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A globally convergent BFGS method with nonmonotone line search for non-convex minimization

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第1期230卷 95-106页

作者： Xiao, Yunhai Sun, Huijuan Wang, Zhiguo Nanjing Univ Dept Math Nanjing 210093 Peoples R China Henan Univ Inst Appl Math Coll Math & Informat Sci Kaifeng 475004 Peoples R China

In this paper, we propose a modified BFGS (Broyden-Fletcher-Goldfarb-Shanno) method with nommonotone line search for unconstrained optimization. Under some mild conditions, we show that the method is globally convergent without a convexity assumption on the objective function. We also report some preliminary numerical results to show the efficiency of the proposed method. Crown Copyright (C) 2008 Published by Elsevier B.V. All rights reserved.

关键词： non-convex minimization Secant equation BFGS method nonmonotone line search Global convergence

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WNC-ECGlet: Weighted non-convex minimization based reconstruction of compressively transmitted ECG using ECGlet

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BIOMEDICAL SIGNAL PROCESSING AND CONTROL 2019年第0期49卷 1-13页

作者： Ansari, Naushad Gupta, Anubha IIIT Delhi Dept ECE SBILab Delhi 110020 India

Data compression is very important in wireless body area network (WBAN), where a huge amount of ECG data is continuously transmitted to the receiver. Recent research has shown that compressive sensing (CS) is a better alternative for ECG data compression over traditional compression methods. In this work, we provide a method to improve CS-based ECG reconstruction that can be directly implemented in WBAN technology and hence, may enhance the use of WBAN technology in healthcare informatics and telemedicine. The methodology is named as WNC-ECGIet and consists of the following key contributions: (1) non-convex minimization is proposed for use in CS based ECG reconstruction, (2) weighted sparsity on wavelet coefficients of ECG signals is proposed to minimize for reconstruction, and (3) wavelet transform learning is proposed for ECG signals, named hereby, as ECGlet. An algorithm is provided to solve weighted non-convex minimization (WNC) problem. ECGIet is learned from an ensemble of ECG signals in the lifting framework rendering learning to be convex having closed-form solution with compactly supported filters that can be easily implemented on hardware. The learned wavelet transform is used as the sparsifying transform in the CS-based reconstruction of ECG signals transmitted compressively to the receiver. Extensive experiments are performed with comparing results of WNC-ECGIet with standard wavelets, I t norm minimization and, Gaussian, Bernoulli and sparse binary sensing matrices on MIT-BIH Arrhythmia ECG dataset. The proposed WNC-ECGlet method is observed to perform better than conventional methods in CS-based ECG signal reconstruction. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Compressed sensing Wavelet transform learning Sparse recovery non-convex minimization

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Global Convergence of a Modified Limited Memory BFGS Method for non-convex minimization

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Acta Mathematicae Applicatae Sinica 2013年第3期29卷 555-566页

作者： Yun-hai XIAO Ting-feng Zeng-xin WEI Department of Mathematics Nanjing University Zhengzhou Technical College College of Mathematics and Information Science Guangxi University

In this paper, a modified limited memory BFGS method for solving large-scale unconstrained optimization problems is proposed. A remarkable feature of the proposed method is that it possesses global convergence property without convexity assumption on the objective function. Under some suitable conditions, the global convergence of the proposed method is proved. Some numerical results are reported which illustrate that the proposed method is efficient.

关键词： non-convex minimization secant equation limited memory BFGS method global convergence

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A PARAMETERIZED THREE-OPERATOR SPLITTING ALGORITHM FOR non-convex minimization PROBLEMS WITH APPLICATIONS

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF nonLINEAR AND VARIATIONAL ANALYSIS 2024年第3期8卷 451-471页

作者： Miao, Liuyi Tang, Yuchao Wang, Changlong Nanchang Univ Dept Math Nanchang 330031 Jiangxi Peoples R China Guangzhou Univ Sch Math & Informat Sci Guangzhou 510006 Peoples R China Xidian Univ Key Lab Elect Informat Countermeasure & Simulat T Minist Educ Xian 710071 Peoples R China

In this paper, we propose a parameterized three-operator splitting algorithm to solve nonconvex minimization problems with the sum of three non-convex functions, where two of them have Lipschitz continuous gradients. We establish the convergence of the proposed algorithm under the Kurdykaojasiewicz assumption by constructing a suitable energy function with a non-increasing property. As applications, we employ the proposed algorithm to solve low-rank matrix recovery and image inpainting problems. Numerical results demonstrate the efficiency and effectiveness of the proposed algorithm compared to other algorithms.

关键词： Image inpainting problems Low rank matrix recovery non-convex minimization Parame- terized Three-operator splitting algorithm

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Sufficient global optimality conditions for non-convex quadratic minimization problems with box constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2006年第3期36卷 471-481页

作者： Jeyakumar, V. Rubinov, A. M. Wu, Z. Y. Univ New S Wales Dept Appl Math Sydney NSW 2052 Australia Univ Ballarat Sch Informat Technol & Math Sci Ballarat Vic 3353 Australia Chongqing Normal Univ Dept Math Chongqing 400047 Peoples R China

In this paper we establish conditions which ensure that a feasible point is a global minimizer of a quadratic minimization problem subject to box constraints or binary constraints. In particular, we show that our conditions provide a complete characterization of global optimality for non-convex weighted least squares minimization problems. We present a new approach which makes use of a global subdifferential. It is formed by a set of functions which are not necessarily linear functions, and it enjoys explicit descriptions for quadratic functions. We also provide numerical examples to illustrate our optimality conditions.

关键词： quadratic optimization global optimality conditions non-convex minimization weighted least squares box constraints bivalent programs

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Space Telescope Image reconstruction based on Combined convex Relaxation and non convex minimization Algorithms

Space Telescope Image reconstruction based on Combined Conve...

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IEEE Conference on Recent Advances in Intelligent Computational Systems (RAICS)

作者： Althaf, A. P. Byjubai, T. P. Govt Engn Coll Dept Appl Elect & Instrumentat Kozhikode Kerala India

ISBN: (纸本)9781538673362

One of the main challenges in space communication is the difficulty in the transmission and processing of large amount of data from satellite to earth. Also, space telescope image reconstruction with sampling rate less than Nyquist rate is preferred in space mission. Compressive sensing (CS) is a sampling theory that allows to capture and represent compressible signals at a rate significantly below Nyquist rate, by exploring the sparsity structure of the signal. Hence a novel image reconstruction technique based on CS to address the problems of space telescope image reconstruction is proposed in this paper. The common reconstruction algorithms for CS which comes under the class of convex Relaxation, non-convex minimization Algorithms, Greedy Iterative Algorithm, and Iterative Thresholding Algorithms are used for space telescope image reconstruction.A combination algorithm of convex relaxation and non-convex minimization Algorithms is proposed to enhance the quality of the reconstructed *** analysis of the proposed method is performed and it is found that the proposed method gives better result than the classical reconstruction algorithms of CS.

关键词： Compressive Sensing convex Relaxation non-convex minimization Iterative Thresholding

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Recovery analysis for weighted mixed l2/lp minimization with 0 < p ≤ 1

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 352卷 210-222页

作者： Zhou, Zhiyong Yu, Jun Umea Univ Dept Math & Math Stat S-90187 Umea Sweden

We study the recovery conditions of weighted mixed l(2)/l(p) (0 < p <= 1) minimization for block sparse signal reconstruction from compressed measurements when partial block support information is available. We show that the block p-restricted isometry property (RIP) can ensure the robust recovery. Moreover, we present the sufficient and necessary condition for the recovery by using weighted block p-null space property. The relationship between the block p-RIP and the weighted block p-null space property has been established. Finally, we illustrate our results with a series of numerical experiments. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Compressive sensing Prior support information Block sparse non-convex minimization

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LOW RANK MATRIX RECOVERY VIA AUGMENTED LAGRANGE MULTIPLIER WITH nonconvex minimization 12

LOW RANK MATRIX RECOVERY VIA AUGMENTED LAGRANGE MULTIPLIER W...

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12th IEEE Image, Video, and Multidimensional Signal Processing Workshop (IVMSP)

作者： Lee, Jieun Choe, Yoonsik Yonsei Univ Seoul South Korea

ISBN: (纸本)9781509019298

In recovery problem, nuclear norm as a convex envelope of rank function is widely used. However, nuclear norm minimization problem tends not to identify optimal solution, so recently, other heuristic surrogate functions such as nonconvex logdet are utilized to recover sparser signal. In this paper, to handle nonconvex optimation problem, a modified Augmented Lagrange Multiplier Method (ALMM) is developed using weighted nuclear norm instead of nuclear norm which conventional ALMM treats for convex optimization. We experiment on real images in Matrix Completion problem with diverse nonconvex, and show that instead of solving a simple convex problem, nonconvex optimization problem can reconstruct a low rank matrix more accurately and the convergence rate is faster with having higher average PSNR.

关键词： Low Rank Matrix Recovery ALMM Nuclear norm minimization non-convex minimization

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Regularized non-local Total Variation and Application in Image Restoration

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2017年第2期59卷 296-317页

作者： Li, Zhi Malgouyres, Francois Zeng, Tieyong Michigan State Univ Dept Computat Math Sci & Engn E Lansing MI 48824 USA Univ Toulouse CNRS Toulouse France Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

In the usual non-local variational models, such as the non-local total variations, the image is regularized by minimizing an energy term that penalizes gray-levels discrepancy between some specified pairs of pixels;a weight value is computed between these two pixels to penalize their dissimilarity. In this paper, we impose some regularity to those weight values. More precisely, we minimize a function involving a regularization term, analogous to an term, on weights. Doing so, the finite differences defining the image regularity depend on their environment. When the weights are difficult to define, they can be restored by the proposed stable regularization scheme. We provide all the details necessary for the implementation of a PALM algorithm with proved convergence. We illustrate the ability of the model to restore relevant unknown edges from the neighboring edges on an image inpainting problem. We also argue on inpainting, zooming and denoising problems that the model better recovers thin structures.

关键词： non-local regularization Proximal alternating linearized minimization non-convex minimization Total variation Image restoration

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Free discontinuity finite element models in two-dimensions for in-plane crack problems

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THEORETICAL AND APPLIED FRACTURE MECHANICS 2007年第3期47卷 274-282页

作者： Fraternali, F. Univ Salerno Dept Civil Engn I-84084 Fisciano SA Italy

Two different free discontinuity finite element models for studying crack initiation and propagation in 2D elastic problems are presented. minimization of an energy functional, composed of bulk and surface terms, is adopted to search for the displacement field and the crack pattern. Adaptive triangulations and embedded or r-adaptive discontinuities are employed. Cracks are allowed to nucleate, propagate, and branch. In order to eliminate rank-deficiency and perform local minimization, a vanishing viscosity regularization of the discrete Euler-Lagrange equations is enforced. Converge properties of the proposed models are examined using arguments of the F-convergence theory. Numerical results for an in-plane crack kinking problem illustrate the main operational features of the free discontinuity approach. (c) 2007 Elsevier Ltd. All rights reserved.

关键词： variational methods crack nucleation and growth free discontinuity problems discontinuous finite elements r-adaption non-convex minimization vanishing viscosity gamma convergence crack kinking

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