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A GENERALIZED proximal point algorithm AND ITS CONVERGENCE RATE

SIAM JOURNAL ON OPTIMIZATION 2014年第4期24卷 1614-1638页

作者： Corman, Etienne Yuan, Xiaoming Ecole Polytech CMAP F-91128 Palaiseau France Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

We propose a generalized proximal point algorithm (PPA) in the generic setting of finding a root of a maximal monotone operator. In addition to the classical PPA, a number of benchmark operator splitting methods in the PDE and optimization literatures can be retrieved by this generalized PPA scheme. We establish the convergence rate of this generalized PPA scheme under different conditions, including estimating its worst-case convergence rate measured by the iteration complexity under mild assumptions and deriving its linear convergence rate under certain stronger conditions. Throughout our discussion, we pay particular attention to the special case where the operator is the sum of two maximal monotone operators and specify our theoretical results in the generic setting to this special case. Our result turns out to be a general and unified study on the convergence rate of a number of existing methods and subsumes some existing results in the literature.

关键词： convex optimization proximal point algorithm operator splitting methods convergence rate

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The proximal point algorithm Revisited

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第2期161卷 478-489页

作者： Dong, Yunda Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China

In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator's value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem's solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Br,zis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly. Furthermore, we also reveal its similarity to Guler's classical results in the context of convex minimization in the sense of strictly convex quadratic functions, and we discuss an application to an I mu-approximation solution of the problem above.

关键词： Monotone operator Convex minimization proximal point algorithm Rate of convergence

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On modified proximal point algorithms for solving minimization problems and fixed point problems in CAT(κ) spaces

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2021年第17期44卷 12369-12382页

作者： Pakkaranang, Nuttapol Kumam, Poom Wen, Ching-Feng Yao, Jen-Chih Cho, Yeol Je KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE Bangkok Thailand KMUTT KMUTTFixed Point Res Lab KMUTT Fixed Point Theory & Applicat Res Grp Dept MathFac Sci Bangkok Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung Taiwan Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung Taiwan China Med Univ Ctr Gen Educ Taichung Taiwan Gyeongsang Natl Univ Dept Math Educ Jinju South Korea

In this paper, we show the existence of solutions of the convex minimization problems and common fixed problems in CAT(1) spaces under some mild conditions. For this, we propose the new modified the proximal point algorithm. Further, we give some applications for the convex minimization problem and the fixed point problem in CAT(kappa) spaces with the bounded positive real number kappa. Our results improve and generalize many recent important results in the literature.

关键词： CAT(1) spaces minimization problem nonexpansive mapping proximal point algorithm resolvent operator

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Manifold proximal point algorithms for Dual Principal Component Pursuit and Orthogonal Dictionary Learning

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 4759-4773页

作者： Chen, Shixiang Deng, Zengde Ma, Shiqian So, Anthony Man-Cho Texas A&M Univ Wm Michael Barnes Dept Ind & Syst Engn 64 College Stn TX 77843 USA Cainiao Network Hangzhou Peoples R China Univ Calif Davis Dept Math Davis CA 95616 USA Chinese Univ Hong Kong Dept Syst Engn & Engn Management Hong Kong Peoples R China

We consider the problem of minimizing the l(1) norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is numerically challenging due to its nonsmooth objective and nonconvex constraint, and its algorithmic aspects have not been well explored. In this paper, we show how the manifold structure of the sphere can be exploited to design fast algorithms with provable guarantees for tackling this problem. Specifically, our contribution is fourfold. First, we present a manifold proximal point algorithm (ManPPA) for the problem and show that it converges at a global sublinear rate. Furthermore, we show that ManPPA can achieve a local quadratic convergence rate when applied to sharp instances of the problem. Second, we develop a semismooth Newton-based inexact augmented Lagrangian method for computing the search direction in each iteration of ManPPA and show that it has an asymptotic superlinear convergence rate. Third, we propose a stochastic variant of ManPPA called StManPPA, which is well suited for large-scale computation, and establish its sublinear convergence rate. Both ManPPA and StManPPA have provably faster convergence rates than existing subgradient-type methods. Fourth, using ManPPA as a building block, we propose a new heuristic method for solving a matrix analog of the problem, in which the sphere is replaced by the Stiefel manifold. The results from our extensive numerical experiments on the ODL and RSR problems demonstrate the efficiency and efficacy of our proposed methods.

关键词： Convergence Manifolds Signal processing algorithms Machine learning Standards Smoothing methods Optimization Riemannian optimization proximal point algorithm dictionary learning subspace recovery

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ON THE CONVERGENCE OF INEXACT proximal point algorithm ON HADAMARD MANIFOLDS

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TAIWANESE JOURNAL OF MATHEMATICS 2014年第2期18卷 419-433页

作者： Ahmadi, P. Khatibzadeh, H. Univ Zanjan Dept Math Zanjan Iran

In this paper we consider the proximal point algorithm to approximate a singularity of a multivalued monotone vector field on a Hadamard manifold. We study the convergence of the sequence generated by an inexact form of the algorithm. Our results extend the results of [3, 25] to Hadamard manifolds as well as the main result of [11] with more general assumptions on the control sequence. We also give some application to optimization.

关键词： proximal point algorithm Maximal monotone operator Resolvent Subdifferential Convergence Hadamard manifold

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The indefinite proximal point algorithms for maximal monotone operators

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OPTIMIZATION 2021年第8期70卷 1759-1790页

作者： Jiang, Fan Cai, Xingju Han, Deren Nanjing Normal Univ Key Lab NSLSCS Jiangsu Prov Sch Math Sci Nanjing Peoples R China Beihang Univ Beijing Adv Innovat Ctr Big Data & Brain Comp BDB Sch Math Sci Beijing Peoples R China

The proximal point algorithm (PPA) has been widely used in convex optimization. Many algorithms fall into the framework of PPA. To guarantee the convergence of PPA, however, existing results conventionally need to ensure the positive definiteness of the corresponding proximal measure. In some senses, this essentially results in tiny step sizes (or over regularization) for the subproblems and thus inevitably decelerates the overall convergence speed of PPA. In this paper, we investigate the possibility of relaxing the positive definiteness requirement of the proximal measure in PPA. An indefinite PPA for finding a zero of maximal monotone operator is thus proposed via choosing an indefinite proximal regularization term, resulting in larger step sizes. Under some suitable conditions, we prove the global convergence of the proposed algorithm and its extension. To make our method more practical, we suggest to solve the subproblem in an approximate manner and propose two flexible inexact criteria. We show that the condition which guarantees the convergence of the proposed indefinite PPA is tight by a simple example. In addition, we show how to apply the indefinite PPA to some convex models. We report some preliminary numerical results, which show the efficiencies of the proposed algorithms.

关键词： proximal point algorithm indefinite proximal term convex optimization global convergence inexact criteria

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A proximal point DUAL NEWTON algorithm FOR SOLVING GROUP GRAPHICAL LASSO PROBLEMS

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SIAM JOURNAL ON OPTIMIZATION 2020年第3期30卷 2197-2220页

作者： Zhang, Yangjing Zhang, Ning Sun, Defeng Toh, Kim-Chuan Natl Univ Singapore Dept Math Singapore 119076 Singapore Dongguan Univ Technol Sch Comp Sci & Technol Dongguan 523808 Peoples R China Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China Natl Univ Singapore Inst Operat Res & Analyt Singapore 119076 Singapore

Undirected graphical models have been especially popular for learning the conditional independence structure among a large number of variables where the observations are drawn independently and identically from the same distribution. However, many modern statistical problems would involve categorical data or time-varying data, which might follow different but related underlying distributions. In order to learn a collection of related graphical models simultaneously, various joint graphical models inducing sparsity in graphs and similarity across graphs have been proposed. In this paper, we aim to propose an implementable proximal point dual Newton algorithm (PPDNA) for solving the group graphical Lasso model, which encourages a shared pattern of sparsity across graphs. Though the group graphical Lasso regularizer is non-polyhedral, the asymptotic superlinear convergence of our proposed method PPDNA can be obtained by leveraging on the local Lipschitz continuity of the Karush-Kuhn-Tucker solution mapping associated with the group graphical Lasso model. A variety of numerical experiments on real data sets illustrates that the PPDNA for solving the group graphical Lasso model can be highly efficient and robust.

关键词： group graphical Lasso proximal point algorithm semismooth Newton method Lipschitz continuity

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Modified proximal point algorithms involving convex combination technique for solving minimization problems with convergence analysis

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OPTIMIZATION 2020年第7-8期69卷 1655-1680页

作者： Wairojjana, Nopparat Pakkaranang, Nuttapol Uddin, Izhar Kumam, Poom Awwal, Aliyu Muhammed Valaya Alongkorn Rajabhat Univ Appl Math Program Fac Sci & Technol Pathum Thani Thailand KMUTT KMUTTFixed Point Res Lab KMUTT Fixed Point Theory & Applicat Res Grp SCL 802 Fixed Point Lab Bangkok Thailand KMUTT Fac Sci Dept Math Bangkok Thailand Jamia Millia Islamia Dept Math Fac Nat Sci New Delhi India KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE Sci Lab Bldg Fac Sci Bangkok Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan Gombe State Univ Dept Math Fac Sci Gombe Nigeria

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding common solutions of their problems, we introduce a new modified proximal point algorithm involving convex combination technique. Under suitable conditions, Delta-convergence and strong convergence theorems are proved. Furthermore, we provide some numerical examples which show the efficiency and implementation of the proposed algorithm as well as show computational overview by comparing it with previously known algorithms.

关键词： proximal point algorithm asymptotically nonexpansive mapping Hadamard space minimization problem iteration process

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A refined proximal algorithm for nonconvex multiobjective optimization in Hilbert spaces

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JOURNAL OF GLOBAL OPTIMIZATION 2025年第1期92卷 187-203页

作者： Bento, G. C. Cruz Neto, J. X. Lopes, J. O. Mordukhovich, B. S. Silva Filho, P. R. Univ Fed Piaui Dept Math CCN Teresina Piaui Brazil Univ Fed Goias Inst Math & Stat IME Goiania Go Brazil Wayne State Univ Dept Math Detroit MI USA

This paper is devoted to general nonconvex problems of multiobjective optimization in Hilbert spaces. Based on limiting/Mordukhovich subgradients, we define a new notion of Pareto critical points for such problems, establish necessary optimality conditions for them, and then employ these conditions to develop a refined version of the vectorial proximal point algorithm providing its detailed convergence analysis. The obtained results largely extend those initiated by Bonnel et al. [SIAM J Optim, 15 (2005), pp. 953-970] for convex vector optimization problems, specifically in the case where the codomain is an m-dimensional space and by Bento et al. [SIAM J Optim, 28 (2018), pp. 1104-1120] for nonconvex finite-dimensional problems in terms of Clarke's generalized gradients.

关键词： Multiobjective programming proximal point algorithm Locally lipschitz-continuous functions Weakly pareto point

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Two accelerated double inertial algorithms for variational inequalities on Hadamard manifolds

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2025年 145卷

作者： Tan, Bing Abass, Hammed Anuoluwapo Li, Songxiao Oyewole, Olawale Kazeem Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China Sefako Makgatho Hlth Sci Univ Dept Math & Appl Math POB 94 ZA-0204 Pretoria South Africa Shantou Univ Dept Math Shantou 515821 Peoples R China Tshwane Univ Technol Dept Math ZA-0007 Pretoria South Africa

Two modified double inertial proximal point algorithms are proposed for solving variational inequality problems with a pseudomonotone vector field in the settings of a Hadamard manifold. Weak convergence of the proposed methods is attained without the requirement of Lipschitz continuity conditions. The convergence efficiency of the proposed algorithms is improved with the help of the double inertial technique and the non-monotonic self-adaptive step size rule. We present a numerical experiment to demonstrate the effectiveness of the proposed algorithm compared to several existing ones. The results extend and generalize many recent methods in the literature.

关键词： Variational inequality problem Double inertial proximal point algorithm Uniformly continuous Hadamard manifold

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