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A modified projective forward-backward splitting algorithm for variational inclusion problems to predict Parkinson's disease

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APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 2024年第1期32卷

作者： Cholamjiak, Watcharaporn Das, Shilpa Univ Phayao Sch Sci Phayao Thailand Kamrup Coll Dept Math Chamata India Kamrup Coll Dept Math Chamata 781306 Nalbari India

This research studies variational inclusion problems, which is a branch of optimization. A modified projective forward-backward splitting algorithm is constructed to solve this problem. The algorithm adds the inertial technique for speeding up the convergence, and the projective method for several regularization machine learning models to meet good model fitting. To evaluate the performance of the classification models employed in this research, four evaluation metrics are computed: accuracy, F1-score, recall, and precision. The highest performance value of 92.86% accuracy, 62.50% precision, 100% recall, and 76.92% F1-score shows that our algorithm performs better than the other machine learning models.

关键词： Variational inclusion problem extreme learning machines (ELM) forward-backward splitting algorithm regularized least squares Parkinson's disease

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Convergence analysis of a variable metric forward-backward splitting algorithm with applications

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2019年第1期2019卷 1-27页

作者： Cui, Fuying Tang, Yuchao Zhu, Chuanxi Nanchang Univ Dept Math Nanchang Jiangxi Peoples R China Nanchang Univ Sch Management Nanchang Jiangxi Peoples R China

The forward-backward splitting algorithm is a popular operator-splitting method for solving monotone inclusion of the sum of a maximal monotone operator and an inverse strongly monotone operator. In this paper, we present a new convergence analysis of a variable metric forward-backward splitting algorithm with extended relaxation parameters in real Hilbert spaces. We prove that this algorithm is weakly convergent when certain weak conditions are imposed upon the relaxation parameters. Consequently, we recover the forward-backward splitting algorithm with variable step sizes. As an application, we obtain a variable metric forward-backward splitting algorithm for solving the minimization problem of the sum of two convex functions, where one of them is differentiable with a Lipschitz continuous gradient. Furthermore, we discuss the applications of this algorithm to the fundamental of the variational inequalities problem, constrained convex minimization problem, and split feasibility problem. Numerical experimental results on LASSO problem in statistical learning demonstrate the effectiveness of the proposed iterative algorithm.

关键词： forward-backward splitting algorithm Monotone inclusion Variable metric Split feasibility problem

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An inertial Mann forward-backward splitting algorithm of variational inclusion problems and its applications

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CHAOS SOLITONS & FRACTALS 2022年 158卷

作者： Peeyada, Pronpat Suparatulatorn, Raweerote Cholamjiak, Watcharaporn Univ Phayao Sch Sci Phayao 56000 Thailand Chiang Mai Univ Fac Sci Dept Math Chiang Mai 50200 Thailand

In this paper, we introduce the inertial Mann forward-backward splitting algorithm for solving variational inclusion problem of the sum of two operators, the one is maximally monotone and the other is monotone and Lipschitz continuous. Under standard assumptions, we prove the weak convergence theorem of the proposed algorithm. We show that the algorithm is flexible to use by choosing the variable stepsizes and two different algorithms are shown by choosing constant stepsize and update stepsize. Moreover, we apply our algorithms to solve data classification using the Wisconsin original breast cancer data set as a training set. We also compare our algorithms with the other two algorithms to show the efficiency of the algorithm and show suitably learns the training dataset and generalizes well to a hold-out dataset of the algorithm by considering overfitting. Finally, we apply our algorithms to solve signal recovery and show the efficiency of the algorithm by compare with the other two algorithms. The results of data classification and signal recovery showed that choosing the right stepsizes of the algorithm would be a good efficient for the different problems. (c) 2022 Elsevier Ltd.

关键词： Variational inclusion problem Inertial methods forward-backward splitting algorithm Data classification Machine learning Signal recovery

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Stochastic forward-backward splitting for Monotone Inclusions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2016年第2期169卷 388-406页

作者： Rosasco, Lorenzo Villa, Silvia Vu, Bang Cong Univ Genoa DIBRIS Genoa Italy Ist Italiano Tecnol & Massachusetts Inst Technol Lab Computat & Stat Learning Cambridge MA USA

We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward-backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as well as almost sure convergence under weaker assumptions. For minimization problems, we recover rates matching those obtained by stochastic extensions of the so-called accelerated methods. Stochastic quasi-Fejer's sequences are a key technical tool to prove almost sure convergence.

关键词： Stochastic first-order methods forward-backward splitting algorithm Monotone inclusions Stochastic Fejer sequences

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A generalized forward-backward splitting method for solving a system of quasi variational inclusions in Banach spaces

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2019年第2期113卷 729-747页

作者： Chang, Shih-sen Wen, Ching-Feng Yao, Jen-Chih China Med Univ Ctr Gen Educ Taichung 40402 Taiwan Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 807 Taiwan Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung 807 Taiwan

The purpose of this paper is by using the generalized forward-backward splitting method and implicit midpoint rule to propose an iterative algorithm for finding a common element of the set of solutions to a system of quasi variational inclusions with accretive mappings and the set of fixed points for a -strict pseudo-contractive mapping in Banach spaces. Some strong convergence theorems of the sequence generated by the algorithm are proved. The results presented in the paper extend and improve some recent results. At the end of the paper, some applications to a system of variational inequalities problem, monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem are presented.

关键词： forward-backward splitting algorithm Implicit middle rules Accretive operator Maximal monotone operator Strict pseudo-contractive mapping splitting method

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Variable metric forward-backward splitting with applications to monotone inclusions in duality

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OPTIMIZATION 2014年第9期63卷 1289-1318页

作者： Combettes, Patrick L. Vu, Bang C. Univ Paris 06 Lab Jacques Louis Lions UMR 7598 CNRS F-75005 Paris France

We propose a variable metric forward-backward splitting algorithm and prove its convergence in real Hilbert spaces. We then use this framework to derive primal-dual splitting algorithms for solving various classes of monotone inclusions in duality. Some of these algorithms are new even when specialized to the fixed metric case. Various applications are discussed.

关键词： cocoercive operator composite operator demiregularity duality forward-backward splitting algorithm monotone inclusion monotone operator primal-dual algorithm quasi-Fejer sequence variable metric

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forward-backward splitting Method for Solving a System of Quasi-Variational Inclusions

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2019年第5期42卷 2169-2189页

作者： Chang, Shih-Sen Wen, Ching-Feng Yao, Jen-Chih China Med Univ Ctr Gen Educ Taichung 40402 Taiwan Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 80708 Taiwan Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung 80708 Taiwan

The purpose of this paper is by using a generalized forward-backward splitting method to propose an iterative algorithm for finding a common element of the set of solutions to a system of quasi-variational inclusions with accretive mappings and the set of fixed points for a lambda-strictly pseudo-contractive mapping in Banach spaces. Some strong convergence theorems of the sequence generated by the algorithm are proved. The results presented in the paper extend and improve some recent results. As applications, we utilize our results to study the approximation problem of solutions to a system of variational inequalities, accretive variational inequality problem and convex minimization problem in Banach spaces.

关键词： forward-backward splitting algorithm Accretive operator Maximal-accretive operator Strictly pseudo-contractive mapping

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Primal-dual fixed point algorithm based on adapted metric method for solving convex minimization problem with application

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APPLIED NUMERICAL MATHEMATICS 2020年 157卷 236-254页

作者： Huang, Wenli Tang, Yuchao Nanchang Univ Dept Math Nanchang 330031 Jiangxi Peoples R China

Optimization problems involving the sum of three convex functions have received much attention in recent years, where one is differentiable with Lipschitz continuous gradient, one is composed of a linear operator and the other is proximity friendly. The primal-dual fixed point algorithm is a simple and effective algorithm for such problems. To exploit the second-order derivatives information of the objective function, we propose a primal-dual fixed point algorithm with an adapted metric method. The proposed algorithm is derived from the idea of establishing a generally fixed point formulation for the solution of the considered problem. Under mild conditions on the iterative parameters, we prove the convergence of the proposed algorithm. Further, we establish the ergodic convergence rate in the sense of primal-dual gap and also derive the linear convergence rate with additional conditions. Numerical experiments on image deblurring problems show that the proposed algorithm outperforms other state-of-the-art primal-dual algorithms in terms of the number of iterations. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Primal-dual algorithm Adapted metric method Proximity operator forward-backward splitting algorithm

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Non-convex Total Variation Regularization for Convex Denoising of Signals

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2020年第6-7期62卷 825-841页

作者： Selesnick, Ivan Lanza, Alessandro Morigi, Serena Sgallari, Fiorella NYU Dept Elect & Comp Engn Brooklyn NY 11220 USA Univ Bologna Dept Math Bologna Italy

Total variation (TV) signal denoising is a popular nonlinear filtering method to estimate piecewise constant signals corrupted by additive white Gaussian noise. Following a 'convex non-convex' strategy, recent papers have introduced non-convex regularizers for signal denoising that preserve the convexity of the cost function to be minimized. In this paper, we propose a non-convex TV regularizer, defined using concepts from convex analysis, that unifies, generalizes, and improves upon these regularizers. In particular, we use the generalized Moreau envelope which, unlike the usual Moreau envelope, incorporates a matrix parameter. We describe a novel approach to set the matrix parameter which is essential for realizing the improvement we demonstrate. Additionally, we describe a new set of algorithms for non-convex TV denoising that elucidate the relationship among them and which build upon fast exact algorithms for classical TV denoising.

关键词： Signal denoising Total variation regularization forward-backward splitting algorithm Convex non-convex regularization

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Compressive Video Sampling With Approximate Message Passing Decoding

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 2012年第9期22卷 1354-1364页

作者： Ma, Jianwei Plonka, Gerlind Hussaini, M. Yousuff Florida State Univ Dept Math Tallahassee FL 32306 USA Harbin Inst Technol Inst Appl Math Harbin 150001 Peoples R China Univ Gottingen Inst Numer & Appl Math D-37083 Gottingen Germany

In this paper, we apply compressed sensing (CS) to video compression. CS techniques exploit the observation that one needs much fewer random measurements than given by the Shannon-Nyquist sampling theory to recover an object if this object is compressible (i.e., sparse in the spatial domain or in a transform domain). In the CS framework, we can achieve sensing, compression, and denoising simultaneously. We propose a fast and simple online encoding by the application of pseudorandom downsampling of the 2-D fast Fourier transform to video frames. For offline decoding, we apply a modification of the recently proposed approximate message passing (AMP) algorithm. The AMP method has been derived using the statistical concept of "state evolution," and it has been shown to considerably accelerate the convergence rate in special CS-decoding applications. We shall prove that the AMP method can be rewritten as a forward-backward splitting algorithm. This new representation enables us to give conditions that ensure convergence of the AMP method and to modify the algorithm in order to achieve higher robustness. The success of reconstruction methods for video decoding also essentially depends on the chosen transform, where sparsity of the video signals is assumed. We propose incorporating the 3-D dual-tree complex wavelet transform that possesses sufficiently good directional selectivity while being computationally less expensive and less redundant than other directional 3-D wavelet transforms.

关键词： 3-D directional wavelets approximate message passing (AMP) algorithm compressed sensing (CS) forward-backward splitting algorithm high-speed jet flow video online compression

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