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SOLVING LOG-DETERMINANT OPTIMIZATION PROBLEMS BY A NEWTON-CG PRIMAL proximal point algorithm

SIAM JOURNAL ON OPTIMIZATION 2010年第6期20卷 2994-3013页

作者： Wang, Chengjing Sun, Defeng Toh, Kim-Chuan SW Jiaotong Univ Coll Math Chengdu 610031 Peoples R China Natl Univ Singapore Dept Math Singapore 119076 Singapore Natl Univ Singapore NUS Risk Management Inst Singapore 119076 Singapore Singapore MIT Alliance Singapore 117576 Singapore

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.

关键词： log-determinant optimization problem sparse inverse covariance selection proximal point algorithm Newton's method

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STRONG CONVERGENCE OF AN INEXACT proximal point algorithm FOR EQUILIBRIUM PROBLEMS IN BANACH SPACES

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2010年第9期31卷 1053-1071页

作者： Mashreghi, Javad Nasri, Mostafa Univ Laval Dept Math & Stat Fac Sci & Genie Quebec City PQ G1V 0A6 Canada

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.

关键词： Bregman distance Bregman projection Equilibrium problem Inexact solution proximal point algorithm Strong convergence

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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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Pseudomonotone operators and the Bregman proximal point algorithm

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JOURNAL OF GLOBAL OPTIMIZATION 2010年第4期47卷 537-555页

作者： Langenberg, Nils Univ Trier Dept Math D-54286 Trier Germany

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).

关键词： Pseudomonotone operators Variational inequalities Bregman distances proximal point algorithm Interior-point-effect

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A proximal point algorithm converging strongly for general errors

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OPTIMIZATION LETTERS 2010年第4期4卷 635-641页

作者： Boikanyo, O. A. Morosanu, G. Cent European Univ Dept Math & Its Applicat H-1051 Budapest Hungary

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.

关键词： proximal point algorithm Prox-Tikhonov algorithm Monotone operator Regularization parameters Strong convergence

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Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2014年第1-2期59卷 135-161页

作者： Gu, Guoyong He, Bingsheng Yuan, Xiaoming Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Nanjing Univ Int Ctr Management Sci & Engn Nanjing 210093 Jiangsu Peoples R China Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

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.

关键词： Convex minimization Saddle-point problem proximal point algorithm Convergence rate Customized algorithms Splitting algorithms

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proximal point algorithms for zero points of nonlinear operators

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FIXED point THEORY AND APPLICATIONS 2014年第1期2014卷 42-42页

作者： Qing, Yuan Cho, Sun Young Hangzhou Normal Univ Dept Math Hangzhou 310036 Zhejiang Peoples R China Gyeongsang Natl Univ Dept Math Jinju 660701 South Korea

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.

关键词： accretive operator fixed point nonexpansive mapping proximal point algorithm zero point

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A new algorithm for linearly constrained c-convex vector optimization with a supply chain network risk application

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2015年第2期247卷 359-365页

作者： Qu, Shaojian Goh, Mark Ji, Ying De Souza, Robert Shanghai Univ Sci & Technol Sch Business Shanghai 200093 Peoples R China Harbin Inst Technol Ctr Nat Sci Harbin Heilongjiang Peoples R China Natl Univ Singapore Logist Inst Asia Pacific Singapore 117548 Singapore Natl Univ Singapore NUS Business Sch Singapore 117548 Singapore

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.

关键词： Multiple objective programming Pareto optimum C-convex proximal point algorithm Supply chain network risk management

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CONVERGENCE RATE ANALYSIS OF PRIMAL-DUAL SPLITTING SCHEMES

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SIAM JOURNAL ON OPTIMIZATION 2015年第3期25卷 1912-1943页

作者： Davis, Damek Univ Calif Los Angeles Dept Math Los Angeles CA 90025 USA Cornell Univ Sch Operat Res & Informat Engn Ithaca NY 14850 USA

Primal-dual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They decompose problems that are built from sums, linear compositions, and infimal convolutions of simple functions so that each simple term is processed individually via proximal mappings, gradient mappings, and multiplications by the linear maps. This leads to easily implementable and highly parallelizable or distributed algorithms, which often obtain nearly state-of-the-art performance. In this paper, we analyze a monotone inclusion problem that captures a large class of primal-dual splittings as a special case. We introduce a unifying scheme and use some abstract analysis of the algorithm to prove convergence rates of the proximal point algorithm, forward-backward splitting, Peaceman-Rachford splitting, and forward-backward-forward splitting applied to the model problem. Our ergodic convergence rates are deduced under variable metrics, stepsizes, and relaxation parameters. Our nonergodic convergence rates are the first shown in the literature. Finally, we apply our results to a large class of primal-dual algorithms that are a special case of our scheme and deduce their convergence rates.

关键词： primal-dual algorithms convergence rates proximal point algorithm forward-backward splitting forward-backward-forward splitting nonexpansive operator Douglas-Rachford splitting Peaceman-Rachford splitting averaged operator fixed-point algorithm

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THE PROX-TIKHONOV-LIKE FORWARD-BACKWARD METHOD AND APPLICATIONS

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TAIWANESE JOURNAL OF MATHEMATICS 2015年第2期19卷 481-503页

作者： Sahu, D. R. Ansari, Q. H. Yao, J. C. Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 804 Taiwan Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 807 Taiwan King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia

It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi [Optimization, 37(1996), 239-252] introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with non-summable error sequence is also discussed.

关键词： Accretive operator Maximal monotone operator Metric projection mapping proximal point algorithm Regularization method Resolvent identity Strong convergence Uniformly Gateaux differentiable norm

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