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proximal THRESHOLDING algorithm FOR MINIMIZATION OVER ORTHONORMAL BASES

SIAM JOURNAL ON OPTIMIZATION 2008年第4期18卷 1351-1376页

作者： Combettes, Patrick L. Pesquet, Jean-Christophe Univ Paris 06 Fac Math Lab Jacques Louis Lions UMR CNRS 7598 F-75005 Paris France Univ Marne la Vallee Inst Gaspard Monge F-77454 Marne La Vallee 2 France Univ Marne la Vallee UMR CNRS 8049 F-77454 Marne La Vallee 2 France

The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing, in particular, several characterizations of such thresholders. We then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and we establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated.

关键词： convex programming deconvolution denoising forward-backward splitting algorithm Hilbert space orthonormal basis proximal algorithm proximal thresholding proximity operator signal recovery soft thresholding strong convergence

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A proximal MINIMIZATION algorithm FOR STRUCTURED NONCONVEX AND NONSMOOTH PROBLEMS

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SIAM JOURNAL ON OPTIMIZATION 2019年第2期29卷 1300-1328页

作者： Bot, Radu Ioan Csetnek, Erno Robert Dang-Khoa Nguyen Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria

We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function (with each of the nonsmooth summands depending on an independent block variable), and a smooth function which couples the two block variables. The algorithm is a full splitting method, which means that the nonsmooth functions are processed via their proximal operators, the smooth function via gradient steps, and the linear operator via matrix times vector multiplication. We provide sufficient conditions for the boundedness of the generated sequence and prove that any cluster point of the latter is a KKT point of the minimization problem. In the setting of the Kurdyka-Lojasiewicz property, we show global convergence and derive convergence rates for the iterates in terms of the Lojasiewicz exponent.

关键词： structured nonconvex and nonsmooth optimization proximal algorithm full splitting scheme Kurdyka-Lojasiewicz property limiting subdifferential

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A proximal distance algorithm for likelihood-based sparse covariance estimation

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BIOMETRIKA 2022年第4期109卷 1047-1066页

作者： Xu, Jason Lange, Kenneth Duke Univ Dept Stat Sci Box 90251 Durham NC 27708 USA Univ Calif Los Angeles Dept Computat Med Box 708822 Los Angeles CA 90095 USA

This paper addresses the task of estimating a covariance matrix under a patternless sparsity assumption. In contrast to existing approaches based on thresholding or shrinkage penalties, we propose a likelihood-based method that regularizes the distance from the covariance estimate to a symmetric sparsity set. This formulation avoids unwanted shrinkage induced by more common norm penalties, and enables optimization of the resulting nonconvex objective by solving a sequence of smooth, unconstrained subproblems. These subproblems are generated and solved via the proximal distance version of the majorization-minimization principle. The resulting algorithm executes rapidly, gracefully handles settings where the number of parameters exceeds the number of cases, yields a positive-definite solution, and enjoys desirable convergence properties. Empirically, we demonstrate that our approach outperforms competing methods across several metrics, for a suite of simulated experiments. Its merits are illustrated on international migration data and a case study on flow cytometry. Our findings suggest that the marginal and conditional dependency networks for the cell signalling data are more similar than previously concluded.

关键词： Distance-to-set penalty Majorization-minimization Penalized likelihood proximal algorithm Sequential unconstrained minimization Sparse estimation

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ProMeSCT: A proximal Metric algorithm for Spectral CT

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IEEE TRANSACTIONS ON RADIATION AND PLASMA MEDICAL SCIENCES 2021年第4期5卷 548-558页

作者： Tairi, Souhil Anthoine, Sandrine Boursier, Yannick Dupont, Mathieu Morel, Christian Aix Marseille Univ CNRS IN2P3 CPPM F-13009 Marseille France Aix Marseille Univ CNRS Cent Marseille 12M F-13009 Marseille France

The acquisition of a set of spectral photon-counting computed tomography (spectral PC-CT) measurements aims at uncovering both the spatial and energetic characteristics of the imaged body, which widens the potential of tomography compared to classical computed tomography (CT). In the preclinical context, the use of polychromatic beams induces spectral mixing and, as a consequence, the reconstruction procedure requires specific algorithmic tools more complex than the standard ones used in CT. In this article, we propose a one-step inversion method to simultaneously separate and reconstruct the physical materials of an object observed in the context of spectral PC-CT. To do so, we carefully consider the underlying polychromatic model of the X-ray beam and combine it with a priori on the materials of the object to reconstruct. The simultaneous separation and reconstruction of materials is done by minimizing the resulting nonconvex ill-posed inverse problem. The dimensionality of the data and object materials worsens the computational complexity of the problem. We propose an efficient optimization algorithm based on a proximal forward-backward algorithm that is accelerated by a metric, which is specifically designed for spectral PC-CT. The efficiency of our method called ProMeSCT is demonstrated on results obtained on 3-D synthetic data with a simple regularization that encompasses the positivity of the quantities of interest.

关键词： One-step inversion photon-counting (PC) computed tomography (CT) proximal algorithm spectral CT

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A connection between Chidume iteration and the contraction-proximal point algorithm

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AFRIKA MATEMATIKA 2015年第1-2期26卷 127-130页

作者： Moudafi, Abdellatif French West Indies Univ Ceregmia DSI Schoelcher Martinique France

In this short note we comment on some recent results on the convergence of two type of iterative algorithms for finding a zero of a maximal monotone operator. The first one is based on a proximal approach and was investigated by many authors, see for example (Boikanyo and Morosanu in Nonlinear Anal TMA 74:544-555, 2011;Kamimura and Takahashi in J Approx Theory 106:226-240, 2(X)0, Marino and Xu in Comm Pure Appl Anal 3:791-808, 2004, Wang and Cui in J Global Optim 54:485-491, 2012, Yao and Noor in J Comput Appl Math 217:46-55, 2008) and the references therein. The second is based on a gradient approach and was essentially developed by Chidume et al. see for instance (Fixed Point Theory 2013, Chidume and Djitte in Abst Appl Anal doi:10.1155/2012/681348 2012, Chidume et al. in J Nonlinear Cony Anal (to appear) 2013, Ofoedu and Zegeye in J Math Anal Appl 372:68-76, 2010) and the references therein. Then, relying on the Yosida approximate, we compare and make a link between these two approaches.

关键词： Maximal monotone operators Gradient method proximal algorithm Regularization Yosida approximate Strong convergence

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proximal algorithms and temporal difference methods for solving fixed point problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2018年第3期70卷 709-736页

作者： Bertsekas, Dimitri P. MIT Dept Elect Engn & Comp Sci Lab Informat & Decis Syst Cambridge MA 02139 USA

In this paper we consider large fixed point problems and solution with proximal algorithms. We show that for linear problems there is a close connection between proximal iterations, which are prominent in numerical analysis and optimization, and multistep methods of the temporal difference type such as TD(), LSTD(), and LSPE(), which are central in simulation-based exact and approximate dynamic programming. One benefit of this connection is a new and simple way to accelerate the standard proximal algorithm by extrapolation towards a multistep iteration, which generically has a faster convergence rate. Another benefit is the potential for integration into the proximal algorithmic context of several new ideas that have emerged in the approximate dynamic programming context, including simulation-based implementations. Conversely, the analytical and algorithmic insights from proximal algorithms can be brought to bear on the analysis and the enhancement of temporal difference methods. We also generalize our linear case result to nonlinear problems that involve a contractive mapping, thus providing guaranteed and potentially substantial acceleration of the proximal and forward backward splitting algorithms at no extra cost. Moreover, under certain monotonicity assumptions, we extend the connection with temporal difference methods to nonlinear problems through a linearization approach.

关键词： proximal algorithm Temporal differences Dynamic programming Convex optimization Fixed point problems

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proximal algorithms for Multicomponent Image Recovery Problems

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2011年第1-2期41卷 3-22页

作者： Briceno-Arias, L. M. Combettes, P. L. Pesquet, J. -C. Pustelnik, N. UPMC Univ Paris 06 CNRS Lab Jacques Louis Lions UMR 7598 F-75005 Paris France UPMC Univ Paris 06 CNRS Equipe Combinatoire & Optimisat FRE 3232 F-75005 Paris France Univ Paris Est CNRS Lab Informat Gaspard Monge UMR 8049 F-77454 Marne La Vallee 2 France

In recent years, proximal splitting algorithms have been applied to various monocomponent signal and image recovery problems. In this paper, we address the case of multicomponent problems. We first provide closed form expressions for several important multicomponent proximity operators and then derive extensions of existing proximal algorithms to the multicomponent setting. These results are applied to stereoscopic image recovery, multispectral image denoising, and image decomposition into texture and geometry components.

关键词： Convex minimization Image recovery Inverse problems Multicomponent images Multichannel images Multispectral images proximal algorithm Sparsity Stereoscopy Wavelets

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Convergence analysis of a primal-dual optimization-by-continuation algorithm

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 457卷

作者： Loris, Ignace Rebegoldi, Simone Univ Libre Bruxellles Math Dept Blvd Triomphe B-1050 Brussels Belgium Univ Modena & Reggio Emilia Dipartimento Sci Fis Informat & Matemat Via Campi 213-B I-41125 Modena Italy

We present a numerical iterative optimization algorithm for the minimization of a cost function consisting of a linear combination of three convex terms, one of which is differentiable, a second one is prox-simple and the third one is the composition of a linear map and a proxsimple function. The algorithm's special feature lies in its ability to approximate, in a single iteration run, the minimizers of the cost function for many different values of the parameters determining the relative weight of the three terms in the cost function. A proof of convergence of the algorithm, based on an inexact variable metric approach, is also provided. As a special case, one recovers a generalization of the primal-dual algorithm of Chambolle and Pock, and also of the proximal-gradient algorithm. Finally, we show how it is related to a primal-dual iterative algorithm based on inexact proximal evaluations of the non-smooth terms of the cost function.

关键词： Primal-dual algorithm proximal algorithm Inverse problem Iterative algorithm

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proximal algorithms for a class of mixed equilibrium problems

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FIXED POINT THEORY AND APPLICATIONS 2012年第1期2012卷 1-15页

作者： Song, Yisheng Zhang, Qingnian Henan Normal Univ Coll Math & Informat Sci Xinxiang 453007 Henan Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China North China Univ Water Resources & Elect Power Coll Math & Informat Sci Zhengzhou 450011 Peoples R China

We present two proximal algorithms for solving the mixed equilibrium problems. Under some simpler framework, the strong and weak convergence of the sequences defined by two general algorithms is respectively obtained. In particular, we deal with several iterative schemes in a united way and apply our algorithms for solving the classical equilibrium problem, the minimization problem, the classical variational inequality problem and the generalized variational inequality problem. Our results properly include some corresponding results in this field as a special case.

关键词： mixed equilibrium problem proximal algorithm variational inequality

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Event-Triggered proximal Online Gradient Descent algorithm for Parameter Estimation

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2024年 72卷 2594-2606页

作者： Zhou, Yaoyao Chen, Gang Chongqing Univ Coll Automat Chongqing 400044 Peoples R China

The constrained composite-convex parameter estimation problem on the networked system, where the composite-convex function consists of a sum of node-specific smooth loss functions and a nonsmooth regularizer, is investigated in this paper. To reduce the communication burden, the event-triggered mechanism is introduced and the novel event-triggered proximal online gradient descent algorithm (EPOGDA) is proposed. The analysis shows that if the event-triggered threshold converges to zero as time tends to infinity and the cumulative difference between consecutive optimal values is sublinear, the dynamic regret of EPOGDA is sublinear. Further, we extend the proposed EPOGDA to the gradient-free scenarios, where the gradients are estimated using the Gaussian smoothed gradient estimator (GSGE). The GSGE-EPOGDA is presented and analyzed, which does not lead to performance degradation as compared to EPOGDA. Finally, the advantages of EPOGDA and GSGE-EPOGDA are verified on a distributed multi-sensor network.

关键词： Heuristic algorithms Signal processing algorithms Sensors Estimation Convex functions Parameter estimation Dynamic scheduling Distributed parameter estimation online convex optimization event-triggered mechanism dynamic regret proximal algorithm

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