one can improve speed of convergence of sequence.
We propose a Newton-CG primal proximal point algorithm (PPA) for solving large scale log-determinant optimization problems. Our algorithm employs the essential ideas of PPA, the Newton method, and the preconditioned C...
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We propose a Newton-CG primal proximal point algorithm (PPA) for solving large scale log-determinant optimization problems. Our algorithm employs the essential ideas of PPA, the Newton method, and the preconditioned CG solver. When applying the Newton method to solve the inner subproblem, we find that the log-determinant term plays the role of a smoothing term as in the traditional smoothing Newton technique. Focusing on the problem of maximum likelihood sparse estimation of a Gaussian graphical model, we demonstrate that our algorithm performs favorably compared to existing state-of-the-art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints.
We develop an inexact proximal point algorithm for solving equilibrium problems in Banach spaces which consists of two principal steps and admits an interesting geometric interpretation. At a certain iterate, first we...
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We develop an inexact proximal point algorithm for solving equilibrium problems in Banach spaces which consists of two principal steps and admits an interesting geometric interpretation. At a certain iterate, first we solve an inexact regularized equilibrium problem with a flexible error criterion to obtain an axillary point. Using this axillary point and the inexact solution of the previous iterate, we construct two appropriate hyperplanes which separate the current iterate from the solution set of the given problem. Then the next iterate is defined as the Bregman projection of the initial point onto the intersection of two halfspaces obtained from the two constructed hyperplanes containing the solution set of the original problem. Assuming standard hypotheses, we present a convergence analysis for our algorithm, establishing that the generated sequence strongly and globally converges to a solution of the problem which is the closest one to the starting point of the algorithm.
This paper develops two implementations of Halpern-type proximalalgorithms(HPA1 and HPA2) solving nonsmooth *** prove their convergence to the solutions of the problems under the new *** this idea to the Basis Pursui...
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ISBN:
(纸本)9781509001668
This paper develops two implementations of Halpern-type proximalalgorithms(HPA1 and HPA2) solving nonsmooth *** prove their convergence to the solutions of the problems under the new *** this idea to the Basis Pursuit model in image/signal processing,a new Halpern-type proximalalgorithm(HPA) for the model is *** show that the Halpern-type proximalalgorithm has better descent property than the usual proximalalgorithm(PA) for the Basis Pursuit model.
To permit the stable solution of ill-posed problems, the proximal point algorithm (PPA) was introduced by Martinet (RIRO 4:154-159, 1970) and further developed by Rockafellar (SIAM J Control Optim 14:877-898, 1976). L...
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To permit the stable solution of ill-posed problems, the proximal point algorithm (PPA) was introduced by Martinet (RIRO 4:154-159, 1970) and further developed by Rockafellar (SIAM J Control Optim 14:877-898, 1976). Later on, the usual proximal distance function was replaced by the more general class of Bregman(-like) functions and related distances;see e.g. Chen and Teboulle (SIAM J Optim 3:538-543, 1993), Eckstein (Math Program 83:113-123, 1998), Kaplan and Tichatschke (Optimization 56(1-2):253-265, 2007), and Solodov and Svaiter (Math Oper Res 25:214-230, 2000). An adequate use of such generalized non-quadratic distance kernels admits to obtain an interior-point-effect, that is, the auxiliary problems may be treated as unconstrained ones. In the above mentioned works and nearly all other works related with this topic it was assumed that the operator of the considered variational inequality is a maximal monotone and paramonotone operator. The approaches of El-Farouq (JOTA 109:311-326, 2001), and Schaible et al. (Taiwan J Math 10(2):497-513, 2006) only need pseudomonotonicity (in the sense of Karamardian in JOTA 18:445-454, 1976);however, they make use of other restrictive assumptions which on the one hand contradict the desired interior-point-effect and on the other hand imply uniqueness of the solution of the problem. The present work points to the discussion of the Bregman algorithm under significantly weaker assumptions, namely pseudomonotonicity [and an additional assumption much less restrictive than the ones used by El-Farouq and Schaible et al. We will be able to show that convergence results known from the monotone case still hold true;some of them will be sharpened or are even new. An interior-point-effect is obtained, and for the generated subproblems we allow inexact solutions by means of a unified use of a summable-error-criterion and an error criterion of fixed-relative-error-type (this combination is also new in the literature).
In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condit...
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In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that Rockafellar (SIAM J Control Optim 14: 877-898, 1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial form, and this restrictive condition on errors has been extensively used so far for different versions of PPA. Thus this Note provides a solution to a long standing open problem and in particular offers new possibilities towards the approximation of the minimum points of convex functionals.
This paper focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective functio...
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This paper focuses on some customized applications of the proximal point algorithm (PPA) to two classes of problems: the convex minimization problem with linear constraints and a generic or separable objective function, and a saddle-point problem. We treat these two classes of problems uniformly by a mixed variational inequality, and show how the application of PPA with customized metric proximal parameters can yield favorable algorithms which are able to make use of the models' structures effectively. Our customized PPA revisit turns out to unify some algorithms including some existing ones in the literature and some new ones to be proposed. From the PPA perspective, we establish the global convergence and a worst-case O(1/t) convergence rate for this series of algorithms in a unified way.
A proximal point algorithm with double computational errors for treating zero points of accretive operators is investigated. Strong convergence theorems of zero points are established in a Banach space.
A proximal point algorithm with double computational errors for treating zero points of accretive operators is investigated. Strong convergence theorems of zero points are established in a Banach space.
We study a class of vector optimization problems with a C-convex objective function under linear constraints. We extend the proximal point algorithm used in scalar optimization to vector optimization. We analyze both ...
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We study a class of vector optimization problems with a C-convex objective function under linear constraints. We extend the proximal point algorithm used in scalar optimization to vector optimization. We analyze both the global and local convergence results for the new algorithm. We then apply the proximal point algorithm to a supply chain network risk management problem under bi-criteria considerations. (C) 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.
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