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Convergence Rate Comparison of proximal algorithms for Non-Smooth Convex Optimization With an Application to Texture Segmentation

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IEEE SIGNAL PROCESSING LETTERS 2022年 29卷 1337-1341页

作者： Briceno-Arias, L. M. Pustelnik, N. Univ Tecn Federico Santa Maria Dept Math Santiago 8940897 Chile Univ Lyon Ens Lyon Univ Lyon 1 Lab PhysCNRS Lyon France UCLouvain ICTEAM Inst ISPGrp INMA B-1348 Louvain La Neuve Belgium

In this paper we provide a theoretical and numerical comparison of convergence rates of forward-backward, Douglas-Rachford, and Peaceman-Rachford algorithms for minimizing the sum of a convex proper lower semicontinuous function and a strongly convex differentiable function with Lipschitz continuous gradient. Our results extend the comparison made in [1], when both functions are smooth, to the context where only one is assumed differentiable. Optimal step-sizes and rates of the three algorithms are compared theoretically and numerically in the context of texture segmentation problem, obtaining very sharp estimations and illustrating the high efficiency of Peaceman-Rachford splitting.

关键词： Convergence Signal processing algorithms Task analysis Optimization Linear programming Fractals Convex functions Convex optimization convergence rates proximal algorithms texture segmentation

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Delay-tolerant distributed Bregman proximal algorithms

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OPTIMIZATION METHODS & SOFTWARE 2024年

作者： Chraibi, S. Iutzeler, F. Malick, J. Rogozin, A. Univ Grenoble Alpes LJK Grenoble France Univ Grenoble Alpes CNRS Grenoble INP LJK Grenoble France Moscow Inst Phys & Technol Moscow Russia Univ Grenoble Alpes LJK F-38000 Grenoble France

Many problems in machine learning write as the minimization of a sum of individual loss functions over the training examples. These functions are usually differentiable but, in some cases, their gradients are not Lipschitz continuous, which compromises the use of (proximal) gradient algorithms. Fortunately, changing the geometry and using Bregman divergences can alleviate this issue in several applications, such as for Poisson linear inverse problems. However, the Bregman operation makes the aggregation of several points and gradients more involved, hindering the distribution of computations for such problems. In this paper, we propose an asynchronous variant of the Bregman proximal-gradient method, able to adapt to any centralized computing system. In particular, we prove that the algorithm copes with arbitrarily long delays and we illustrate its behaviour on distributed Poisson inverse problems.

关键词： Distributed optimization asynchronous methods proximal algorithms Bregman divergence Poisson inverse problems

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proximal algorithms for Large-Scale Statistical Modeling and Sensor/Actuator Selection

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2020年第8期65卷 3441-3456页

作者： Zare, Armin Mohammadi, Hesameddin Dhingra, Neil K. Georgiou, Tryphon T. Jovanovic, Mihailo R. Univ Texas Dallas Dept Mech Engn Dallas TX 75219 USA Univ Southern Calif Dept Elect & Comp Engn Los Angeles CA 90089 USA Numerica Corp Ft Collins CO 80528 USA Univ Calif Irvine Dept Mech & Aerosp Engn Irvine CA 92697 USA

Several problems in modeling and control of stochastically driven dynamical systems can be cast as regularized semidefinite programs. We examine two such representative problems and show that they can be formulated in a similar manner. The first, in statistical modeling, seeks to reconcile observed statistics by suitably and minimally perturbing prior dynamics. The second seeks to optimally select a subset of available sensors and actuators for control purposes. To address modeling and control of large-scale systems, we develop a unified algorithmic framework using proximal methods. Our customized algorithms exploit problem structure and allow handling statistical modeling, as well as sensor and actuator selection, for substantially larger scales than what is amenable to current general-purpose solvers. We establish linear convergence of the proximal gradient algorithm, draw contrast between the proposed proximal algorithms and the alternating direction method of multipliers, and provide examples that illustrate the merits and effectiveness of our framework.

关键词： Sensors Actuators Mathematical model Covariance matrices Stochastic processes Perturbation methods Heuristic algorithms Actuator selection method of multipliers (MM) nonsmooth convex optimization proximal algorithms regularization for design semidefinite programming sensor selection sparsity-promoting estimation and control structured covariances

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Union Averaged Operators with Applications to proximal algorithms for Min-Convex Functions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第1期181卷 61-94页

作者： Dao, Minh N. Tam, Matthew K. Univ Newcastle CARMA Callaghan NSW 2308 Australia Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany

In this paper, we introduce and study a class of structured set-valued operators, which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations, where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions, where the first is convex and the second can be expressed as the minimum of finitely many convex functions.

关键词： Admissible control Averaged operator Fixed point iteration Local convergence proximal algorithms Set-valued map

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Optimization on Spheres: Models and proximal algorithms with Computational Performance Comparisons

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SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE 2019年第3期1卷 408-445页

作者： Luke, D. Russell Sabach, Shoham Teboulle, Marc Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany Technion Israel Inst Technol Fac Ind Engn & Management IL-3200003 Haifa Israel Tel Aviv Univ Sch Math Sci IL-69978 Ramat Aviv Israel

We present a unified treatment of the abstract problem of finding the best approximation between a cone and spheres in the image of affine transformations. Prominent instances of this problem are phase retrieval and source localization. The common geometry binding these problems permits a generic application of algorithmic ideas and abstract convergence results for nonconvex optimization. We organize variational models for this problem into three different classes and derive the main algorithmic approaches within these classes (13 in all). We identify the central ideas underlying these methods and provide thorough numerical benchmarks comparing their performance on synthetic and laboratory data. The software and data of our experiments are all publicly accessible. We also introduce one new algorithm, a cyclic relaxed Douglas-Rachford algorithm, which outperforms all other algorithms by every measure: speed, stability, and accuracy. The analysis of this algorithm remains open.

关键词： phase retrieval nonconvex optimization nonsmooth optimization proximal algorithms feasibility fixed points source localization

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Ptychographic phase retrieval by proximal algorithms

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NEW JOURNAL OF PHYSICS 2020年第2期22卷

作者： Yan, Hanfei Brookhaven Natl Lab Natl Synchrotron Light Source 2 Upton NY 11973 USA

We derive a set of ptychography phase-retrieval iterative engines based on proximal algorithms originally developed in convex optimization theory, and discuss their connections with existing ones. The use of proximal operator creates a simple frame work that allows us to incorporate the effect of noise from a maximum-likelihood (ML) principle. We focus on three particular algorithms, namely proximal minimization, alternating direction method of multiplier and accelerated proximal gradient (APG). We benchmark their performance with numerical simulations, and discuss their optimal conditions for convergence and accuracy. An experimental dataset is used to demonstrate their effectiveness as well, in which case an array of cubic Au nanoparticles with a size of 50 nm is imaged. We show that with the presence of Poisson noise, a dataset with photon counts up to 10(4) at one detector pixel already requires ML-based methods to achieve a stable convergence. Among the three algorithms derived in this work, APG method is reported first time for its application in ptychographic reconstruction and shows superior performance in terms of both accuracy and convergence rate with a noisy dataset.

关键词： ptychographic phase-retrieval proximal algorithms x-ray imaging maximum-likelihood

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MODULAR-proximal GRADIENT algorithms IN VARIABLE EXPONENT LEBESGUE SPACES

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2022年第6期44卷 A3463-A3489页

作者： Lazzaretti, Marta Calatroni, Luca Estatico, Claudio Univ Cote Azur CNRS Lab I3S F-06900 Sophia Antipolis France Univ Genoa Dipartimento Matemat I-16146 Genoa Italy

We consider structured optimization problems defined in terms of the sum of a smooth and convex function and a proper, lower semicontinuous (l.s.c.), convex (typically nonsmooth) function in reflexive variable exponent Lebesgue spaces Lp(\cdot )(\Omega ). Due to their intrinsic space-variant properties, such spaces can be naturally used as solution spaces and combined with space-variant functionals for the solution of ill-posed inverse problems. For this purpose, we propose and analyze two instances (primal and dual) of proximal-gradient algorithms in Lp(\cdot )(\Omega ), where the proximal step, rather than depending on the natural (nonseparable) Lp(\cdot )(\Omega ) norm, is defined in terms of its modular function, which, thanks to its separability, allows for the efficient computation of algorithmic iterates. Convergence in function values is proved for both algorithms, with convergence rates depending on problem/space smoothness. To show the effectiveness of the proposed modeling, some numerical tests highlighting the flexibility of the space Lp(\cdot )(\Omega ) are shown for exemplar deconvolution and mixed noise removal problems. Finally, a numerical verification of the convergence speed and computational costs of both algorithms in comparison with analogous ones defined in standard Lp(\Omega ) spaces is presented.

关键词： proximal algorithms variable exponent Lebesgue spaces modular proximity sep-arability thresholding functions

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Data-driven proximal algorithms for the design of structured optimal feedback gains

Data-driven proximal algorithms for the design of structured...

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American Control Conference (ACC)

作者： Hassan-Moghaddam, Sepideh Jovanovic, Mihailo R. Meyn, Sean Univ Southern Calif Dept Elect & Comp Engn Los Angeles CA 90089 USA Univ Florida Dept Elect & Comp Engn Gainesville FL 32611 USA

ISBN: (纸本)9781538679265

Distributed feedback design and complexity constrained control are examples of problems posed within the domain of structured optimal feedback synthesis. The optimal feedback gain is typically a non-convex function of system primitives. However, in recent years, algorithms have been proposed to obtain locally optimal solutions. In applications to large-scale distributed control, the major obstacle is computational complexity. This paper addresses complexity through a combination of linear-algebraic techniques and computational methods adapted from both machine learning and reinforcement learning. It is shown that for general classes of optimal control problems, the objective function and its gradient can be computed from data. Transformations borrowed from the theory of reinforcement learning are adapted to obtain simulation-based algorithms for computing the structured optimal H-2 feedback gain. Customized proximal algorithms based on gradient descent and incremental gradient are tested in computational experiments and their relative merits are discussed.

关键词： Data-driven feedback design large-scale systems optimization proximal algorithms reinforcement learning structured and sparsity-promoting optimal control

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A Dynamical Systems Perspective to Convergence Rate Analysis of proximal algorithms 55

A Dynamical Systems Perspective to Convergence Rate Analysis...

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55th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

作者： Fazlyab, Mahyar Ribeiro, Alejandro Morari, Manfred Preciado, Victor M. Univ Penn Dept Elect & Syst Engn 200 South 33rd St Philadelphia PA 19104 USA

ISBN: (纸本)9781538632666

In this paper, we develop a semidefinite programming (SDP) framework for convergence rate analysis of proximal algorithms designed to solve convex composite optimization problems. We first represent these algorithms as linear dynamical systems interconnected with a nonlinear component. We then propose a family of time-varying nonquadratic Lyapunov functions that are particularly useful for establishing arbitrary (exponential or subexponential) convergence rates. Using Integral Quadratic Constraints (IQCs) to describe the class of allowable nonlinearities in the interconnection, we derive sufficient conditions for Lyapunov stability of proximal algorithms in terms of Linear Matrix Inequalities (LMIs). We show how the developed LMI-based framework can be used to establish convergence rates by specializing it to the proximal gradient method and its accelerated variant for both convex and strongly convex problems.

关键词： Composite Convex Optimization proximal algorithms Nesterov's Accelerated Method Semidefinite Programming Integral Quadratic Constraints

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GASPACHO: a Generic Automatic Solver using proximal algorithms for Convex Huge Optimization problems 17

GASPACHO: a Generic Automatic Solver using Proximal Algorith...

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Conference on Wavelets and Sparsity XVII

作者： Goossens, Bart Luong, Hiep Philips, Wilfried Univ Ghent Dept Telecommun & Informat Proc Imec IPI UGent Sint Pietersnieuwstr 41 B-9000 Ghent Belgium

ISBN: (数字)9781510612464

ISBN: (纸本)9781510612464;9781510612457

Many inverse problems (e.g., demosaicking, deblurring, denoising, image fusion, HDR synthesis) share various similarities: degradation operators are often modeled by a specific data fitting function while image prior knowledge (e.g., sparsity) is incorporated by additional regularization terms. In this paper, we investigate automatic algorithmic techniques for evaluating proximal operators. These algorithmic techniques also enable efficient calculation of adjoints from linear operators in a general matrix-free setting. In particular, we study the simultaneous-direction method of multipliers (SDMM) and the parallel proximal algorithm (PPXA) solvers and show that the automatically derived implementations are well suited for both single-GPU and multi-GPU processing. We demonstrate this approach for an Electron Microscopy (EM) deconvolution problem.

关键词： convex optimization automatic differentiation inverse problems proximal algorithms

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