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Interior proximal extragradient method for equilibrium problems

OPTIMIZATION 2015年第10期64卷 2145-2161页

作者： Langenberg, Nils Univ Trier Fachbereich 4 Abt Math Trier Germany

The Bregman function-based proximal point algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additional feature consists in an application of zone coercive regularizations. The latter allows to treat the generated subproblems as unconstrained ones, albeit with a certain precaution in numerical experiments. However, compared to the (classical) proximal point algorithm for equilibrium problems, convergence results require additional assumptions which may be seen as the price to pay for unconstrained subproblems. Unfortunately, they are quite demanding - for instance, as they imply a sort of unique solvability of the given problem. The main purpose of this paper is to develop a modification of the BPPA, involving an additional extragradient step with adaptive (and explicitly given) stepsize. We prove that this extragradient step allows to leave out any of the additional assumptions mentioned above. Hence, though still of interior proximal type, the suggested method is applicable to an essentially larger class of equilibrium problems, especially including non-uniquely solvable ones.

关键词： 65K10 65J20 91A10 91A06 fixed-point problems interior-point effect proximal point algorithm Equilibrium problems rescaled Bregman distances

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PPA-Like Contraction Methods for Convex Optimization:A Framework Using Variational Inequality Approach

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Journal of the Operations Research Society of China 2015年第4期3卷 391-420页

作者： Bing-Sheng He Department of Mathematics Nanjing UniversityNanjing 210093China

Linearly constrained convex optimization has many *** first-order optimal condition of the linearly constrained convex optimization is a monotone variational inequality(VI).For solving VI,the proximal point algorithm(PPA)in Euclideannorm is classical but ***,the classical PPA only plays an important theoretical role and it is rarely used in the practical scientific *** this paper,we give a review on the recently developed customized PPA in Hnorm(H is a positive definite matrix).In the frame of customized PPA,it is easy to construct the contraction-type methods for convex optimization with different linear *** each iteration of the proposed methods,we need only to solve the proximal subproblems which have the closed form solutions or can be efficiently solved up to a high *** novel applications and numerical experiments are ***,the original primaldual hybrid gradient method is modified to a convergent algorithm by using a prediction-correction uniform *** the variational inequality approach,the contractive convergence and convergence rate proofs of the framework are more general and quite simple.

关键词： Linearly constrained convex optimization Variational inequality proximal point algorithm

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Composite iterative schemes for maximal monotone operators in reflexive Banach spaces

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FIXED point THEORY AND APPLICATIONS 2011年第1期2011卷 1-10页

作者： Cholamjiak, Prasit Cho, Yeol Je Suantai, Suthep Gyeongsang Natl Univ Dept Math Educ Chinju 660701 South Korea Gyeongsang Natl Univ RINS Chinju 660701 South Korea Univ Phayao Sch Sci Phayao 56000 Thailand Chiang Mai Univ Fac Sci Dept Math Chiang Mai 50200 Thailand

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.

关键词： Maximal monotone operator Shrinking projection method proximal point algorithm Bregman projection Totally convex function Legendre function

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A component-wise EM algorithm for mixtures

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2001年第4期10卷 697-712页

作者： Celeux, G Chrétien, S Forbes, F Mkhadri, A INRIA Rhone Alpes ZIRST F-38334 Saint Ismier France Univ Franche Comte Dept Math F-25030 Besancon France Univ Cadi Ayyad Marrahesh Fac Sci Semlalie Dept Math Marrakech Morocco

Maximum likelihood estimation in finite mixture distributions is typically approached as an incomplete data problem to allow application of the expectation-maximization (EM) algorithm. In its general formulation, the EM algorithm involves the notion of a complete data space, in which the observed measurements and incomplete data are embedded. An advantage is that many difficult estimation problems are facilitated when viewed in this way. One drawback is that the simultaneous update used by standard EM requires overly informative complete data spaces, which leads to slow convergence in some situations. In the incomplete data context, it has been shown that the use of less informative complete data spaces, or equivalently smaller missing data spaces, can lead to faster convergence without sacrifying simplicity. However, in the mixture case, little progress has been made in speeding up EM. In this article we propose a component-wise EM for mixtures. It uses, at each iteration, the smallest admissible missing data space by intrinsically decoupling the parameter updates. Monotonicity is maintained, although the estimated proportions may not sum to one during the course of the iteration. However, we prove that the mixing proportions will satisfy this constraint upon convergence. Our proof of convergence relies on the interpretation of our procedure as a proximal point algorithm. For performance comparison, we consider standard EM as well as two other algorithms based on missing data space reduction, namely the SAGE and AECME algorithms. We provide adaptations of these general procedures to the mixture case. We also consider the ECME algorithm, which is not a data augmentation scheme but still aims at accelerating EM. Our numerical experiments illustrate the advantages of the component-wise EM algorithm relative to these other methods.

关键词： EM algorithm missing data space reduction mixture estimation proximal point algorithm SAGE algorithm

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proximal interior point approach in convex programming (ILL-posed problems)

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Optimization 1999年第1-4期45卷 117-147页

作者： Kaplan, A. Tichatschke, R. Department of Mathematics University Trier D-54290 Trier Universitätsring 15 Germany

The paper deals with the theoretical analysis of a regularized logarithmic barrier method for solving ill-posed convex programming problems. In this method a multi-step proximal regularization of the auxiliary problems is performed in the framework of the interior point approach. The termination of the proximal iterations for each auxiliary problem is controlled by means of estimates, characterizing the efficiency of the iterations. A special deleting rule permits to use only a part of the constraints of the original problem for constructing the auxiliary problems. Convergence and rate of convergence of the method suggested are studied as well as its stability with respect to data perturbations. An example is given showing the behavior of the classical barrier method in the case of ill-posed convex programming problems. © 1999 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint.

关键词： Convex optimization Ill-posed problem Interior point method proximal point algorithm

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A DECOMPOSITION METHOD FOR CONVEX MINIMIZATION PROBLEMS AND ITS APPLICATION

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Acta Mathematicae Applicatae Sinica 2001年第1期17卷 20-28页

作者：徐以汎吴方 School of Management Fudan University Shanghai China he Academy of Mathematics and Systems Sciences Institute of Applied Mathematics Beijing China Institute of Applied Mathematics the Chinese Academy of Sciences Beijing China

In this paper, we present a modified decomposition algorithm and its bundle style variant for convex programming problems with separable structure. We prove that these methods are globally and linearly convergent and ... 详细信息

关键词： Convex programming bundle decomposition method proximal point algorithm

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proximal DECOMPOSITION ON THE GRAPH OF A MAXIMAL MONOTONE OPERATOR

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SIAM JOURNAL ON OPTIMIZATION 1995年第2期5卷 454-466页

作者： MAHEY, P OUALIBOUCH, S TAO, PD INST INFORMAT & INTELLIGENCE ARTIFICIELLE CH-2000 NEUCHATELSWITZERLAND INSA LMAIF-76131 MONT ST AIGNANFRANCE

We present an algorithm to solve: Find (x, y) epsilon A x A perpendicular to such that y epsilon Tx, where A is a subspace and T is a maximal monotone operator. The algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover Spingarn's decomposition method. We give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case. Numerical results performed on quadratic problems confirm the robust behaviour of the algorithm.

关键词： proximal point algorithm PARTIAL INVERSE CONVEX PROGRAMMING

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An implementation of Halpern-type proximal algorithms for nonsmooth problems

An implementation of Halpern-type proximal algorithms for no...

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2016 IEEE Advanced Information Management,Communicates,Electronic and Automation Control Conference(IMCEC 2016)

作者： Lei Sun Heng Chen Xiaodong Fan College of Information Science and Technology Bohai University 78046 Uint Chinese People's Liberation Army Department of Mathematics Bohai University

ISBN: (纸本)9781509001668

This paper develops two implementations of Halpern-type proximal algorithms(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 proximal algorithm(HPA) for the model is *** show that the Halpern-type proximal algorithm has better descent property than the usual proximal algorithm(PA) for the Basis Pursuit model.

关键词： proximal point algorithm maximal monotone operator nonsmooth problem Halpern type algorithm Subdifferential

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ACCELERATED proximal algorithmS FOR L1-MINIMIZATION PROBLEM 11

ACCELERATED PROXIMAL ALGORITHMS FOR L1-MINIMIZATION PROBLEM

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11th International Computer Conference on Wavelet Active Media Technology and Information Processing (ICCWAMTIP)

作者： Zhang, Xiao-Ya Wang, Hong-Xia Zhang, Hui Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China

ISBN: (纸本)9781479972081

Linearized Bregman algorithm is effective on solving l1-minimization problem, but its parameter's selection must rely on prior information. In order to ameliorate this weakness, we proposed a new algorithm in this paper, which combines the proximal point algorithm and the linearized Bregman iterative method. In the second part of the paper, the proposed algorithm is further accelerated through Nestrove's accelerated scheme and parameters' reset skills. Compared with the original linearized Bregman algorithm, the accelerated algorithms have better convergent speed while avoiding selecting model parameter. Simulations on sparse recovery problems show the new algorithms really have robust parameter's selections, and improve the convergent precision at the same time.

关键词： l(1)-minimization proximal point algorithm linearized Bregman iteration signal recovery Nestrove's acceleration Reset

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Variational Convexity and the Local Monotonicity of Subgradient Mappings

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VIETNAM JOURNAL OF MATHEMATICS 2019年第3期47卷 547-561页

作者： Rockafellar, R. Tyrrell Univ Washington Dept Math Box 354350 Seattle WA 98195 USA

It is well known that the subgradient mapping associated with a lower semicontinuous function is maximal monotone if and only if the function is convex, but what characterization can be given for the case in which a subgradient mapping is only maximal monotone locally instead of globally? That question is answered here in terms of a condition more subtle than local convexity. Applications are made to the tilt stability of a local minimum and to the local execution of the proximal point algorithm in optimization.

关键词： Second-order variational analysis Local maximal monotonicity Variational convexity Local optimality Tilt stability proximal point algorithm

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