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An inexact regularized proximal Newton method without line search

COMPUTATIONAL optimization AND APPLICATIONS 2024年第3期89卷 585-624页

作者： vom Dahl, Simeon Kanzow, Christian Univ Wurzburg Inst Math Emil Fischer Str 30 D-97074 Wurzburg Germany

In this paper, we introduce an inexact regularized proximal Newton method (IRPNM) that does not require any line search. The method is designed to minimize the sum of a twice continuously differentiable function f and a convex (possibly non-smooth and extended-valued) function phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}. Instead of controlling a step size by a line search procedure, we update the regularization parameter in a suitable way, based on the success of the previous iteration. The global convergence of the sequence of iterations and its superlinear convergence rate under a local H & ouml;lderian error bound assumption are shown. Notably, these convergence results are obtained without requiring a global Lipschitz property for del f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \nabla f $$\end{document}, which, to the best of the authors' knowledge, is a novel contribution for proximal Newton methods. To highlight the efficiency of our approach, we provide numerical comparisons with an IRPNM using a line search globalization and a modern FISTA-type method.

关键词： nonsmooth and nonconvex optimization Global and local convergence Regularized proximal Newton method H & ouml lderian local error bound

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nonsmooth optimization-Based WADC Tuning in Large Delayed Cyber-Physical Power System by Interarea Mode Tracking and Gradient Sampling

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IEEE TRANSACTIONS ON POWER SYSTEMS 2019年第1期34卷 668-679页

作者： Mou, Qianying Ye, Hua Liu, Yutian Shandong Univ Key Lab Power Syst Intelligent Dispatch & Control Minist Educ Jinan 250061 Shandong Peoples R China

This paper presents a nonsmooth optimization-based method to optimally tune wide-area damping controllers in large delayed cyber-physical power system (DCPPS). The objective is to maximize the damping ratios of the weakly damped interarea oscillation modes under multiple operating conditions. An eigenvalue-perturbation and receding-tracking method is then proposed to reliably trace the targeted interarea oscillation modes during the optimization process. The Broyden-Fletcher-GoldfarbShanno method combined with the gradient sampling technique is employed to efficiently solve the presented nonsmooth, nonconvex, and nonlinear eigenvalue optimization-based problem. The effectiveness of the proposed method is validated by the significantly improved damping of the interarea oscillation modes in the two-area four-machine test system and a real-life large DCPPS.

关键词： Cyber-physical power system eigen-analysis gradient sampling nonsmooth and nonconvex optimization perturbation time delay wide-area damping control

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A BUNDLE TRUST REGION ALGORITHM FOR MINIMIZING LOCALLY LIPSCHITZ FUNCTIONS

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SIAM JOURNAL ON optimization 2023年第1期33卷 319-337页

作者： Monjezi, Najmeh Hoseini Univ Isfahan Fac Math & Stat Dept Appl Math & Comp Sci Esfahan 73441 Iran Inst Res Fundamental Sci IPM Sch Math Tehran 193955746 Iran

We propose a new algorithm for minimizing locally Lipschitz functions that combines both the bundle and trust region techniques. Based on the bundle methods the objective function is approximated by a piecewise linear working model which is updated by adding cutting planes at unsuccessful trial steps. The algorithm defines, at each iteration, a new trial point by solving a subproblem that employs the working model in the objective function subject to a region, which is called the trust region. The algorithm is studied from both theoretical and practical points of view. Under a lower -C1 assumption on the objective function, global convergence of it is verified to stationary points. In order to demonstrate the reliability and efficiency of the proposed algorithm, a MATLAB implementation of it is prepared and numerical experiments have been made using some academic nonsmooth test problems. Computational results show that the developed method is efficient for solving nonsmooth and nonconvex optimization problems.

关键词： nonsmooth and nonconvex optimization trust region method bundle algorithm

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Robust Design of Power System Controllers Based on optimization of Pseudospectral Functions

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IEEE TRANSACTIONS ON POWER SYSTEMS 2013年第2期28卷 1756-1765页

作者： Dill, Gustavo K. Silva, Aguinaldo S. e Univ Fed Santa Catarina Dept Elect Engn Florianopolis SC Brazil

Robust design methods for power systems must deal with the fixed-structure of controllers and the system large dimension. This limits the applicability of the available design methods. In this paper, the power system stabilizer (PSS) design is posed as a multiobjective optimization problem combining performance and robustness requirements. Robustness is dealt with using concepts of pseudospectra. Evolutionary algorithms are used to solve the multiobjective optimization problem. The method is applied to a small test system, in order to clarify its characteristics and to the New England New York test system, selected for comparison purposes since it is a widely used test system.

关键词： Control robustness multiobjective optimization nonsmooth and nonconvex optimization pseudospectra power system stabilizer (PSS) and power system control

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Asynchronous Activity Detection for Cell-Free Massive MIMO: From Centralized to Distributed Algorithms

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2023年第4期22卷 2477-2492页

作者： Li, Yang Lin, Qingfeng Liu, Ya-Feng Ai, Bo Wu, Yik-Chung Shenzhen Res Inst Big Data Shenzhen 518172 Peoples R China Beijing Jiaotong Univ State Key Lab Rail Traff Control & Safety Beijing 100044 Peoples R China Univ Hong Kong Dept Elect & Elect Engn Hong Kong Peoples R China Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math & Sci Engn Comp State Key Lab Sci & Engn Comp Beijing 100190 Peoples R China Peng Cheng Lab Shenzhen 518055 Peoples R China Zhengzhou Univ Henan Joint Int Res Lab Intelligent Networking & D Zhengzhou 450001 Peoples R China

Device activity detection in the emerging cell-free massive multiple-input multiple-output (MIMO) systems has been recognized as a crucial task in machine-type communications, in which multiple access points (APs) jointly identify the active devices from a large number of potential devices based on the received signals. Most of the existing works addressing this problem rely on the impractical assumption that different active devices transmit signals synchronously. However, in practice, synchronization cannot be guaranteed due to the low-cost oscillators, which brings additional discontinuous and nonconvex constraints to the detection problem. To address this challenge, this paper reveals an equivalent reformulation to the asynchronous activity detection problem, which facilitates the development of a centralized algorithm and a distributed algorithm that satisfy the highly nonconvex constraints in a gentle fashion as the iteration number increases, so that the sequence generated by the proposed algorithms can get around bad stationary points. To reduce the capacity requirements of the fronthauls, we further design a communication-efficient accelerated distributed algorithm. Simulation results demonstrate that the proposed centralized and distributed algorithms outperform state-of-the-art approaches, and the proposed accelerated distributed algorithm achieves close detection performance to that of the centralized algorithm but with a much smaller number of bits to be transmitted on the fronthaul links.

关键词： Asynchronous activity detection cell-free massive multiple-input multiple-output (MIMO) grant-free random access Internet-of-Things (IoT) machine-type communications (MTC) nonsmooth and nonconvex optimization

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Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

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COMPUTATIONAL optimization AND APPLICATIONS 2021年第3期78卷 705-740页

作者： Geiersbach, Caroline Scarinci, Teresa Weierstrass Inst Mohrenstr 39 D-10117 Berlin Germany Univ Aquila Dept Informat Engn Comp Sci & Math Via Vetoio Loc Coppito I-67010 Laquila Italy

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a L-1-penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation.

关键词： Stochastic programming nonsmooth and nonconvex optimization Differential inclusions Mathematical programming methods Partial differential equations with randomness Optimal control problems involving partial differential equations

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Convergence of the reweighted l₁ minimization algorithm for l₂-l_p minimization

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COMPUTATIONAL optimization AND APPLICATIONS 2014年第1-2期59卷 47-61页

作者： Chen, Xiaojun Zhou, Weijun Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Changsha Univ Sci & Technol Dept Math Changsha 410004 Hunan Peoples R China

The iteratively reweighted a"" (1) minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the a"" (2)-a"" (p) minimization problem with 0 < p < 1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm.

关键词： l(p) minimization Stationary points nonsmooth and nonconvex optimization Pseudo convex Global convergence

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Properties of approximated empirical mode decomposition and optimal design of its system kernel matrix for signal decomposition

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SIGNAL IMAGE AND VIDEO PROCESSING 2019年第6期13卷 1173-1181页

作者： Tian, Nili Wang, Xiaoling Ling, Bingo Wing-Kuen Sakalli, Mustafa Guangdong Univ Technol Sch Informat Engn Guangzhou Higher Educ Mega Ctr 100 Waihuan Xi Rd Guangzhou Guangdong Peoples R China Marmara Univ CSE Engn Istanbul Turkey

An approximated empirical mode decomposition generates a set of approximated intrinsic mode functions via a linear, nonadaptive but iterative approach. The decomposition was found to be very useful for a content-independent pattern recognition application. As the process is characterized by a system kernel matrix and performed iteratively, the approximated intrinsic mode functions can be understood as the original signals processed by a set of mask operations. Here, some properties of the decomposition are studied and an optimal design of the system kernel matrix is proposed. It is found that there is only one approximated intrinsic mode function if the exact perfect reconstruction condition is satisfied. Obviously, the approximated intrinsic mode function is the original signal. Therefore, the decomposition is practically not meaningful. To address this issue, the infinite number of iterations in the algorithm is truncated to a finite number of iterations. Also, the design of the system kernel matrix is formulated as an optimization problem. In particular, the exact perfect reconstruction error between the sum of the approximated intrinsic mode functions and the original signal is minimized and the total absolute sum of the difference between any two different eigenvalues of the iterative matrix is maximized subject to a stability condition. Here, the stability condition refers to the eigenvalues of the iterative matrix being between zero and one. Since the optimization problem is nonsmooth and nonconvex, a genetic algorithm is employed for finding its near global optimal solution. Compared to the conventional approximated empirical mode decomposition, computer numerical simulations show that our proposed approach can achieve more than one approximated intrinsic mode function with each approximated intrinsic mode function corresponding to an output of a more meaningful mask operation.

关键词： Approximated Empirical mode decomposition System kernel matrix nonsmooth and nonconvex optimization Genetic algorithm Nilpotent matrix

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Low-Rank Matrix Recovery With Scaled Subgradient Methods: Fast and Robust Convergence Without the Condition Number

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 2396-2409页

作者： Tong, Tian Ma, Cong Chi, Yuejie Carnegie Mellon Univ Dept Elect & Comp Engn Pittsburgh PA 15213 USA Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA

Many problems in data science can be treated as estimating a low-rank matrix from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are optimized via first-order methods over a smooth loss function, such as the residual sum of squares. While tremendous progress has been made in recent years, the natural smooth formulation suffers from two sources of ill-conditioning, where the iteration complexity of gradient descent scales poorly both with the dimension as well as the condition number of the low-rank matrix. Moreover, the smooth formulation is not robust to corruptions. In this paper, we propose scaled subgradient methods to minimize a family of nonsmooth and nonconvex formulations-in particular, the residual sum of absolute errors-which is guaranteed to converge at a fast rate that is almost dimension-free and independent of the condition number, even in the presence of corruptions. We illustrate the effectiveness of our approach when the observation operator satisfies certain mixed-norm restricted isometry properties, and derive state-of-the-art performance guarantees for a variety of problems such as robust low-rank matrix sensing and quadratic sampling.

关键词： Sensors Convergence Signal processing algorithms Robustness Gradient methods Computational complexity Velocity measurement Low-rank matrix recovery nonsmooth and nonconvex optimization residual sum of absolute errors scaled subgradient methods

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Efficient Reconstruction of Piecewise Constant Images Using nonsmooth nonconvex Minimization

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SIAM JOURNAL ON IMAGING SCIENCES 2008年第1期1卷 2-25页

作者： Nikolova, Mila Ng, Michael K. Zhang, Shuqin Ching, Wai-Ki Hong Kong Baptist Univ Ctr Math Imaging & Vis Kowloon Hong Kong Peoples R China ENS Cachan Ctr Math & Leurs Applicat F-94235 Cachan France Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

We consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies {J(epsilon)}, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each J(epsilon) is nonsmooth and is expressed as the sum of an l(1) regularization term and a smooth nonconvex function. Furthermore, the local minimization of each J(epsilon) is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the p

关键词： image restoration regularization nonsmooth and nonconvex optimization continuation interior point method unsupervised segmentation inverse problems deblurring constrained optimization graduated nonconvexity (GNC)

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