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

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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On some steplength approaches for proximal algorithms

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APPLIED MATHEMATICS AND COMPUTATION 2015年 253卷 345-362页

作者： Porta, Federica Loris, Ignace Univ Modena & Reggio Emilia Dipartimento Sci Fis Informat & Matemat Modena Italy Univ Libre Bruxelles Dept Math Brussels Belgium

We discuss a number of novel steplength selection schemes for proximal-based convex optimization algorithms. In particular, we consider the problem where the Lipschitz constant of the gradient of the smooth part of the objective function is unknown. We generalize two optimization algorithms of Khobotov type and prove convergence. We also take into account possible inaccurate computation of the proximal operator of the non-smooth part of the objective function. Secondly, we show convergence of an iterative algorithm with Armijo-type steplength rule, and discuss its use with an approximate computation of the proximal operator. Numerical experiments show the efficiency of the methods in comparison to some existing schemes. (C) 2014 Elsevier Inc. All rights reserved.

关键词： proximal algorithms Steplength selection Non-smooth optimization Signal recovering

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Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods

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MATHEMATICAL PROGRAMMING 2013年第1-2期137卷 91-129页

作者： Attouch, Hedy Bolte, Jerome Svaiter, Benar Fux Univ Montpellier 2 UMR CNRS 5149 I3M F-34095 Montpellier France Univ Toulouse 1 GREMAQ TSE F-31042 Toulouse France IMPA BR-22460320 Rio De Janeiro Brazil

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Aojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward-backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss-Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.

关键词： Nonconvex nonsmooth optimization Semi-algebraic optimization Tame optimization Kurdyka-Lojasiewicz inequality Descent methods Relative error Sufficient decrease Forward-backward splitting Alternating minimization proximal algorithms Iterative thresholding Block-coordinate methods o-minimal structures

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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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Approximate iterations in Bregman-function-based proximal algorithms

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MATHEMATICAL PROGRAMMING 1998年第1期83卷 113-123页

作者： Eckstein, J Rutgers State Univ Fac Management Piscataway NJ 08854 USA Rutgers State Univ RUTCOR Piscataway NJ 08854 USA

This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately. The problem being solved is modeled as a general maximal monotone operator, and need not reduce to minimization of a function. The accuracy conditions on the iterates resemble those required for the classical "linear" proximal point algorithm, but are slightly stronger;they should be easier to verify or enforce in practice than conditions given in earlier analyses of approximate generalized proximal methods. Subjects to these practically enforceable accuracy restrictions, convergence is obtained under the same conditions currently established for exact Bregman-function-based proximal methods. (C) 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

关键词： proximal algorithms Bregman functions monotone operators

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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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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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Local Search proximal algorithms as Decision Dynamics with Costs to Move

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SET-VALUED AND VARIATIONAL ANALYSIS 2011年第1期19卷 157-177页

作者： Attouch, H. Soubeyran, A. Univ Montpellier 2 CNRS UMR 5149 I3M F-34095 Montpellier France Univ Mediterranee Chateau Lafarge CNRS GREQAM UMR 6579 F-13290 Les Milles France

Acceptable moves for the "worthwhile-to-move" incremental principle are such that "advantages-to-move" are higher than some fraction of "costs-to-move". When combined with optimization, this principle gives raise to adaptive local search proximal algorithms. Convergence results are given in two distinctive cases, namely low local costs-to-move and high local costs-to-move. In this last case, one obtains a dynamic cognitive approach to Ekeland's I mu-variational principle. Introduction of costs-to-move in the algorithms yields robustness and stability properties.

关键词： Costs-to-move Decision dynamics Exploration process Friction Inertia Local optimization Local search algorithms proximal algorithms Worthwhile-to-move incremental process

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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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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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