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A product space reformulation with reduced dimension for splitting algorithms

COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2022年第1期83卷 319-348页

作者： Campoy, Ruben Univ Valencia Dept Stat & Operat Res C Dr Moliner 50 E-46100 Valencia Spain

In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra's classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments.

关键词： Pierra's product space reformulation splitting algorithm Douglas-Rachford algorithm Monotone inclusions Feasibility problem Projection methods

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Monotone splitting Sequential Quadratic Optimization algorithm with Applications in Electric Power Systems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2020年第1期186卷 226-247页

作者： Jian, Jinbao Zhang, Chen Yin, Jianghua Yang, Linfeng Ma, Guodong Guangxi Univ Nationalities Coll Sci Nanning 530006 Peoples R China Guangxi Univ Coll Elect Engn Nanning 530004 Peoples R China Inner Mongolia Univ Sch Math Sci Hohhot 010021 Peoples R China Yulin Normal Univ Sch Math & Stat Yulin 537000 Peoples R China

In this paper, we propose a new sequential quadratic optimization algorithm for solving two-block nonconvex optimization with linear equality and generalized box constraints. First, the idea of the splitting algorithm is embedded in the method for solving the quadratic optimization approximation subproblem of the discussed problem, and then, the subproblem is decomposed into two independent low-dimension quadratic optimization subproblems to generate a search direction for the primal variable. Second, a deflection of the steepest descent direction of the augmented Lagrangian function with respect to the dual variable is considered as the search direction of the dual variable. Third, using the augmented Lagrangian function as the merit function, a new primal-dual iterative point is generated by Armijo line search. Under mild conditions, the global convergence of the proposed algorithm is proved. Finally, the proposed algorithm is applied to solve a series of mid-to-large-scale economic dispatch problems for power systems. Comparing the numerical results demonstrates that the proposed algorithm possesses superior numerical effects and good robustness.

关键词： Linear equality and box constraints Two-block nonconvex optimization Sequential quadratic optimization splitting algorithm Electric power systems

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splitting algorithms for the high-order compact finite-difference schemes in wave-equation modeling

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GEOPHYSICS 2016年第6期81卷 T295-T302页

作者： Zhang, Cai Sun, Bingbing Ma, Jianwei Yang, Huizhu Hu, Ying Tsinghua Univ Sch Aerosp Beijing Peoples R China Res Inst Petr Explorat & Dev Dept Geophys Beijing Peoples R China Harbin Inst Technol Dept Math Harbin Peoples R China

We have developed efficient splitting algorithms for high-order compact finite-difference methods to approximate second-order space derivatives. In general, the methods' high-order compact finite-difference schemes require the inversion of a multidiagonal matrix that is commonly less efficient. To solve this problem, we used ideas from splitting algorithms in one-way wave-equation migration that work by decomposing the multidiagonal matrix into a series of tridiagonal matrices and then subsequently solving the tridiagonal matrices. This approach results in more efficient algorithms with little loss of accuracy. The splitting algorithms can be implemented in three different ways. Our computational complexity analysis verifies that our methods can reduce the calculation burden from exponential to linear growth. Numerical experiments demonstrate the correctness and effectiveness of our algorithms.

关键词： compact finite difference wave equation seismic modeling splitting algorithm

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The Use of Reliability-Based splitting algorithms to Improve Distributed Estimation in WSNs

The Use of Reliability-Based Splitting Algorithms to Improve...

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IEEE Military Communications Conference

作者： Laitrakun, Seksan Coyle, Edward J. Georgia Inst Technol Sch Elect & Comp Engn Atlanta GA 30332 USA

ISBN: (纸本)9780769551241

We consider distributed estimation algorithms in a large Wireless Sensor Network (WSN) with a star topology. The local estimates computed by each sensor must be collected in a fixed amount of time by the network's Fusion Center (FC) and processed to produce a global estimate. When the amount of time available is not sufficient to collect a local estimate from every node, a strategy that enables the most reliable estimates to be collected first must be developed. In this paper, we develop such an algorithm for a WSN whose FC uses a Best Linear Unbiased Estimator (BLUE) and collects the local estimates on a collision channel. The proposed schemes use population splitting that is based on the reliabilities of the local estimates. The time available for data collection is divided into frames and each frame is subdivided into slots. The slots in the first frame are used to collect the most reliable local estimates;the next frame of slots is used to collect the next most reliable set of local estimates;etc. As the performance of the schemes depends on the reliability thresholds, the number of bits representing the estimates, and the number of time slots in a frame, we formulate time-constrained optimization problems and derive methods to obtain the optimal values of these parameters. An interesting result shows that the optimal reliability thresholds do not maximize the channel's throughput.

关键词： BLUE Distributed estimation censoring sensor cross-layer design frame slotted ALOHA ordered transmssion random access splitting algorithm wireless sensor networks

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A proximal fully parallel splitting method with a relaxation factor for separable convex programming

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APPLIED NUMERICAL MATHEMATICS 2024年 195卷 17-38页

作者： Yin, Jianghua Jian, Jinbao Jiang, Xianzhen Wu, Jiansheng Ma, Guodong Guangxi Minzu Univ Ctr Appl Math Guangxi Sch Math & Phys Guangxi Key Lab Hybrid Computat & IC Design Anal Nanning 530006 Peoples R China Guangxi Sci & Technol Normal Univ Coll Math & Comp Sci Laibin 546199 Peoples R China

In this paper, we propose a proximal fully parallel splitting method with a relaxation factor for solving separable convex minimization model with linear constraints, where the objective function is the sum of m individual functions without coupled variables. With a full Jacobian decomposition, we decompose the subproblem associated with the augmented Lagrangian method into m smaller subproblems and then add a quadratic proximal term to each decomposed subproblem, which makes the resulting ones easier to solve for many applications. In order to accelerate the numerical performance, we attach a positive relaxation factor to update the Lagrange multiplier, which also allows more flexibility in the design of algorithms. Moreover, we refine the step size of the underrelaxation step, which enlarges several existing ones in the literature. We prove that the proposed method is globally convergent, and show the worst-case O(1/root K) convergence rate in a nonergodic sense. Finally, the efficiency and robustness of the proposed method are also demonstrated by solving the & ell;(1) norm problem, the & ell;(1)-regularized least squares problem, the exchange problem and the total variation image restoration problem.

关键词： Augmented Lagrangian method splitting algorithm Relaxation factor Step size Convergence rate

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Strengthened splitting methods for computing resolvents

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2021年第2期80卷 549-585页

作者： Aragon Artacho, Francisco J. Campoy, Ruben Tam, Matthew K. Univ Alicante Dept Math Alicante Spain Univ Valencia Dept Stat & Operat Res Valencia Spain Univ Melbourne Sch Math & Stat Parkville Vic 3010 Australia

In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the "strengthening" of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs.

关键词： Monotone operator Resolvent splitting algorithm Strengthening

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Convergence of a splitting inertial proximal method for monotone operators

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2003年第2期155卷 447-454页

作者： Moudafi, A Oliny, M Univ Antilles Guyane DSI GRIMAAG Schoelcher 97200 Martinique France

A forward-backward inertial procedure for solving the problem of finding a zero of the sum of two maximal monotone operators is proposed and its convergence is established under a cocoercivity condition with respect t... 详细信息

关键词： monotone operators enlargements proximal point algorithm cocoercivity splitting algorithm projection convergence

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A NOTE ON THE PAPER BY ECKSTEIN AND SVAITER ON "GENERAL PROJECTIVE splitting METHODS FOR SUMS OF MAXIMAL MONOTONE OPERATORS"

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2009年第4期48卷 2513-2515页

作者： Bauschke, Heinz H. Univ British Columbia Okanagan Irving K Barber Sch Kelowna BC V1V 1V7 Canada

In their recent paper from 2009 [SIAM J. Control Optim., 48 (2009), pp. 787-811], J. Eckstein and B. F. Svaiter proposed a very general and flexible splitting framework for finding a zero of the sum of finitely many maximal monotone operators. In this short note, we provide a technical result that allows for the removal of Eckstein and Svaiter's assumption that the sum of the operators be maximal monotone or that the underlying Hilbert space be finite-dimensional.

关键词： firmly nonexpansive mapping maximal monotone operator nonexpansive mapping proximal algorithm splitting algorithm

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Resolvent splitting for sums of monotone operators with minimal lifting

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MATHEMATICAL PROGRAMMING 2023年第1-2期201卷 231-262页

作者： Malitsky, Yura Tam, Matthew K. K. Linkoping Univ Dept Math S-58183 Linkoping Sweden Univ Melbourne Sch Math & Stat Parkville Vic 3010 Australia

In this work, we study fixed point algorithms for finding a zero in the sum of n >= 2 maximally monotone operators by using their resolvents. More precisely, we consider the class of such algorithms where each resolvent is evaluated only once per iteration. For any algorithm from this class, we show that the underlying fixed point operator is necessarily defined on a d-fold Cartesian product space with d >= n - 1. Further, we show that this bound is unimprovable by providing a family of examples for which d = n - 1 is attained. This family includes the Douglas-Rachford algorithm as the special case when n = 2. Applications of the new family of algorithms in distributed decentralised optimisation and multi-block extensions of the alternation direction method of multipliers (ADMM) are discussed.

关键词： Monotone operator splitting algorithm Decentralised optimisation ADMM

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On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2009年第2期359卷 508-513页

作者： Moudafi, Abdellatif [a]CEREGMIA Université des Antilles-Guyane Département Scientifique Interfacultaire Campus de Schoelcher 97230 Cedex Martinique (F.W.I.)

Two splitting procedures for solving equilibrium problems involving the sum of two bifunctions are proposed and their convergence is established under mild assumptions. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Equilibrium problem splitting algorithm Optimization Variational inequality Fixed-point Proximal algorithm Ergodic convergence

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