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...
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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.
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 Lip...
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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.
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 pre...
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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.
This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski-Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski-Mann i...
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This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski-Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski-Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward-backwardsplitting methods and Douglas-Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski-Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others.
First-order methods such as proximal gradient, which use forward-backwardsplitting techniques have proved to be very effective in solving nonsmooth convex minimization problem, which is useful in solving various prac...
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First-order methods such as proximal gradient, which use forward-backwardsplitting techniques have proved to be very effective in solving nonsmooth convex minimization problem, which is useful in solving various practical problems in different fields such as machine learning and image processing. In this paper, we propose few new forward-backward splitting algorithms, which consume less number of iterations to converge to an optimum. In addition, we derive convergence rates for the proposed formulations and show that the speed of convergence of these algorithms is significantly better than the traditional forward-backwardalgorithm. To demonstrate the practical applicability, we apply them to two real-world problems of machine learning and image processing. The first issue deals with the regression on high-dimensional datasets, whereas the second one is the image deblurring problem. Numerical experiments have been conducted on several publicly available real datasets to verify the obtained theoretical results. Results demonstrate the superiority of our algorithms in terms of accuracy, the number of iterations required to converge and the rate of convergence against the classical first-order methods.
The purpose of this paper is by using the generalized forward-backwardsplitting method and implicit midpoint rule to propose an iterative algorithm for finding a common element of the set of solutions to a system of ...
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The purpose of this paper is by using the generalized forward-backwardsplitting 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.
The purpose of this paper is by using a generalized forward-backwardsplitting method to propose an iterative algorithm for finding a common element of the set of solutions to a system of quasi-variational inclusions ...
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The purpose of this paper is by using a generalized forward-backwardsplitting 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.
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 ...
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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.
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...
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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.
In dictionary learning, sparse regularization is used to promote sparsity and has played a major role in the developing of dictionary learning algorithms. l(1)-norm is of the most popular sparse regularization due to ...
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In dictionary learning, sparse regularization is used to promote sparsity and has played a major role in the developing of dictionary learning algorithms. l(1)-norm is of the most popular sparse regularization due to its convexity and the related tractable convex optimization problems. However, l(1)-norm leads to biased solutions and provides inferior performance on certain applications compared with nonconvex sparse regularizations. In this work, we propose a generalized minimax-concave (GMC) sparse regularization, which is nonconvex, to promote sparsity to design dictionary learning model. Applying the alternate optimization scheme, we use the forward-backwardsplitting (FBS) algorithm to solve the sparse coding problem. As the improvement, we incorporate Nesterov's acceleration technique and adaptive threshold scheme into the FBS algorithm to improve the convergence efficiency and performance. In the dictionary update step, we apply the difference of convex functions (DC) programming and the DC algorithm (DCA) to address the dictionary update. Two dictionary update algorithms are designed;one updates the dictionary atoms one by one, and the other one updates the dictionary atoms simultaneously. The presented dictionary learning algorithms perform robustly in dictionary recovery. Numerical experiments are designed to verify the performance of proposed algorithms and to compare with the state-of-the-art algorithms. (C) 2020 Published by Elsevier B.V.
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