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A Second-Order TGV Discretization with 90° Rotational Invariance Property

JOURNAL OF MATHEMATICAL IMAGING AND VISION 2025年第2期67卷 1-33页

作者： Hosseini, Alireza Bredies, Kristian Univ Tehran Coll Sci Sch Math Stat & Comp SciStat & Comp SciStat & Comp POB 14115-175 Tehran Iran Karl Franzens Univ Graz Dept Math & Sci Comp Heinrich str 36 A-8010 Graz Austria

In this work, we propose a new discretization for second-order total generalized variation (TGV) with some distinct properties compared to existing discrete formulations. The introduced model is based on the same design principles as Condat's discrete total variation model (Condat in SIAM J Imaging Sci 10(3):1258-1290, 2017) and shares its benefits, particularly improved solution quality for imaging problems. We propose an algorithm for general discrete inverse problems with second-order TGV using the new discretization. Numerical results obtained with this algorithm for denoising and upscaling demonstrate the advantages of the discretization. Moreover, to assess the invariance properties of the new model, we compare the results of the proposed TGV and the classic discrete TGV for original data and 90 degrees rotated versions. Additionally, we provide an algorithm for calculating the TGV value with respect to the new discretization model.

关键词： Image processing Total generalized variation TGV discretization Inverse problem primal-dual algorithm

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Exponential convergence of distributed primal-dual convex optimization algorithm without strong convexity

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AUTOMATICA 2019年 105卷 298-306页

作者： Liang, Shu Wang, Le Yi Yin, George Univ Sci & Technol Beijing Minist Educ Key Lab Knowledge Automat Ind Proc Sch Automat & Elect Engn Beijing 100083 Peoples R China Wayne State Univ Dept Elect & Comp Engn Detroit MI 48202 USA Wayne State Univ Dept Math Detroit MI 48202 USA

This paper establishes exponential convergence rates for a class of primal-dual gradient algorithms in distributed optimization without strong convexity. The convergence analysis is based on a carefully constructed Lyapunov function. By evaluating metric subregularity of the primal-dual gradient map, we present a general criterion under which the algorithm achieves exponential convergence. To facilitate practical applications of this criterion, several simplified sufficient conditions are derived. We also prove that although these results are developed for the continuous-time algorithms, they carry over in a parallel manner to the discrete-time algorithms constructed by using Euler's approximation method. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Distributed optimization Convex optimization without strong convexity primal-dual algorithm Exponential convergence Rate of convergence Metric subregularity Variational analysis

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Proximal primal-dual best approximation algorithm with memory

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2018年第3期71卷 767-794页

作者： Bednarczuk, E. M. Jezierska, A. Rutkowski, K. E. Polish Acad Sci Syst Res Inst Warsaw Poland Warsaw Univ Technol Fac Math & Informat Sci Warsaw Poland Gdansk Univ Technol Fac Elect Telecommun & Informat Gdansk Poland

We propose a new modified primal-dual proximal best approximation method for solving convex not necessarily differentiable optimization problems. The novelty of the method relies on introducing memory by taking into account iterates computed in previous steps in the formulas defining current iterate. To this end we consider projections onto intersections of halfspaces generated on the basis of the current as well as the previous iterates. To calculate these projections we are using recently obtained closed-form expressions for projectors onto polyhedral sets. The resulting algorithm with memory inherits strong convergence properties of the original best approximation proximal primal-dual algorithm. Additionally, we compare our algorithm with the original (non-inertial) one with the help of the so called attraction property defined below. Extensive numerical experimental results on image reconstruction problems illustrate the advantages of including memory into the original algorithm.

关键词： Proximal algorithm with memory primal-dual algorithm Best approximation of the Kuhn-Tucker set Inclusions with maximally monotone operators Attraction property Image reconstruction

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primal-dual stochastic distributed algorithm for constrained convex optimization

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JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 2019年第16期356卷 9763-9787页

作者： Niu, Youcheng Wang, Haijing Wang, Zheng Xia, Dawen Li, Huaqing Southwest Univ Coll Elect & Informat Engn Chongqing Key Lab Nonlinear Circuits & Intelligen Chongqing 400715 Peoples R China Guizhou Minzu Univ Coll Data Sci & Informat Engn Guiyang 550025 Guizhou Peoples R China

This paper investigates distributed convex optimization problems over an undirected and connected network, where each node's variable lies in a private constrained convex set, and overall nodes aim at collectively minimizing the sum of all local objective functions. Motivated by a variety of applications in machine learning problems with large-scale training sets distributed to multiple autonomous nodes, each local objective function is further designed as the average of moderate number of local instantaneous functions. Each local objective function and constrained set cannot be shared with others. A primal-dual stochastic algorithm is presented to address the distributed convex optimization problems, where each node updates its state by resorting to unbiased stochastic averaging gradients and projects on its private constrained set. At each iteration, for each node the gradient of one local instantaneous function selected randomly is evaluated and the average of the most recent stochastic gradients is used to approximate the true local gradient. In the constrained case, we show that with strong-convexity of the local instantaneous function and Lipschitz continuity of its gradient, the algorithm converges to the global optimization solution almost surely. In the unconstrained case, an explicit linear convergence rate of the algorithm is provided. Numerical experiments are presented to demonstrate correctness of the theoretical results. (C) 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

关键词： Constrained convex optimization Machine learning primal-dual algorithm Stochastic averaging gradients Linear convergence

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A primal-dual Forward-Backward Splitting algorithm for Distributed Convex Optimization

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IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTATIONAL INTELLIGENCE 2023年第1期7卷 278-284页

作者： Li, Huaqing Su, Enbing Wang, Chengbo Liu, Jiawei Zheng, Zuqing Wang, Zheng Xia, Dawen Southwest Univ Chongqing Key Lab Nonlinear Circuits & Intelligen Coll Elect & Informat Engn Chongqing 400715 Peoples R China Southwest Univ Westa Coll Chongqing 400715 Peoples R China Univ New South Wales Sch Elect Engn & Telecommun Sydney NSW 2052 Australia Guizhou Minzu Univ Coll Data Sci & Informat Engn Guiyang 550025 Peoples R China

Motivated by modern large-scale information processing problems in engineering, this paper concentrates on studying distributed constrained convex optimization problems over a connected undirected network. The problem involves a sum of a differentiable convex function with Lipschitz continuous gradient and two non-smooth convex functions with a linear operator. To solve such a problem, we propose a novel distributed primal-dual forward-backward splitting algorithm, called D-PDFBS. Each agent locally computes the Lipschitz continuous gradient and two proximal operators, and exchanges information with its neighbors. D-PDFBS adopts non-identical stepsizes, and we reveal the relationship between selection of stepsizes and parameters of objective functions. The simulation results verify the feasibility of D-PDFBS and the correctness of theoretical findings.

关键词： Optimization Convex functions Convergence Wireless sensor networks Wireless communication Symmetric matrices Smart grids Forward-backward splitting method distributed optimization primal-dual algorithm non-smooth function

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primal-dual fixed point algorithm based on adapted metric method for solving convex minimization problem with application

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APPLIED NUMERICAL MATHEMATICS 2020年 157卷 236-254页

作者： Huang, Wenli Tang, Yuchao Nanchang Univ Dept Math Nanchang 330031 Jiangxi Peoples R China

Optimization problems involving the sum of three convex functions have received much attention in recent years, where one is differentiable with Lipschitz continuous gradient, one is composed of a linear operator and the other is proximity friendly. The primal-dual fixed point algorithm is a simple and effective algorithm for such problems. To exploit the second-order derivatives information of the objective function, we propose a primal-dual fixed point algorithm with an adapted metric method. The proposed algorithm is derived from the idea of establishing a generally fixed point formulation for the solution of the considered problem. Under mild conditions on the iterative parameters, we prove the convergence of the proposed algorithm. Further, we establish the ergodic convergence rate in the sense of primal-dual gap and also derive the linear convergence rate with additional conditions. Numerical experiments on image deblurring problems show that the proposed algorithm outperforms other state-of-the-art primal-dual algorithms in terms of the number of iterations. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： primal-dual algorithm Adapted metric method Proximity operator Forward-backward splitting algorithm

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A globally and quadratically convergent primal-dual augmented Lagrangian algorithm for equality constrained optimization

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OPTIMIZATION METHODS & SOFTWARE 2017年第1期32卷 1-21页

作者： Armand, Paul Omheni, Riadh Univ Limoges France XLIM Lab UMR CNRS 7252 Limoges France

We present a primal-dual augmented Lagrangian method to solve an equality constrained minimization problem. This is a Newton-like method applied to a perturbation of the optimality system that follows from a reformulation of the initial problem by introducing an augmented Lagrangian function. An important aspect of this approach is that, by a choice of suitable updating rules of parameters, the algorithm reduces to a regularized Newton method applied to a sequence of optimality systems. The global convergence is proved under mild assumptions. An asymptotic analysis is also presented and quadratic convergence is proved under standard regularity assumptions. Some numerical results show that the method is very efficient and robust.

关键词： equality constrained minimization primal-dual algorithm augmented Lagrangian method quadratic convergence

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An adaptive fractional-order regularization primal-dual image denoising algorithm based on non-convex function

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APPLIED MATHEMATICAL MODELLING 2024年 131卷 67-83页

作者： Li, Minmin Bi, Shaojiu Cai, Guangcheng Kunming Univ Sci & Technol Dept Math Kunming 650500 Yunnan Peoples R China

In this paper, a novel non-convex fractional-order image denoising model is proposed to suppress the staircase effect produced by the TV model while maintaining a neat contour. The model combines l(q) (0algorithm. To obtain the optimal approximate solution of the model, an enhanced primal-dual algorithm is adopted and the complexity and convergence of the algorithm are theoretically analyzed. Finally, the effectiveness of the proposed method is demonstrated through numerical experiments.

关键词： Image denoising Diffusion coefficient Fractional-order Non-convex function primal-dual algorithm Complexity analysis

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Use of primal-dual technique in the network algorithm for two-way contingency tables

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JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2005年第1期22卷 133-145页

作者： Suzuki, T Aoki, S Murota, K Univ Tokyo Dept Math Informat Grad Sch Informat Sci & Technol Tokyo 1138656 Japan

An algorithm for the longest-path subproblem in the network algorithm of Mehta and Patel for Fisher's exact test in two-way contingency tables is proposed. Use of the primal-dual technique for convex-cost flows an... 详细信息

关键词： Fisher's exact test contingency table discrete convexity minimum cost flow primal-dual algorithm

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A primal-dual prediction-correction algorithm for saddle point optimization

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JOURNAL OF GLOBAL OPTIMIZATION 2016年第3期66卷 573-583页

作者： He, Hongjin Desai, Jitamitra Wang, Kai Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Zhejiang Peoples R China Nanyang Technol Univ Sch Mech & Aerosp Engn 50 Nanyang Ave Singapore 639798 Singapore

In this paper, we introduce a new primal-dual prediction-correction algorithm for solving a saddle point optimization problem, which serves as a bridge between the algorithms proposed in Cai et al. (J Glob Optim 57:1419-1428, 2013) and He and Yuan (SIAM J Imaging Sci 5:119-149, 2012). An interesting byproduct of the proposed method is that we obtain an easily implementable projection-based primal-dual algorithm, when the primal and dual variables belong to simple convex sets. Moreover, we establish the worst-case convergence rate result in an ergodic sense, where t represents the number of iterations.

关键词： Saddle point problem primal-dual algorithm Prediction-correction algorithm Projection method Convergence rate

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