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A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

NUMERICAL algorithmS 2023年第1期93卷 103-130页

作者： Aragon-Artacho, Francisco J. Bot, Radu, I Torregrosa-Belen, David Univ Alicante Dept Math Alicante 03690 Spain Univ Vienna Fac Math A-1090 Vienna Austria

In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.

关键词： Monotone operator Monotone inclusion splitting algorithm Primal-dual algorithm Minimal lifting

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A Direct Proof of Convergence of Davis-Yin splitting algorithm Allowing Larger Stepsizes

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第3期30卷 1011-1029页

作者： Aragon-Artacho, Francisco J. Torregrosa-Belen, David Univ Alicante Dept Math Alicante Spain

This note is devoted to the splitting algorithm proposed by Davis and Yin (Set-valued Var. Anal. 25(4), 829-858, 2017) for computing a zero of the sum of three maximally monotone operators, with one of them being cocoercive. We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin. As a by-product, the same conclusion applies to the forward-backward splitting algorithm. Further, we use the notion of "strengthening" of a set-valued operator to derive a new splitting algorithm for computing the resolvent of the sum. Last but not least, we provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms.

关键词： Monotone inclusion Resolvent splitting algorithm Forward-backward Strengthening

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Conservation laws with discontinuous flux function on networks: a splitting algorithm

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NETWORKS AND HETEROGENEOUS MEDIA 2023年第1期18卷 1-28页

作者： Friedrich, Jan Gottlich, Simone Uphoff, Annika Rhein Westfal TH Aachen Inst Appl Math D-52064 Aachen Germany Univ Mannheim Dept Math D-68131 Mannheim Germany

In this article, we present an extension of the splitting algorithm proposed in [22] to networks of conservation laws with piecewise linear discontinuous flux functions in the unknown. We start with the discussion of a suitable Riemann solver at the junction and then describe a strategy how to use the splitting algorithm on the network. In particular, we focus on two types of junctions, i.e., junctions where the number of outgoing roads does not exceed the number of incoming roads (dispersing type) and junctions with two incoming and one outgoing road (merging type). Finally, numerical examples demonstrate the accuracy of the splitting algorithm by comparisons to the exact solution and other approaches used in the literature.

关键词： traffic flow networks LWR model conservation laws with discontinuous flux numerical schemes splitting algorithm

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On the convergence of splitting algorithm for mixed equilibrium problems on Hadamard manifolds

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JOURNAL OF MATHEMATICAL CHEMISTRY 2020年第4期58卷 799-815页

作者： Khammahawong, K. Kumam, P. Chaipunya, P. KMUTT Dept Math 126 Pracha Uthit Rd Bangkok 10140 Thailand KMUTT KMUTTFixed Point Res Lab KMUTTFixed Point Theory & Applicat Res Grp Dept MathFac Sci 126 Pracha Uthit Rd Bangkok 10140 Thailand KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE SCL 802 Fixed Point Lab Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand

The aim of this paper is to introduce a splitting algorithm to solving mixed equilibrium problems on a Hadamard manifold. The convergence of the sequence generated by the proposed algorithm is established under suitab... 详细信息

关键词： Equilibrium problems Hadamard manifolds Monotonicity Proximal algorithm splitting algorithm

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A PARAMETERIZED THREE-OPERATOR splitting algorithm AND ITS EXPANSION

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2021年第2期5卷 211-226页

作者： Zhang, Cuijie Chen, Jiajia Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

Based on the parameterized Douglas-Rachford algorithm and the three-operator splitting algorithm, we propose the parameterized three-operator splitting algorithm for finding the least norm solution for the sum of three maximally monotone operators, with one of which is a cocoercive operator. In order to improve the convergence speed of the parameterized three-operator splitting algorithm, the inertial version of the algorithm is studied. In addition, two convergence theorems are established under some conditions. Finally, we illustrate the feasibility and superiority of our algorithm via a numerical example.

关键词： Inertial algorithm Maximally monotone operator Resolvent Superiority splitting algorithm

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A splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions and application to cournot-nash model

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RAIRO-OPERATIONS RESEARCH 2021年第1-Sup期55卷 S1395-S1410页

作者： Duc, Phung Minh Le, Xuan Thanh Ho Chi Minh Univ Nat Resources & Environm Ho Chi Minh City Vietnam Vietnam Acad Sci & Technol Inst Math Hanoi Vietnam

In this paper we propose a splitting subgradient algorithm for solving equilibrium problems involving the sum of two bifunctions. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each component bifunction. In contrast to the splitting algorithms previously proposed in Anh and Hai (Numer. algorithms 76 (2017) 67-91) and Hai and Vinh (Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 111 (2017) 1051-1069), our algorithm is convergent for paramonotone and strongly pseudomonotone bifunctions without any Lipschitz type as well as Holder continuity condition of the bifunctions involved. Furthermore, we show that the ergodic sequence defined by the algorithm iterates converges to a solution without paramonotonicity property. Some numerical experiments on differentiated Cournot-Nash models are presented to show the behavior of our proposed algorithm with and without ergodic.

关键词： Monotone equilibria splitting algorithm ergodic sequence

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A Correlation-Aware splitting algorithm for Opportunistic Selection

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IEEE TRANSACTIONS ON COMMUNICATIONS 2018年第3期66卷 1250-1261页

作者： Isaac, Reneeta Sara Mehta, Neelesh B. Indian Inst Sci Dept Elect Commun Engn Bengaluru 560012 Karnataka India

Opportunistic selection is a key technique to improve the performance of wireless systems. In it, one among the available users is selected on the basis of their channel gains or local parameters, such as battery energy state. Formally, each user possesses a real-valued metric that only it knows, and the goal is to select the best user, which has the highest metric. The splitting algorithm is a popular, fast, and scalable algorithm to implement opportunistic selection;it is distributed and guarantees selection of the best user. We show that this algorithm, which has thus far been designed assuming that the metrics are independent and identically distributed, is no longer scalable when the metrics are correlated. We then propose a novel correlation-aware splitting algorithm (CASA) and show how it can be applied to practically motivated probability distributions and correlation models. We present computationally feasible techniques for pre-computing the thresholds that CASA specifies, thereby ensuring that CASA can be implemented in practice. We benchmark the performance of CASA with the conventional algorithm, and show that it reduces the average selection time significantly as the number of users or the correlation among them increases.

关键词： Opportunistic selection splitting algorithm correlation multiple access protocols

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A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems

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JOURNAL OF FIXED POINT THEORY AND APPLICATIONS 2018年第3期20卷 1-16页

作者： Le Dung Muu Xuan Thanh Le Thang Long Univ Inst Math & Appl Sci Hanoi Vietnam Vietnam Acad Sci & Technol Inst Math Hanoi Vietnam

We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz-type condition as well as Holder continuity of the bifunctions involved.

关键词： Monotone equilibria Fixed point Common solution splitting algorithm

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A splitting algorithm for a novel regularization of Perona-Malik and application to image restoration

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EURASIP JOURNAL ON ADVANCES IN SIGNAL PROCESSING 2017年第1期2017卷 1页

作者： Karami, Fahd Ziad, Lamia Sadik, Khadija Univ Cadi Ayyad Ecole Super Technol Essaouira BP 383 Essaouira El Jadida Essaouira Morocco

In this paper, we focus on a numerical method of a problem called the Perona-Malik inequality which we use for image denoising. This model is obtained as the limit of the Perona-Malik model and the p-Laplacian operator with p -> infinity. In Atlas et al., (Nonlinear Anal. Real World Appl 18:57-68, 2014), the authors have proved the existence and uniqueness of the solution of the proposed model. However, in their work, they used the explicit numerical scheme for approximated problem which is strongly dependent to the parameter p. To overcome this, we use in this work an efficient algorithm which is a combination of the classical additive operator splitting and a nonlinear relaxation algorithm. At last, we have presented the experimental results in image filtering show, which demonstrate the efficiency and effectiveness of our algorithm and finally, we have compared it with the previous scheme presented in Atlas et al., (Nonlinear Anal. Real World Appl 18:57-68, 2014).

关键词： Image restoration Perona-Malik inequality p-Laplacian splitting algorithm

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Some ideas for bias and variance reduction in the splitting algorithm for diffusion processes

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JOURNAL OF COMPUTATIONAL SCIENCE 2015年 11卷 58-68页

作者： Jacquemart, Damien Morio, Jerome Le Gland, Francois Onera French Aerosp Lab F-91123 Palaiseau France Onera French Aerosp Lab F-31055 Toulouse France INRIA Rennes ASPI Applicat Interacting Particle Syst Stat F-35042 Rennes France

In this article, we highlight a bias induce by the discretization of the sample Markov paths in the splitting algorithm. Consequently we propose to correct this bias using a deformation of the intermediate regions. Moreover, we propose two estimation methods of the optimal regions in the splitting algorithm to minimise the splitting variance. One is connected with partial differential equation approach, the other one is based on the discretization of the state space of the process. (C) 2015 Elsevier B.V. All rights reserved.

关键词： splitting algorithm Markov process Rare event estimation Bias Variance Stochastic differential equation

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