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On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization

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MATHEMATICAL PROGRAMMING 2025年第1-2期211卷 181-206页

作者： Hu, Yaohua Hu, Xinlin Yang, Xiaoqi Shenzhen Univ Sch Math Sci Shenzhen 518060 Peoples R China Shenzhen Univ WeBank Inst Fintech Shenzhen Audencia Business Sch Shenzhen 518060 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _p$$\end{document} penalty (0 <= p <= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le p \le 1$$\end{document}) and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems. The established results include the existing convergence theory for l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} or l0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _0$$\end{document} regularization problems for finding a true sparse solution as special c

关键词： Nonconvex sparse optimization Iterative thresholding algorithm proximal gradient algorithm Sparse solution Global solution

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Continuous-Time Distributed proximal gradient algorithms for Nonsmooth Resource Allocation Over General Digraphs

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IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING 2021年第2期8卷 1733-1744页

作者： Zhu, Yanan Wen, Guanghui Yu, Wenwu Yu, Xinghuo Nanjing Univ Informat Sci & Technol Sch Automat Nanjing 210044 Peoples R China Southeast Univ Sch Math Jiangsu Prov Key Lab Networked Collect Intelligen Nanjing 210096 Peoples R China RMIT Univ Sch Engn Melbourne Vic 3001 Australia

This paper studies a nonsmooth resource allocation problem with network resource constraints and local set constraints, where the interaction graphs among agents are generally strongly connected digraphs. First, we design a centralized continuous-time proximal gradient algorithm, where each agent uses the global Lagrangian multipliers and the global values of constraint functions. For the case that the agents' private information could not be leaked and the global Lagrangian multipliers are not available, the agents are endowed with some additional variables to estimate those global information via consensus protocols. Then, we construct a class of continuous-time distributed proximal gradient algorithms by using a two-time scale mechanism to integrate the proposed proximal gradient algorithm and consensus protocols. By adopting Lyapunov stability theory and convex optimization theory, we prove that the decision variables asymptotically converge to the optimal solution of the nonsmooth resource allocation problem. Finally, numerical simulations are applied to illustrate the effectiveness of the proposed algorithms.

关键词： Nonsmooth resource allocation proximal gradient algorithm consensus protocol two-time scale mechanism

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Convergence rates of accelerated proximal gradient algorithms under independent noise

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NUMERICAL algorithmS 2019年第2期81卷 631-654页

作者： Sun, Tao Barrio, Roberto Jiang, Hao Cheng, Lizhi Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China Univ Zaragoza Dept Matemat Aplicada IUMA Grp Pedro Cerbuna 12 E-50009 Zaragoza Spain Univ Zaragoza Dept Matemat Aplicada CoDy Grp Pedro Cerbuna 12 E-50009 Zaragoza Spain Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China Natl Univ Def Technol State Key Lab High Performance Computat Changsha 410073 Hunan Peoples R China

We consider an accelerated proximal gradient algorithm for the composite optimization with independent errors (errors little related with historical information) for solving linear inverse problems. We present a new inexact version of FISTA algorithm considering deterministic and stochastic noises. We prove some convergence rates of the algorithm and we connect it with the current existing catalyst framework for many algorithms in machine learning. We show that a catalyst can be regarded as a special case of the FISTA algorithm where the smooth part of the function vanishes. Our framework gives a more generic formulation that provides convergence results for the deterministic and stochastic noise cases and also to the catalyst framework. Some of our results provide simpler alternative analysis of some existing results in literature, but they also extend the results to more generic situations.

关键词： proximal gradient algorithm Inexact algorithm Independent noise Convergence rates Acceleration

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Inertial self-adaptive algorithms for solving non-smooth convex optimization problems

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NUMERICAL algorithmS 2025年第1期98卷 133-163页

作者： Chen, Xin Duan, Peichao Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China

In this paper, for two different forms of non-smooth convex optimization problems, we investigate the self-adaptive algorithms with inertia acceleration. Firstly, we propose a self-adaptive proximal gradient algorithm with an inertial step. Under reasonable parameters, the strong convergence theorem is established. Secondly, we propose a self-adaptive split proximal algorithm with inertial acceleration. We prove that our algorithm converges strongly under suitable conditions. Notably, both inertial algorithms are extended to multi-step inertial version to accelerate the convergence of the algorithms. Finally, numerical results illustrate the performances of our algorithms.

关键词： Convex optimization Self-adaptive algorithm Inertial acceleration proximal gradient algorithm Split proximal algorithm

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proximal gradient/Semismooth Newton Methods for Projection onto a Polyhedron via the Duality-Gap-Active-Set Strategy

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JOURNAL OF SCIENTIFIC COMPUTING 2023年第1期97卷 1-34页

作者： Wang, Yunlong Shen, Chungen Zhang, Lei-Hong Yang, Wei Hong Shanghai Jiao Tong Univ Antai Coll Econ & Management Shanghai 200030 Peoples R China Univ Shanghai Sci & Technol Coll Sci Shanghai 200093 Peoples R China Soochow Univ Sch Math Sci Suzhou 215006 Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China

The polyhedral projection problem arises in various applications. To efficiently solve the dual problem, one of the crucial issues is to safely identify zero-elements as well as the signs of nonzero elements at the optimal solution. In this paper, relying on its nonsmooth dual problem and active set techniques, we first propose a Duality-Gap-Active-Set strategy (DGASS) to effectively identify the indices of zero-elements and the signs of nonzero entries of the optimal solution. Serving as an efficient acceleration strategy, DGASS can be embedded into certain iterative methods. In particular, by applying DGASS to both the proximal gradient algorithm (PGA) and the proximal semismooth Newton algorithm (PSNA), we propose the method of PGA-DGASS and PSNA-DGASS, respectively. Global convergence and local quadratic convergence rate are discussed. We report on numerical results over both synthetic and real data sets to demonstrate the high efficiency of the two DGASS-accelerated methods.

关键词： Polyhedral projection problem Active set Duality gap proximal gradient algorithm proximal semismooth Newton method

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Incremental proximal gradient scheme with penalization for constrained composite convex optimization problems

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OPTIMIZATION 2021年第5-6期70卷 1307-1336页

作者： Petrot, Narin Nimana, Nimit Naresuan Univ Fac Sci Dept Math Phitsanulok Thailand Naresuan Univ Fac Sci Ctr Excellence Nonlinear Anal & Optimizat Phitsanulok Thailand Khon Kaen Univ Fac Sci Dept Math Khon Kaen Thailand

We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with smooth penalization technique is proposed. We show the convergence of the generated sequence of iterates to an optimal solution of the optimization problems, provided that a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. Finally, the functionality of the method is illustrated by some numerical experiments addressing image inpainting problems and generalized Heron problems with least squares constraints.

关键词： Convex optimization Fenchel conjugate incremental proximal method penalization proximal gradient algorithm

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Fast Total Variation-Based Image Restoration Using Blockwise Accelerated proximal gradient Approach 9

Fast Total Variation-Based Image Restoration Using Blockwise...

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9th International Conference on Graphic and Image Processing (ICGIP)

作者： Xiao, Gang Yang, Zifei Lv, Weigang Liu, Shan 211 Hosp PLA Harbin Heilongjiang Peoples R China Harbin Inst Technol Harbin Heilongjiang Peoples R China Ocean Univ China Qingdao Peoples R China

ISBN: (数字)9781510617421

ISBN: (纸本)9781510617421

This paper proposes a blockwise accelerated proximal gradient (BAPG) approach. It chooses a block diagonal Lipschitz matrix in the generalized APG algorithm, such that the subproblems can be solved either by fast Fourier transform (FFT) or in closed forms. Experiments verify the great speed advantage of BAPG for total variation-based image restoration.

关键词： Image Restoration Total Variation Fast Fourier Transform proximal gradient algorithm

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proximal gradient Methods with Adaptive Subspace Sampling

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MATHEMATICS OF OPERATIONS RESEARCH 2021年第4期46卷 1303-1323页

作者： Grishchenko, Dmitry Iutzeler, Franck Malick, Jerome Univ Grenoble Alpes Lab Jean Kuntzmann F-38402 St Martin Dheres France Univ Grenoble Alpes Lab Informat Grenoble F-38401 St Martin Dheres France CNRS F-75016 Paris France

Many applications in machine learning or signal processing involve nonsmooth optimization problems. This nonsmoothness brings a low-dimensional structure to the optimal solutions. In this paper, we propose a randomized proximal gradient method harnessing this underlying structure. We introduce two key components: (i) a random subspace proximal gradient algorithm;and (ii) an identification-based sampling of the subspaces. Their interplay brings a significant performance improvement on typical learning problems in terms of dimensions explored.

关键词： nonsmooth optimization identification proximal gradient algorithm randomized methods

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A Generalized Accelerated proximal gradient Approach for Total-Variation-Based Image Restoration

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2011年第10期20卷 2748-2759页

作者： Zuo, Wangmeng Lin, Zhouchen Harbin Inst Technol Sch Comp Sci & Technol Harbin 150001 Peoples R China Microsoft Res Asia Beijing 100080 Peoples R China

This paper proposes a generalized accelerated proximal gradient (GAPG) approach for solving total variation (TV)-based image restoration problems. The GAPG algorithm generalizes the original APG algorithm by replacing the Lipschitz constant with an appropriate positive-definite matrix, resulting in faster convergence. For TV-based image restoration problems, we further introduce two auxiliary variables that approximate the partial derivatives. Constraints on the variables can easily be imposed without modifying the algorithm much, and the TV regularization can be either isotropic or anisotropic. As compared with the recently developed APG-based methods for TV-based image restoration, i.e., monotone version of the two-step iterative shrinkage/thresholding algorithm (MTwIST) and monotone version of the fast IST algorithm (MFISTA), our GAPG is much simpler as it does not require to solve an image denoising subproblem. Moreover, the convergence rate of O(k(-2)) is maintained by our GAPG, where k s the number of iterations;the cost of each iteration in GAPG is also lower. As a result, in our experiments, our GAPG approach can be much faster than MTwIST and MFISTA. The experiments also verify that our GAPG converges faster than the original APG and MTwIST when they solve identical problems.

关键词： Convex optimization image restoration proximal gradient algorithm regularization total variation (TV)

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DME Interference mitigation for L-DACS1 based on system identification and sparse representation

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Chinese Journal of Aeronautics 2016年第6期29卷 1762-1773页

作者： Li Douzhe Wu Zhijun School of Electronic and Information Engineering Tianjin University School of Electronic and Information Engineering Civil Aviation University of China

L-band digital aeronautical communication system 1(L-DACS1) is a promising candidate data-link for future air-ground communication, but it is severely interfered by the pulse pairs(PPs) generated by distance measure equipment. A novel PP mitigation approach is proposed in this paper. Firstly, a deformed PP detection(DPPD) method that combines a filter bank, correlation detection, and rescanning is proposed to detect the deformed PPs(DPPs) which are caused by multiple filters in the receiver. Secondly, a finite impulse response(FIR) model is used to approximate the overall characteristic of filters, and then the waveform of DPP can be acquired by the original waveform of PP and the FIR model. Finally, sparse representation is used to estimate the position and amplitude of each DPP, and then reconstruct each DPP. The reconstructed DPPs will be subtracted from the contaminated signal to mitigate interference. Numerical experiments show that the bit error rate performance of our approach is about 5 dB better than that of recent works and is closer to interference-free environment.

关键词： DME interference L-DACS1 Least square approximations proximal gradient algorithm Sparse representation

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