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A Novel algorithm and Its Convergence Analysis for Solving the Generalized Split Inverse Problem

MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第5期48卷 5822-5837页

作者： Eslamian, Mohammad Univ Sci & Technol Mazandaran Dept Math Behshahr Iran

In this paper, we consider a bilevel problem: Variational inequalities over the solution set of a general split inverse problem consists of a monotone variational inclusion problem. We propose a relaxed inertial forward-backward-forward splitting algorithm with a new step size rule for finding an approximate solution of this problem in real Hilbert spaces. Under some mild conditions, we prove a strong convergence theorem for the algorithm produced by the method. Also, we apply our result to study certain classes of bilevel optimization problems, and split inverse problems. Finally, we present some numerical experiments and application in signal recovery problem to demonstrate the efficiency of the proposed algorithm.

关键词： forward-backward-forward splitting algorithm inertial algorithm split inverse problem variational inclusion problem variational inequality problem

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inertial algorithms for Solving Nonmonotone Variational Inequality Problems

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TAIWANESE JOURNAL OF MATHEMATICS 2024年第2期28卷 397-421页

作者： Tuyen, Bien Thanh Manh, Hy Duc Van Dinh, Bui Le Quy Don Tech Univ Fac Informat Technol Dept Math Hanoi Vietnam

In this paper, we will construct an inertial algorithm without using the embedded projection method to find a solution of variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. The iterative sequences generated by algorithms under the main assumption SM not equal 0 are proved that they converge to a solution of the corresponding problems. In addition, numerical experiments are provided to show the effectiveness of the algorithm.

关键词： variational inequality nonmonotone inertial algorithm projection

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Superiorization methodology and perturbation resilience of inertial proximal gradient algorithm with application to signal recovery

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JOURNAL OF SUPERCOMPUTING 2020年第12期76卷 9456-9477页

作者： Pakkaranang, Nuttapol Kumam, Poom Berinde, Vasile Suleiman, Yusuf I. KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE KMUTT Fixed Point Theory & Applicat Res Grp KMUTT Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand KMUTT KMUTTFixed Point Res Lab Dept Math Fac Sci Room SCL 802 Fixed Point LabSci Lab Bldg Bangkok 10140 Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung 40402 Taiwan Tech Univ Cluj Napoca Dept Math & Comp Sci North Univ Ctr Baia Mare Victorie 76 Baia Mare 430072 Romania Kano Univ Sci & Technol Dept Math PMB 3042 Kano Nigeria

In this paper, we construct a novel algorithm for solving non-smooth composite optimization problems. By using inertial technique, we propose a modified proximal gradient algorithm with outer perturbations, and under standard mild conditions, we obtain strong convergence results for finding a solution of composite optimization problem. Based on bounded perturbation resilience, we present our proposed algorithm with the superiorization method and apply it to image recovery problem. Finally, we provide the numerical experiments to show efficiency of the proposed algorithm and comparison with previously known algorithms in signal recovery.

关键词： Composite optimization problems Proximal gradient algorithm inertial algorithm Superiorization methodology Bounded perturbation resilience

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QUASI-NONEXPANSIVE ITERATIONS ON THE AFFINE HULL OF ORBITS: FROM MANN'S MEAN VALUE algorithm TO inertial METHODS

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SIAM JOURNAL ON OPTIMIZATION 2017年第4期27卷 2356-2380页

作者： Combettes, Patrick L. Glaudin, Lilian E. North Carolina State Univ Dept Math Raleigh NC 27695 USA UPMC Univ Paris 06 Sorbonne Univ UMR 7598 Lab Jacques Louis Lions F-75005 Paris France

Fixed point iterations play a central role in the design and the analysis of a large number of optimization algorithms. We study a new iterative scheme in which the update is obtained by applying a composition of quasi-nonexpansive operators to a point in the affine hull of the orbit generated up to the current iterate. This investigation unifies several algorithmic constructs, including Mann's mean value method, inertial methods, and multilayer memoryless methods. It also provides a framework for the development of new algorithms, such as those we propose for solving monotone inclusion and minimization problems.

关键词： averaged operator fixed point iteration forward-backward algorithm inertial algorithm mean value iterations monotone operator splitting nonsmooth minimization Peaceman-Rachford algorithm proximal algorithm

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An inertial primal-dual fixed point algorithm for composite optimization problems

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2022年第17期45卷 10628-10639页

作者： Wen, Meng Tang, Yuchao Cui, Angang Peng, Jigen Xian Polytech Univ Sch Sci Xian Peoples R China Guangzhou Univ Sch Math & Informat Sci Guangzhou 510006 Peoples R China Nanchang Univ Dept Math Nanchang Jiangxi Peoples R China Yulin Univ Sch Math & Stat Yulin Peoples R China

In this paper, we consider an inertial primal-dual fixed point algorithm (IPDFP) to compute the minimizations of the sum of a non-smooth convex function and a finite family of composite non-smooth convex functions, each one of which is composed of a non-smooth convex function and a bounded linear operator. This is a full splitting approach, in the sense that non-smooth functions are processed individually via their proximity operators. The convergence of the IPDFP is obtained by reformulating the problem to the sum of three non-smooth convex functions. Furthermore, we propose a preconditioning technique for the IPDFP. The key idea of the preconditioning technique is that the constant iterative parameters are updated self-adaptively in the iteration process. What's more, we also give a simple and easy way to choose the diagonal preconditioners while the convergence of the iterative algorithms is maintained. This work brings together and notably extends several classical splitting schemes, like the primal-dual method proposed by Chambolle and Pock, and the recent proximity algorithms of Charles et al. designed for the L1$$ {L}_1 $$/TV image denoising model. The iterative algorithm is used for solving non-differentiable convex optimization problems arising in image processing.

关键词： composite optimization inertial algorithm operator splitting proximity operator

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Convergence of an inertial Shadow Douglas-Rachford Splitting algorithm for Monotone Inclusions

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2021年第14期42卷 1627-1644页

作者： Fan, Jingjing Qin, Xiaolong Tan, Bing Univ Elect Sci & Technol China Inst Fundamental & Frontier Sci Chengdu Peoples R China Zhejiang Normal Univ Dept Math Jinhua Zhejiang Peoples R China

An inertial shadow Douglas-Rachford splitting algorithm for finding zeros of the sum of monotone operators is proposed in Hilbert spaces. Moreover, a three-operator splitting algorithm for solving a class of monotone inclusion problems is also concerned. The weak convergence of the algorithms is investigated under mild assumptions. Some numerical experiments are implemented to illustrate our main convergence results.

关键词： inertial algorithm Monotone inclusion Shadow Douglas-Rachford splitting algorithm Three-operator splitting

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inertial self-adaptive algorithm for solving split feasible problems with applications to image restoration

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2019年第18期42卷 7268-7284页

作者： Suparatulatorn, Raweerote Charoensawan, Phakdi Poochinapan, Kanyuta Chiang Mai Univ Dept Math Res Ctr Math & Appl Math Fac Sci Chiang Mai 50200 Thailand CHE Ctr Excellence Math Si Ayutthaya Rd Bangkok 10400 Thailand

We introduce a new self-adaptive algorithm for applications to image restoration problems. In order to study an image restoration, we consider the algorithm that contains inertial effects and step sizes, which is independent from the norm of the bounded linear operator. With some control conditions, the strong convergence to the minimum norm solution of the algorithm is obtained. Convergence analysis of the proposed algorithm is also discussed. Moreover, numerical results of image restoration problems illustrate that the proposed algorithm is efficient and outperforms other ones.

关键词： demicontractive operator image restoration problem inertial algorithm self-adaptive algorithm

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inertial viscosity forward-backward splitting algorithm for monotone inclusions and its application to image restoration problems

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2020年第1-2期97卷 482-497页

作者： Kitkuan, Duangkamon Kumam, Poom Martinez-Moreno, Juan Sitthithakerngkiet, Kanokwan KMUTT Fac Sci Dept Math KMUTT Fixed Point Res LabKMUTT Fixed Point Theor 126 Pracha Uthit Rd Bangkok 10140 Thailand KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand Univ Jaen Fac Expt Sci Dept Math Jaen Spain King Mongkuts Univ Technol North Bangkok Fac Appl Sci Dept Math Bangkok Thailand

In this research, we are interested about the monotone inclusion problems in the scope of the real Hilbert spaces by using an inertial forward-backward splitting algorithm. In addition, we have discussed the applicati... 详细信息

关键词： inertial algorithm inertial splitting algorithm forward-backward algorithm monotone inclusion problems

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Two-step inertial Bregman alternating minimization algorithm for nonconvex and nonsmooth problems

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第4期84卷 941-966页

作者： Zhao, Jing Dong, Qiao-Li Rassias, Michael Th Wang, Fenghui Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China Hellen Mil Acad Dept Math & Engn Sci Vari Attikis 16673 Greece Inst Adv Study Program Interdisciplinary Studies 1 Einstein Dr Princeton NJ 08540 USA Luoyang Normal Univ Dept Math Luoyang Peoples R China

In this paper, we propose an algorithm combining Bregman alternating minimization algorithm with two-step inertial force for solving a minimization problem composed of two nonsmooth functions with a smooth one in the absence of convexity. For solving nonconvex and nonsmooth problems, we give an abstract convergence theorem for general descent methods satisfying a sufficient decrease assumption, and allowing a relative error tolerance. Our result holds under the assumption that the objective function satisfies the Kurdyka-Lojasiewicz inequality. The proposed algorithm is shown to satisfy the requirements of our abstract convergence theorem. The convergence is obtained provided an appropriate regularization of the objective function satisfies the Kurdyka-Lojasiewicz inequality. Finally, numerical results are reported to show the effectiveness of the proposed algorithm.

关键词： Nonconvex optimization inertial algorithm Bregman distance Kurdyka-Lojasiewicz inequality Alternating minimization

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A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem

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NUMERICAL algorithmS 2020年第2期84卷 485-512页

作者： Alecsa, Cristian Daniel Laszlo, Szilard Csaba Viorel, Adrian Romanian Acad Tiberiu Popoviciu Inst Numer Anal Cluj Napoca Romania Babes Bolyai Univ Dept Math Cluj Napoca Romania Tech Univ Cluj Napoca Dept Math Str Memorandumului 28 Cluj Napoca 400114 Romania

We investigate an algorithm of gradient type with a backward inertial step in connection with the minimization of a nonconvex differentiable function. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. Further, we provide convergence rates for the generated sequences and the objective function values formulated in terms of the Lojasiewicz exponent. Finally, some numerical experiments are presented in order to compare our numerical scheme with some algorithms well known in the literature.

关键词： inertial algorithm Nonconvex optimization Kurdyka-Lojasiewicz inequality Convergence rate

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