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Alternating proximal algorithms for linearly constrained variational inequalities: Application to domain decomposition for PDE's

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第18期74卷 7455-7473页

作者： Attouch, H. Cabot, A. Frankel, P. Peypouquet, J. Univ Montpellier 2 Dept Math F-34095 Montpellier 5 France Univ Tecn Federico Santa Maria Dept Matemat Valparaiso Chile

Let X, Y, Z be real Hilbert spaces, let f : X -> R boolean OR {+infinity}, g : Y -> R boolean OR {+infinity} be closed convex functions and let A : X. Z, B : Y. Z be linear continuous operators. Let us consider the constrained minimization problem (P) min{f (x) + g(y) : Ax = By}. Given a sequence (gamma(n)) which tends toward 0 as n -> +infinity, we study the following alternating proximal algorithm (A) {x(n+1) = argmin {gamma(n+1) f (zeta) + 1/2 parallel to A zeta - By(n) parallel to(2)(Z) + alpha/2 parallel to zeta - x(n)parallel to(2)(X);zeta is an element of X} y(n+1) = argmin {gamma(n+1) g(eta) + 1/2 parallel to Ax(n+1) - B eta parallel to(2)(Z) + nu/2 parallel to eta - y(n)parallel to(2)(Y);eta is an element of Y}, where alpha and. are positive parameters. It is shown that if the sequence (gamma(n)) tends moderately slowly toward 0, then the iterates of (A) weakly converge toward a solution of (P). The study is extended to the setting of maximal monotone operators, for which a general ergodic convergence result is obtained. Applications are given in the area of domain decomposition for PDE's. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Convex minimization Alternating minimization proximal algorithm Variational inequalities Monotone inclusions Domain decomposition for PDE's

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Descentwise inexact proximal algorithms for smooth optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2012年第3期53卷 755-769页

作者： Fuentes, Marc Malick, Jerome Lemarechal, Claude CNRS LJK Grenoble F-38334 Montbonnot St Martin France Inria Rhone Alpes F-38334 Montbonnot St Martin France

The proximal method is a standard regularization approach in optimization. Practical implementations of this algorithm require (i) an algorithm to compute the proximal point, (ii) a rule to stop this algorithm, (iii) an update formula for the proximal parameter. In this work we focus on (ii), when smoothness is present-so that Newton-like methods can be used for (i): we aim at giving adequate stopping rules to reach overall efficiency of the method. Roughly speaking, usual rules consist in stopping inner iterations when the current iterate is close to the proximal point. By contrast, we use the standard paradigm of numerical optimization: the basis for our stopping test is a "sufficient" decrease of the objective function, namely a fraction of the ideal decrease. We establish convergence of the algorithm thus obtained and we illustrate it on some ill-conditioned problems. The experiments show that combining the proposed inexact proximal scheme with a standard smooth optimization algorithm improves the numerical behaviour of the latter for those ill-conditioned problems.

关键词： proximal algorithm Quasi-Newton algorithms Convex analysis

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A Note on the Complexity of proximal Iterative Hard Thresholding algorithm

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Journal of the Operations Research Society of China 2015年第4期3卷 459-473页

作者： Xue Zhang Xiao-Qun Zhang Department of Mathematics Shanghai Jiao Tong UniversityShanghai 200240China Institute of Natural Sciences Shanghai Jiao Tong UniversityShanghai 200240China

The iterative hard thresholding(IHT)algorithm is a powerful and efficient algorithm for solving l_(0)-regularized problems and inspired many applications in sparse-approximation and image-processing ***,some convergence results are established for the proximal scheme of IHT,namely proximal iterative hard thresholding(PIHT)algorithm(Blumensath and Davies,in J Fourier Anal Appl 14:629–654,2008;Hu et al.,Methods 67:294–303,2015;Lu,Math Program 147:125–154,2014;Trzasko et al.,IEEE/SP 14th Workshop on Statistical Signal Processing,2007)on solving the related l_(0)-optimization ***,the complexity analysis for the PIHT algorithm is not well *** this paper,we aim to provide some complexity estimations for the PIHT *** particular,we show that the complexity of the sequential iterate error is at o(1/k).Under the assumption that the objective function is composed of a quadratic convex function and l_(0)regularization,we show that the PIHT algorithm has R-linear convergence ***,we illustrate some applications of this algorithm for compressive sensing reconstruction and sparse learning and validate the estimated error bounds.

关键词： l_(0)Regularization Iterative hard thresholding proximal algorithm Convergence rate R-linear

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Alternating proximal algorithms with asymptotically vanishing coupling. Application to domain decomposition for PDE's

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OPTIMIZATION 2012年第3期61卷 307-325页

作者： Cabot, A. Frankel, P. Univ Montpellier 2 Dept Math F-34095 Montpellier 5 France

Let X, Y, Z be real Hilbert spaces, let f : X -> R boolean OR {+infinity}, g : Y -> R boolean OR {+infinity} be closed convex functions and let A: X -> Z, B : Y -> Z be linear continuous operators. Given a sequence (gamma(n)) which increases towards infinity as n -> +infinity, we study the following alternating proximal algorithm: (A) x(n+1) = argmin {f(zeta) + 1/2 gamma(n+1) parallel to A zeta - By(n)parallel to(2)(z) + alpha/2 parallel to zeta - x(n)parallel to(2)(x);zeta is an element of x} y(n+1) = argmin {g(eta) + 1/2 gamma(n+1) parallel to Ax(n+1) - B eta parallel to(2)(z) + nu/2 parallel to eta - y(n)parallel to(2)(y);eta is an element of y}, where alpha and nu are positive parameters. If the sequence (gamma(n)) increases moderately slowly towards infinity, the algorithm (A) tends to minimize the function (x, y) bar right arrow parallel to Ax - By parallel to(2)(Z) over the set C = argmin f x argmin g (assumed to be nonempty). An illustration of this result is given in the area of domain decomposition for PDE's.

关键词： convex minimization alternating minimization proximal algorithm hierarchical minimization domain decomposition for PDE's

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Geometry-Texture Decomposition/Reconstruction Using a proximal Interior Point algorithm 10

Geometry-Texture Decomposition/Reconstruction Using a Proxim...

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10th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM)

作者： Corbineau, Marie-Caroline Chouzenoux, Emilie Pesquet, Jean-Christophe Univ Paris Saclay INRIA Cent Supelec CVN F-91190 Gif Sur Yvette France Univ Paris Est Marne Vallee CNRS LIGM UMR 8049 Champs Sur Marne France

ISBN: (纸本)9781538647523

The geometry-texture decomposition of images produced by X-Ray Computed Tomography (CT) is a challenging inverse problem which is usually performed in two steps: reconstruction and decomposition. Decomposition can be used for instance to produce an approximate segmentation of the image, but this one can be compromised by artifacts and noise arising from the acquisition and reconstruction processes. We propose a geometry-texture decomposition based on a TV-Laplacian model, well-suited for segmentation and edge detection. The corresponding joint reconstruction and decomposition task from CT data is then formulated as a convex constrained minimization problem. We use our recently introduced proximal interior point method to solve this inverse problem in a reliable manner. Numerical experiments on realistic images of material samples illustrate the practical efficiency of the proposed approach. Our algorithm indeed compares favorably with a state-of-the-art method.

关键词： Geometry-texture decomposition interior point method proximal algorithm computed tomography

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Low-complexity proximal Gauss-Newton algorithm for Nonnegative Matrix Factorization 7

Low-complexity Proximal Gauss-Newton Algorithm for Nonnegati...

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7th IEEE Global Conference on Signal and Information Processing (IEEE GlobalSIP)

作者： Huang, Kejun Fu, Xiao Univ Florida Dept CISE Gainesville FL 32611 USA Oregon State Univ Sch EECS Corvallis OR 97331 USA

ISBN: (纸本)9781728127231

In this paper we propose a quasi-Newton algorithm for the celebrated nonnegative matrix factorization (NMF) problem. The proposed algorithm falls into the general framework of Gauss-Newton and Levenberg-Marquardt methods. However, these methods were not able to handle constraints, which is present in NME One of the key contributions in this paper is to apply alternating direction method of multipliers (ADMM) to obtain the iterative update from this Gauss-Newton-like algorithm. Furthermore, we carefully study the structure of the Jacobian Gramian matrix given by the Gauss-Newton updates, and designed a way of exactly inverting the matrix with complexity (O(mnk), which is a significant reduction compared to the naive implementation of complexity (O((m + n)(3)k(3)). The resulting algorithm, which we call NLS-ADMM, enjoys fast convergence rate brought by the quasi-Newton algorithmic framework, while maintaining low per-iteration complexity similar to that of alternating algorithms. Numerical experiments on synthetic data confirms the efficiency of our proposed algorithm.

关键词： nonnegative matrix factorization Gauss-Newton proximal algorithm alternating direction method of multipliers low-complexity

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A model-based variable step-size strategy for proximal multitask diffusion LMS algorithm

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DIGITAL SIGNAL PROCESSING 2021年 117卷 103199-103199页

作者： Zhang, Yuge Jin, Danqi Chen, Jie Northwestern Polytech Univ Sch Marine Sci & Technol Ctr Intelligent Acoust & Immers Commun Xian Peoples R China

Several practical applications, such as distributed spectrum sensing and channel identification in underwater communication networks with multiple sensors, can be modeled as a distributed network with jointly sparse structure. For such a network, the proximal multitask diffusion least mean square (LMS) algorithm has been proposed in the literature, and its performance has been studied thoroughly. Due to the trade-off between convergence speed and steady-state performance in the proximal multitask diffusion LMS algorithm, it is important but not trivial, to set the step-size parameter properly. To address this issue, a variable step-size strategy for the proximal multitask diffusion LMS algorithm is proposed in this paper. Based on the transient model of the proximal multitask diffusion LMS algorithm, and by minimizing an upper-bound of the excess mean-square error (EMSE) at each iteration on the basis of a white input assumption, we obtain a closed-form expression of the step-size parameter. Simulation results illustrate the effectiveness of the proposed strategy and highlight its performance through comparison with other existing variable step-size strategies, in the cases of white and moderately correlated inputs. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Distributed optimization Diffusion strategy Variable step-size strategy proximal algorithm Transient model

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Nonsmooth Resource Allocation of Multiagent Systems With Disturbances: A proximal Approach

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IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS 2021年第3期8卷 1454-1464页

作者： Zhu, Yanan Wen, Guanghui Yu, Wenwu Yu, Xinghuo Nanjing Univ Informat Sci & Technol Sch Automat Nanjing 210044 Peoples R China Southeast Univ Sch Math Jiangsu Prov Key Lab Networked Collect Intelligen Nanjing 210096 Peoples R China RMIT Univ Sch Engn Melbourne Vic 3001 Australia

This article aims to solve the nonsmooth resource allocation problem in the presence of a global network resource constraint and local set constraints in the framework of multiagent system optimization. It is assumed that multiagent systems are subject to some external disturbances, and the control inputs of the agents satisfy Lispchitz continuity. These two distinguished features render the existing distributed optimization algorithms, especially the subgradient-based algorithms inapplicable due to the employment of discontinuity of subgradients. To solve such a challenging resource allocation problem, a new kind of continuous-time proximal algorithm is designed with the aid of convex optimization theory and the internal-model technique. The proximal algorithm is further augmented by introducing an event-based communication scheme such that the continuous-time communication among the agents is avoided successfully. The theoretical analysis shows that the multiagent systems under the proposed algorithms can converge to the optimal solution of the considered problem, while the external disturbances are rejected. Besides, the Zeno behavior can be excluded for the proximal algorithm with event-based communication. Finally, the numerical simulations are given to verify the established theoretical results.

关键词： Disturbance rejection event-based communication multiagent systems proximal algorithm resource allocation

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proximal Splitting algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists

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SIAM REVIEW 2023年第2期65卷 375-435页

作者： Condat, Laurent Kitahara, Daichi Contreras, Andres Hirabayashi, Akira King Abdullah Univ Sci & Technol KAUST Visual Comp Ctr Thuwal 239556900 Saudi Arabia Osaka Univ Div Elect Elect & Infocommun Engn Suita Japan UC Temuco Fac Engn Dept Math & Phys Sci Temuco La Araucania Chile Ritsumeikan Univ Coll Informat Sci & Engn Kusatsu 5258577 Japan

Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they consist of simple operations, handling the terms in the objective function separately. In this overview, we demystify a selection of recent proximal splitting algorithms: we present them within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metrics. Along the way, we easily derive new variants of the algorithms and revisit existing convergence results, extending the parameter ranges in several cases. In particular, we emphasize that when the smooth term in the objective function is quadratic, e.g., for least-squares problems, convergence is guaranteed with larger values of the relaxation parameter than previously known. Such larger values are usually beneficial for the convergence speed in practice.

关键词： large-scale convex optimization nonsmooth optimization splitting proximal algorithm primal-dual algorithm

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Evaluation of Sparse proximal Multi-Task Learning for Genome-Wide Prediction

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IEEE ACCESS 2024年 12卷 51665-51675页

作者： Fan, Yuhua Launonen, Ilkka Sillanpaa, Mikko J. Waldmann, Patrik Univ Oulu Res Unit Math Sci Oulu 90014 Finland

Multi-task learning (MTL) is a learning paradigm whose aim is to leverage information shared across related tasks to improve the generalization of models. Motivated by the success of proximal optimization algorithms and single-task learning regression models, sparse proximal multi-task learning (SPMTL) for genome-wide prediction (GWP) should be explored. This study investigates proximal gradient descent splitting algorithms with five non-smooth sparsity-inducing norm regularizers, including the novel L-2,(1)/2norm for GWP. Additionally, two popular methods based on Markov chain Monte Carlo (MCMC) are examined. To improve the computational efficiency, parallel Bayesian optimization strategy is employed for efficient hyperparameter tuning. Evaluation is conducted on three different real-world genomic datasets from mice, pigs and wheat, each associated with two, five, and four traits, respectively. Performance is assessed using mean squared error (MSE) and correlation coefficient between predicted and observed trait values in test sets. Experimental results reveal that the L-2,(1)/2 regularizer consistently achieves the best out-of-sample prediction across all datasets, demonstrating the effectiveness of SPMTL in leveraging shared information for improved GWP accuracy. Furthermore, the influence of different regularizers on sparsity and other properties of the SPMTL model are also explored.

关键词： Multi-task learning genome-wide prediction regularization proximal algorithm sparsity Bayesian optimization

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