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Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in CAT (0) spaces

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Fixed Point Theory and Applications 2016年第1期2016卷 1-11页

作者： Chang, Shih-sen Yao, Jen-Chih Wang, Lin Qin, Li Juan Center for General Education China Medical University Taichung 40402 Taiwan College of Statistics and Mathematics Yunnan University of Finance and Economics Kunming 650221 Yunnan China Department of Mathematics Kunming University Kunming 650214 Yunnan China

In this paper, a new modified proximal point algorithm involving fixed point iterates of asymptotically nonexpansive mappings in CAT (0) spaces is proposed and the existence of a sequence generated by our iterative process converging to a minimizer of a convex function and a common fixed point of asymptotically nonexpansive mappings is proved. © 2016, Chang et al.

关键词： asymptotically nonexpansive mapping CAT (0) space convex minimization problem proximal point algorithm resolvent identity

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A Novel Proximity Algorithm for an Unconstrained convex Segmentation Model 11

A Novel Proximity Algorithm for an Unconstrained Convex Segm...

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2019第十一届数字图像处理国际会议

作者： Weibin Li Yanxia Du Xian Yi Honglin Ma Chao Song Computational Aerodynamics Institute China Aerodynamics Research and Development Center Key Laboratory of Aircraft Icing and Anti/De-Icing China Aerodynamics Research and Development Center

ISBN: (数字)9781510630765

ISBN: (纸本)9781510630758

A new proximity algorithm for an active contour model is proposed in this *** order to derive a mathematical form of the energy,the level set method is *** the new energy,a penalty term is introduced to make sure that the level set function can be restricted in the interval [-1,1].By introducing this term,the energy still keeps convex and is easy to construct its minimization *** on the proximity operator and the corresponding theories,we deduce a proximity algorithm to minimize the *** results demonstrate that the proposed model is powerful in its segmentation ability and *** comparisons with other popular algorithms show that the proposed algorithm is more efficient.

关键词： Image Segmentation active contour model convex minimization problem proximity algorithm

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A MODIFIED HYBRID ALGORITHM FOR SOLVING A COMPOSITE minimization problem IN BANACH SPACES

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2019年第3期3卷 357-364页

作者： Chang, Shih-Sen Liu, Min Zhao, Liangcai Tang, Jinfang Yunnan Univ Finance & Econ Coll Stat & Math Kunming Yunnan Peoples R China Yibin Univ Coll Math Yibin Peoples R China

The purpose of this paper is to use a modified hybrid algorithm for finding a minimizer of a non-smooth composite minimization problem in Banach spaces. Without the assumptions that the potential function is Frechet differentiable and its gradient is L-Lipschitz continuous, we prove that the iterative sequence generated by the hybrid algorithm converges strongly to a minimizer of the composite optimization problem in Banach spaces.

关键词： Banach space Composite optimization convex minimization problem Lower semi-continuous function

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Strong convergence and bounded perturbation resilience of a modified proximal gradient algorithm

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2018年第1期2018卷 103-103页

作者： Guo, Yanni Cui, Wei Civil Aviat Univ China Coll Sci Tianjin Peoples R China

The proximal gradient algorithm is an appealing approach in finding solutions of non-smooth composite optimization problems, which may only has weak convergence in the infinite-dimensional setting. In this paper, we introduce a modified proximal gradient algorithm with outer perturbations in Hilbert space and prove that the algorithm converges strongly to a solution of the composite optimization problem. We also discuss the bounded perturbation resilience of the basic algorithm of this iterative scheme and illustrate it with an application.

关键词： Strong convergence Bounded perturbation resilience Modified proximal gradient algorithm Viscosity approximation convex minimization problem

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An accelerated alternating direction method of multiplier for MRI with TV regularisation

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MAGNETIC RESONANCE IMAGING 2024年 114卷 110249-110249页

作者： Zhu, ZhiBin Ding, YueHong Liu, Ying Huang, JiaQi Guilin Univ Elect Technol Sch Math & Comp Sci Guilin 541002 Peoples R China Ctr Appl Math Guangxi GUET Guilin 541002 Peoples R China Guangxi Coll & Univ Key Lab Data Anal & Computat Guilin 541002 Peoples R China

Compressed Sensing (CS) is important in the field of image processing and signal processing, and CS-Magnetic Resonance Imaging (MRI) is used to reconstruct image from undersampled k-space data. Total Variation (TV) regularisation is a common technique to improve the sparsity of image, and the Alternating Direction Multiplier Method (ADMM) plays a key role in the variational image processing problem. This paper aims to improve the quality of MRI and shorten the reconstruction time. We consider MRI to solve a linear inverse problem, we convert it into a constrained optimization problem based on TV regularisation, then an accelerated ADMM is established. Through a series of theoretical derivations, we verify that the algorithm satisfies the convergence ( rate of O 1/k2) /k 2 ) under the condition that one objective function is quadratically convex and the other is strongly convex. We select five undersampled templates for testing in MRI experiment and compare it with other algorithms, experimental results show that our proposed method not only improves the running speed but also gives better reconstruction results.

关键词： Alternating direction method of multiplier convex minimization problem Nesterov acceleration TV regularisation Magnetic resonance imaging

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A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert space and a Banach space

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JOURNAL OF FIXED POINT THEORY AND APPLICATIONS 2018年第4期20卷 157-157页

作者： Izuchukwu, Chinedu Okeke, Chibueze Christian Isiogugu, Felicia Obiageli Univ Kwazulu Natal Sch Math Stat & Comp Sci Durban South Africa Univ Witwatersrand DST NRF Ctr Excellence Math & Stat Sci CoE MaSS Johannesburg South Africa Univ Nigeria Dept Math Nsukka Nigeria

The main purpose of this paper is to introduce a viscosity-type iterative algorithm for approximating a common solution of a split variational inclusion problem and a fixed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating a common solution of a split variational inclusion problem and a fixed point problem for a multivalued quasi-nonexpansive mapping between a Hilbert space and a Banach space. Furthermore, we applied our results to study a split convex minimization problem. Also, a numerical example of our result is given. Our results extend and improve the results of Byrne et al. (J. Nonlinear convex Anal. 13, 759-775, 2012), Moudafi (J. Optim. Theory Appl. 150, 275-283, 2011), Takahashi and Yao (Fixed Point Theory Appl. 2015, 87, 2015), and a host of other important results in this direction.

关键词： Variational inclusion problem viscosity iterative method convex minimization problem quasi-nonexpansive mapping maximal monotone mapping resolvent operators

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A GENERALIZED HYBRID STEEPEST DESCENT METHOD AND APPLICATIONS

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2017年第1期1卷 111-126页

作者： Sahu, D. R. Yao, J. C. Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India China Med Univ Ctr Gen Educ Taichung 40402 Taiwan

The purpose of this paper is to investigate a generalized hybrid steepest descent method and develop a convergence theory for solving monotone variational inequality over the fixed point set of a mapping which is not necessarily Lipschitz continuous. Using this result, we consider the convex minimization problem for a continuously differentiable convex function whose gradient is not necessarily Lipschitzian.

关键词： convex minimization problem Fixed point Hybrid steepest descent method Monotone variational inequality Nearly asymptotically nonexpansive mapping

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An innovative inertial extra-proximal gradient algorithm for solving convex optimization problems with application to image and signal processing

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HELIYON 2023年第10期9卷 e20513页

作者： Olilima, Joshua Mogbademu, Adesanmi Memon, M. Asif Obalalu, Adebowale Martins Akewe, Hudson Seidu, Jamel Augustine Univ Dept Math Sci Lagos Nigeria Univ Lagos Dept Math Lagos Nigeria Sukkur IBA Univ Dept Math & Social Sci Sukkur 65200 Pakistan Univ Mines & Technol UMaT Essikado Sch Railways & Infrastructure Dev Sekondi Takoradi Ghana

This study introduces an innovative approach to address convex optimization problems, with a specific focus on applications in image and signal processing. The research aims to develop a selfadaptive extra proximal algorithm that incorporates an inertial term to effectively tackle challenges in convex optimization. The study's significance lies in its contribution to advancing optimization techniques in the realm of image deblurring and signal reconstruction. The proposed methodology involves creating a novel self-adaptive extra proximal algorithm, analyzing its convergence rigorously to ensure reliability and effectiveness. Numerical examples, including image deblurring and signal reconstruction tasks using only 10% of the original signal, illustrate the practical applicability and advantages of the algorithm. By introducing an inertial term within the extra proximal framework, the algorithm demonstrates potential for faster convergence and improved optimization outcomes, addressing real-world challenges of image enhancement and signal reconstruction. The algorithm's incorporation of an inertial term showcases its potential for faster convergence and improved optimization outcomes. This research significantly contributes to the field of optimization techniques, particularly in the context of image and signal processing applications.

关键词： Forward-backward algorithm Signal & image processing convex minimization problem Weak convergence Inertial method

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Modified Proximal Point Algorithms for Solving Constrained minimization and Fixed Point problems in Complete CAT(0) Spaces

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2018年第3期15卷 97-97页

作者： Phuengrattana, Withun Onjai-uea, Nawitcha Cholamjiak, Prasit Nakhon Pathom Rajabhat Univ Fac Sci & Technol Dept Math Nakhon Pathom 73000 Thailand Nakhon Pathom Rajabhat Univ Res & Dev Inst Res Ctr Pure & Appl Math Nakhon Pathom 73000 Thailand Univ Phayao Sch Sci Phayao 56000 Thailand

In this paper, we propose a new modified proximal point algorithm for a countably infinite family of nonexpansive mappings in complete CAT(0) spaces and prove strong convergence theorems for the proposed process under suitable conditions. We also apply our results to solving linear inverse problems and minimization problems. Several numerical examples are given to show the efficiency of the presented method.

关键词： convex minimization problem Fixed point problems Proximal point algorithm Nonexpansive mappings CAT(0) spaces

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Split monotone variational inclusion with errors for image-feature extraction with multiple-image blends problem

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FIXED POINT THEORY AND ALGORITHMS FOR SCIENCES AND ENGINEERING 2023年第1期2023卷 5-5页

作者： Tianchai, Pattanapong Maejo Univ Fac Sci Chiangmai 50290 Thailand

In this paper, we introduce a new iterative forward-backward splitting algorithm with errors for solving the split monotone variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for this problem is obtained. We also apply our main result to image-feature extraction with the multiple-image blends problem, the split minimization problem, and the convex minimization problem, and provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique.

关键词： Split monotone variational inclusion problem convex minimization problem Maximal monotone operator Forward-backward method Iterative shrinkage thresholding Image-feature extraction Multiple-image blends problem

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