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Proximal algorithms for a Class of Abstract Convex Functions

SET-VALUED AND VARIATIONAL ANALYSIS 2025年第1期33卷 1-44页

作者： Bednarczuk, Ewa Lorenz, Dirk Tran, The Hung Warsaw Univ Technol Fac Math & Informat Sci Koszykowa 75 PL-00662 Warsaw Poland Univ Bremen Ctr Ind Math Postfach 330440 D-28334 Bremen Germany PAS Syst Res Inst Dept Modeling & Optimizat Dynam Syst Newelska 6 PL-01447 Warsaw Poland

In this paper, we analyze a class of nonconvex optimization problems from the viewpoint of abstract convexity. Using the respective generalizations of the subgradient, we propose an abstract notion of a proximal operator and derive several algorithms, namely abstract proximal point method, abstract forward-backward method, and abstract projected subgradient method. Global convergence results for all algorithms are discussed, and numerical examples are given.

关键词： Abstract convexity Proximal operator forward-backward algorithm Global convergence Proximal subgradient

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A Nonlinearly Preconditioned forward-backward Splitting Method and Applications

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2022年第16期42卷 1880-1895页

作者： Vu, Bang Cong Papadimitriou, Dimitri Huawei Belgium Res Ctr BeRC 3NLab Leuven Belgium

In this paper, we prove the weak convergence of the iterates generated by the nonlinearly preconditioned forward-backward splitting method for the sum of a maximally hypermonotone operator A and a hypercocoercive operator B under several suitable conditions. We provide several choices of the nonlinear preconditioners for solving nonlinearly composed inclusions. In particular, the backward-forward splitting method is recovered by the nonlinearly preconditioned forward-backward splitting method with a special choice of the nonlinear preconditioner.

关键词： Cocoercive composite operator forward-backward algorithm hypercocoercive hypermonotone monotone inclusion monotone operator operator splitting preconditioning

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Finite mixture of hidden Markov models for tensor-variate time series data

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ADVANCES IN DATA ANALYSIS AND CLASSIFICATION 2024年第3期18卷 545-562页

作者： Asilkalkan, Abdullah Zhu, Xuwen Sarkar, Shuchismita Univ Alabama Dept Informat Syst Stat & Management Sci Tuscaloosa AL 35487 USA Bowling Green State Univ Dept Appl Stat & Operat Res Bowling Green OH 43403 USA

The need to model data with higher dimensions, such as a tensor-variate framework where each observation is considered a three-dimensional object, increases due to rapid improvements in computational power and data storage capabilities. In this study, a finite mixture of hidden Markov model for tensor-variate time series data is developed. Simulation studies demonstrate high classification accuracy for both cluster and regime IDs. To further validate the usefulness of the proposed model, it is applied to real-life data with promising results.

关键词： Finite mixture model Hidden Markov model forward-backward algorithm Tensor-variate time series

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DOUBLE INERTIAL PROXIMAL GRADIENT algorithmS FOR CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS

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Acta Mathematica Scientia 2023年第3期43卷 1462-1476页

作者： Kunrada KANKAM Prasit CHOLAMJIAK School of Science University of PhayaoPhayao56000Thailand

In this paper, we propose double inertial forward-backward algorithms for solving unconstrained minimization problems and projected double inertial forward-backward algorithms for solving constrained minimization problems. We then prove convergence theorems under mild conditions. Finally, we provide numerical experiments on image restoration problem and image inpainting problem. The numerical results show that the proposed algorithms have more efficient than known algorithms introduced in the literature.

关键词： weak convergence forward-backward algorithm convex minimization inertial technique

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Adaptive Doppler blood flow estimation in ultrasound with enhanced spectral resolution and contrast using limited observation windows

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ULTRASONICS 2025年 154卷 107678页

作者： Majd, Seyed Mohammad Mahdi Tabatabaei Eslami, Leila Asl, Babak Mohammadzadeh Tarbiat Modares Univ Dept Biomed Engn Tehran 14115111 Iran

Measuring blood velocities during the acceleration and deceleration phases of the systolic period can be challenging due to the trade-off between spectral and temporal resolution. This can significantly affect the accuracy of spectrogram reproduction. When temporal samples are reduced, the spectral width may broaden over time, especially during systole. Additionally, shorter observation windows negatively impact factors such as frequency resolution and contrast. This study hypothesizes that a more accurate ultrasound spectrogram can be generated using a new blood velocity estimator with a minimal observation window length of N = 2. The spectrogram's accuracy is assessed using various criteria, including spectral resolution, contrast, and spectral broadening over time. The proposed adaptive method integrates a new coherence-based post-filter with the Eigenspace-based forward-backward Amplitude Spectrum Capon (ESB-FBASC) technique. The method's performance was evaluated in different conditions, including simulations of the femoral artery, stationary and complex flow, and in vivo data. Under rapid flow conditions simulated over three heartbeats in 0.2 s, the proposed method demonstrated better temporal resolution compared to the Welch-Ref estimator, effectively capturing rapid velocity changes and reducing spectral broadening, despite using only N = 2 slow-time samples. For clinical data on the hepatic vein, the proposed estimator improved spectral resolution by 24 %, 44 %, and 67 %, and increased contrast by 79.8 dB, 120.8 dB, and 155.5 dB compared to MASC, ***, and Capon, respectively, for N = 2. Furthermore, the narrowest power spectrum width at 40 dB was achieved with the proposed method, showing an improvement of 38 % and 75 % compared to MASC and ***, respectively. As a result, the proposed method effectively reduces power spectrum width and enhances spectrogram accuracy by improving spectral resolution and contrast, all while using the limited observat

关键词： Blood velocity estimation Coherence-based post-filter Amplitude spectrum Capon forward-backward algorithm Observation window Spectral resolution Spectrogram

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Conical averagedness and convergence analysis of fixed point algorithms

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第2期82卷 351-373页

作者： Bartz, Sedi Dao, Minh N. Phan, Hung M. Univ Massachusetts Dept Math Sci Kennedy Coll Sci Lowell MA 01854 USA Federat Univ Australia Sch Engn Informat Technol & Phys Sci Ballarat Vic 3353 Australia

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.

关键词： Adaptive Douglas-Rachford algorithm Cocoercivity Conically averaged operator forward-backward algorithm Proximal point algorithm Strong monotonicity Weak monotonicity

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A Reflected forward-backward Splitting Method for Monotone Inclusions Involving Lipschitzian Operators

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SET-VALUED AND VARIATIONAL ANALYSIS 2021年第1期29卷 163-174页

作者： Cevher, Volkan Bang Cong Vu Ecole Polytech Fed Lausanne Lab Informat & Inference Syst Lausanne Switzerland

In this paper, we propose a novel splitting method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. The weak convergence the method is proved in real Hilbert spaces. Applying the proposed method to composite monotone inclusions involving parallel sums yields a new primal-dual splitting which is different from the existing methods. Connections to existing works are clearly stated. We also provide an application of the proposed method to the image denoising by the total variation.

关键词： Monotone inclusion Monotone operator Operator splitting Cocoercive forward-backward-forward method forward-backward algorithm Composite operator Duality Primal-dual algorithm

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Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration

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OPTIMIZATION METHODS & SOFTWARE 2022年第6期37卷 2225-2240页

作者： Cheval, Horaiu Leustean, Laureniu Univ Bucharest Fac Math & Comp Sci Res Ctr Log Optimizat & Secur LOS Dept Comp Sci Acad 14 Bucharest 010014 Romania Romanian Acad Simion Stoilow Inst Math Calea Grivitei 21 Bucharest 010702 Romania

In this paper, we compute quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration in W-hyperbolic spaces. This iteration is an extension to a nonlinear setting of the modified Mann iteration defined recently by Bo, Csetnek and Meier in Hilbert spaces. Furthermore, we show that the Douglas-Rachford and forward-backward algorithms with Tikhonov regularization terms are special cases, in Hilbert spaces, of our Tikhonov-Mann iteration.

关键词： Mann iteration Tikhonov regularization rates of asymptotic regularity Douglas-Rachford algorithm forward-backward algorithm proof mining

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Convergence analysis and applications of the inertial algorithm solving inclusion problems

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APPLIED NUMERICAL MATHEMATICS 2022年 175卷 1-17页

作者： Tang, Yan Lin, Honghua Gibali, Aviv Cho, Yeol Je Chongqing Technol & Business Univ Sch Math & Stat Chongqing 400067 Peoples R China Sichuan Univ Coll Math Chengdu 610065 Peoples R China Southwest Univ Polit Sci & Law Business Sch Chongqing 401120 Peoples R China ORT Braude Coll Dept Math IL-2161002 Karmiel Israel Univ Haifa Ctr Math & Sci Computat Mt Carmel Haifa 3498838 Israel Gyeongsang Natl Univ Dept Math Educ Jinju 52828 South Korea China Med Univ Ctr Gen Educ Taichung 40402 Taiwan

Nonlinear operator theory is an important area of nonlinear functional analysis. This area encompasses diverse nonlinear problems in many areas of mathematics, the physical sciences and engineering such as monotone operator equations, fixed point problems and more. In this work we are concern with the problem of finding a common solution of a monotone operator equation and fixed point of a nonexpansive mapping in real Hilbert spaces. Derived from dynamical systems, a simple inertial forward-backward splitting method for solving the problem is presented and analyzed under mild and standard assumptions. Some numerical examples in real-world and comparisons with related works, illustrate the theoretical advantages as well the potential applicability of the proposed scheme. (c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Monotone operator equation Fixed point forward-backward algorithm Inertial technique Nonexpansive mapping

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A hidden Markov modeling approach for identifying tumor subclones in next-generation sequencing studies

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BIOSTATISTICS 2022年第1期23卷 69-82页

作者： Choo-Wosoba, Hyoyoung Albert, Paul S. Zhu, Bin NCI Biostat Branch Div Canc Epidemiol & Genet 9609 Med Ctr Dr Rockville MD 20850 USA

Allele-specific copy number alteration (ASCNA) analysis is for identifying copy number abnormalities in tumor cells. Unlike normal cells, tumor cells are heterogeneous as a combination of dominant and minor subclones with distinct copy number profiles. Estimating the clonal proportion and identifying mainclone and subclone genotypes across the genome are important for understanding tumor progression. Several ASCNA tools have recently been developed, but they have been limited to the identification of subclone regions, and not the genotype of subclones. In this article, we propose subHMM, a hidden Markov model-based approach that estimates both subclone region and region-specific subclone genotype and clonal proportion. We specify a hidden state variable representing the conglomeration of clonal genotype and subclone status. We propose a two-step algorithm for parameter estimation, where in the first step, a standard hidden Markov model with this conglomerated state variable is fit. Then, in the second step, region-specific estimates of the clonal proportions are obtained by maximizing region-specific pseudo-likelihoods. We apply subHMM to study renal cell carcinoma datasets in The Cancer Genome Atlas. In addition, we conduct simulation studies that show the good performance of the proposed approach. The R source code is available online at https://***/tools/analysis/subhmm. Expectation-Maximization algorithm;forward-backward algorithm;Somatic copy number alteration;Tumor subclones.

关键词： E-M algorithm forward-backward algorithm Somatic copy number alteration Tumor subclones

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