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About the relaxed cocoercivity and the convergence of the proximal point algorithm

Journal of the Egyptian Mathematical Society 2013年第3期21卷 281-284页

作者： Abdellatif Moudafi Zhenyu Huang French West Indies University Ceregmia-Scientific Department 97200 Schelcher Martinique France Department of Mathematics Nanjing University Nanjing 210093 PR China

The aim of this paper is to study the convergence of two proximal algorithms via the notion of ( α , r )-relaxed cocoercivity without Lipschitzian continuity. We will show that this notion is enough to obtain some interesting convergence theorems without any Lipschitz-continuity assumption. The relaxed cocoercivity case is also investigated.

关键词： Primary 49J53 65K10 Secondary 49M37 90C25 Relaxed cocoercivity Relaxed monotonicity proximal point algorithm Strong convergence Variational inequalities

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A Projection proximal-point algorithm for 1 Minimization

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2010年第2期31卷 172-190页

作者： Lorenz, Dirk A. TU Braunschweig Inst Anal & Algebra D-38092 Braunschweig Germany

The problem of the minimization of least squares functionals with 1 penalties is considered in an infinite dimensional Hilbert space setting. Though there are several algorithms available in the finite dimensional setting there are only a few of them that come with a proper convergence analysis in the infinite dimensional setting. In this work we provide an algorithm from a class that has not been considered for 1 minimization before, namely, a proximal-point method in combination with a projection step. We show that this idea gives a simple and easy-to-implement algorithm. We present experiments that indicate that the algorithm may perform better than other algorithms if we employ them without any special tricks. Hence, we may conclude that the projection proximal-point idea is a promising idea in the context of 1 minimization.

关键词： proximal point algorithm Sparse reconstruction Sparsity constraint

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BREGMAN proximal point algorithm REVISITED: A NEW INEXACT VERSION AND ITS INERTIAL VARIANT

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SIAM JOURNAL ON OPTIMIZATION 2022年第3期32卷 1523-1554页

作者： Yang, L. E., I Toh, Kim-chuan Hong Kong Polytech Univ Dept Appl Math Hung Hom Kowloon Hong Kong Peoples R China Natl Univ Singapore Dept Math Singapore 119076 Singapore Natl Univ Singapore Inst Operat Res & Analyt Singapore 119076 Singapore

We study a general convex optimization problem, which covers various classic problems in different areas and particularly includes many optimal transport related problems arising in recent years. To solve this problem, we revisit the classic Bregman proximal point algorithm (BPPA) and introduce a new inexact stopping condition for solving the subproblems, which can circumvent the underlying feasibility difficulty often appearing in existing inexact conditions when the problem has a complex feasible set. Our inexact condition also covers several existing inexact conditions as special cases and hence makes our inexact BPPA (iBPPA) more flexible to fit different scenarios in practice. As an application to the standard optimal transport (OT) problem, our iBPPA with the entropic proximal term can bypass some numerical instability issues that usually plague the popular Sinkhorn's algorithm in the OT community, since our iBPPA does not require the proximal param-eter to be very small for obtaining an accurate approximate solution. The iteration complexity of O(1/k) and the convergence of the sequence are also established for our iBPPA under some mild conditions. Moreover, inspired by Nesterov's acceleration technique, we develop an inertial variant of our iBPPA, denoted by V-iBPPA, and establish the iteration complexity of O(1/k\lambda), where \lambda\geq 1 is a quadrangle scaling exponent of the kernel function. In particular, when the proximal parameter is a constant and the kernel function is strongly convex with Lipschitz continuous gradient (hence \lambda = 2), our V-iBPPA achieves a faster rate of O(1/k(2)) just as existing accelerated inexact proximal point algorithms. Some preliminary numerical experiments for solving the standard OT problem are conducted to show the convergence behaviors of our iBPPA and V-iBPPA under different inexactness settings. The experiments also empirically verify the potential of our V-iBPPA for improving the convergence speed.

关键词： proximal point algorithm Bregman distance inexact condition Nesterov's accel-eration optimal transport

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A Self-adaptive proximal point algorithm for Signal Reconstruction in Compressive Sensing

A Self-adaptive Proximal Point Algorithm for Signal Reconstr...

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2016 IEEE International Conference on Signal and Image Processing

作者： Kaizhan Huai Yejun Li Mingfang Ni Zhanke Yu Xiaoguo Wang Institute of Communications Engineering PLA University of Science and Technology Xi'an Communications Institute

Compressive sensing(CS) is a new framework for simulations sensing and *** to reconstruct a sparse signal from limited measurements is the key problem in *** solving the reconstruction problem of a sparse signal, we proposed a self-adaptive proximal point algorithm(PPA).This algorithm can handle the sparse signal reconstruction by solving a substituted problem—l *** last, the numerical results shows that the proposed method is more effective compared with the compressive sampling matching pursuit(CoSaMP).

关键词： compressive sensing signal reconstruction proximal point algorithm self-adaptive

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Δ-convergence for proximal point algorithm and fixed point problem in CAT(0) spaces

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Fixed point Theory and Applications 2019年第1期2019卷 8-8页

作者： Husain, Shamshad Singh, Nisha Department of Applied Mathematics Aligarh Muslim University Aligarh India

In this paper, we prove the Δ-convergence of a modified proximal point algorithm for common fixed points in a CAT(0) space for different classes of generalized nonexpansive mappings including a total asymptotically nonexpansive mapping, a multivalued mapping, and a minimizer of a convex function. The results in this paper generalize the corresponding results given by some authors. Moreover, numerical example is given to illustrate and show the Δ-convergence of the proposed algorithm for supporting our result. © 2019, The Author(s).

关键词： Nonexpansive multivalued mapping proximal point algorithm Total asymptotically nonexpansive mapping Δ-convergence

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proximal point algorithms for nonsmooth convex optimization with fixed point constraints

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2016年第2期253卷 503-513页

作者： Iiduka, Hideaki Meiji Univ Dept Comp Sci Tama Ku 1-1-1 Higashimita Kawasaki Kanagawa 2148571 Japan

The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem. With the focus on fixed point approximation techniques, two algorithms are devised for solving the problem. One blends an incremental subgradient method, which is a useful algorithm for nonsmooth convex optimization, with a Halpern-type fixed point iteration algorithm. The other is based on an incremental subgradient method and the Krasnosel'skiT-Mann fixed point algorithm. It is shown that any weak sequential cluster point of the sequence generated by the Halpern -type algorithm belongs to the solution set of the problem and that there exists a weak sequential cluster point of the sequence generated by the Krasnosel'skii-Mann-type algorithm, which also belongs to the solution set. Numerical comparisons of the two proposed algorithms with existing subgradient methods for concrete nonsmooth convex optimization show that the proposed algorithms achieve faster convergence. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Fixed point Halpern algorithm Incremental subgradient method Krasnosel'skit-Mann algorithm proximal point algorithm

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proximal point algorithmS AND FOUR RESOLVENTS OF NONLINEAR OPERATORS OF MONOTONE TYPE IN BANACH SPACES

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TAIWANESE JOURNAL OF MATHEMATICS 2008年第8期12卷 1883-1910页

作者： Takahashi, Wataru Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku Tokyo 1528552 Japan

In this article, motivated by Rockafellar's proximal point algorithm in Hilbert spaces, we discuss various weak and strong convergence theorems for resolvents of accretive operators and maximal monotone operators which are connected with the proximal point algorithm. We first deal with proximal point algorithms in Hilbert spaces. Then, we consider weak and strong convergence theorems for resolvents of accretive operators in Banach spaces which generalize the results in Hilbert spaces. Further, we deal with weak and strong convergence theorems for three types of resolvents of maximal monotone operators in Banach spaces which are related to proximal point algorithms. Finally, in Section 7, we apply some results obtained in Banach spaces to the problem of finding minimizers of convex functions in Banach spaces.

关键词： Banach space proximal point algorithm Resolvent Nonexpansive mapping Maximal monotone operator Retraction Projection Convex optimization

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proximal point algorithms based onS-iterative technique for nearly asymptotically quasi-nonexpansive mappings and applications

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NUMERICAL algorithmS 2021年第4期86卷 1561-1590页

作者： Sahu, D. R. Kumar, Ajeet Kang, Shin Min Banaras Hindu Univ Inst Sci Dept Math Varanasi 221005 Uttar Pradesh India Gyeongsang Natl Univ Dept Math Jinju 52828 South Korea China Med Univ Ctr Gen Educ Taichung 40402 Taiwan

In this paper, we combine theS-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal.,8(1), 61-792007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim.,14, 877-8981976) to propose a new modified proximal point algorithm based on theS-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the o-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl.,56, 2572-25792008), Khan and Abbas (Comput. Math. Appl.,61, 109-1162011), Abbas et al. (Math. Comput. Modelling,55, 1418-14272012) and many more.

关键词： Convex minimization problem Fixed point problem Nearly asymptotically quasi-nonexpansive mapping Mean nonexpansive mapping S-iteration process proximal point algorithm CAT(0) space o-convergence

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proximal point algorithms for Multi-criteria Optimization with the Difference of Convex Objective Functions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2016年第1期169卷 280-289页

作者： Ji, Ying Goh, Mark de Souza, Robert Univ Shanghai Sci & Technol Sch Business 516 Jungong Rd Shanghai 200093 Peoples R China Harbin Inst Technol Acad Fundamental & Interdisciplinary Sci Harbin 150006 Peoples R China Natl Univ Singapore Logist Inst Asia Pacific Singapore 117548 Singapore Natl Univ Singapore Sch Business Singapore 117548 Singapore

This paper focuses on solving a class of multi-criteria optimization with the difference of convex objective functions. proximal point algorithms, extensively studied for scalar optimization, are extended to our setting. We show that the proposed algorithms are well posed and globally convergent to a critical point. For an application, the new methods are used to a multi-criteria model arising in portfolio optimization. The numerical results show the efficiency of our methods.

关键词： Multi-criteria optimization DC proximal point algorithm Critical point Portfolio optimization

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proximal point algorithms for a Hybrid Pair of Nonexpansive Single-Valued and Multi-Valued Mappings in Geodesic Metric Spaces

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2017年第2期14卷 1-17页

作者： Suantai, Suthep Phuengrattana, Withun Chiang Mai Univ Fac Sci Dept Math Chiang Mai 50200 Thailand Nakhon Pathom Rajabhat Univ Fac Sci & Technol Dept Math Mueang Nakhon Pathom 730000 Nakhon Pathom Thailand

In this paper, we propose a new proximal point algorithm for finding a common element of the set of fixed points of nonexpansive single-valued mappings, the set of fixed points of nonexpansive multivalued mappings, and the set of minimizers of convex and lower semicontinuous functions. We obtain Delta-convergence and strong convergence of the proposed algorithm to a common element of the three sets in CAT(0) spaces. Furthermore, we apply our convergence results to obtain in a special space of CAT(0) spaces, so-called R-tree, under the gate condition. A numerical example to support our main results is also given.

关键词： Geodesic metric spaces fixed point proximal point algorithm nonexpansive mappings

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