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A Convergent 3-Block Semi-Proximal ADMM for convex minimization problems with One Strongly convex Block

ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 2015年第4期32卷 1550024-1550024页

作者： Li, Min Sun, Defeng Toh, Kim-Chuan Southeast Univ Sch Econ & Management Nanjing 210096 Jiangsu Peoples R China Natl Univ Singapore Dept Math Singapore 117548 Singapore Natl Univ Singapore Risk Management Inst Singapore 117548 Singapore

In this paper, we present a semi-proximal alternating direction method of multipliers (sPADMM) for solving 3-block separable convex minimization problems with the second block in the objective being a strongly convex function and one coupled linear equation constraint. By choosing the semi-proximal terms properly, we establish the global convergence of the proposed sPADMM for the step-length tau is an element of (0, (1 + root 5)/2) and the penalty parameter sigma is an element of (0, +infinity). In particular, if sigma > 0 is smaller than a certain threshold and the first and third linear operators in the linear equation constraint are injective, then all the three added semi-proximal terms can be dropped and consequently, the convergent 3-block sPADMM reduces to the directly extended 3-block ADMM with tau is an element of (0, (1 + root 5)/2).

关键词： convex minimization problems alternating direction method of multipliers semi-proximal strongly convex

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On convergence and complexity analysis of an accelerated forward-backward algorithm with linesearch technique for convex minimization problems and applications to data prediction and classification

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2021年第1期2021卷 1-20页

作者： Sarnmeta, Panitarn Inthakon, Warunun Chumpungam, Dawan Suantai, Suthep Chang Mai Univ Fac Sci Data Sci Res Ctr Dept Math Chang Mai 56200 Thailand Chang Mai Univ Fac Sci Dept Math Res Ctr Math & Appl Math Chang Mai 56200 Thailand

In this work, we introduce a new accelerated algorithm using a linesearch technique for solving convex minimization problems in the form of a summation of two lower semicontinuous convex functions. A weak convergence of the proposed algorithm is given without assuming the Lipschitz continuity on the gradient of the objective function. Moreover, the convexity of this algorithm is also analyzed. Some numerical experiments in machine learning are also discussed, namely regression and classification problems. Furthermore, in our experiments, we evaluate the convergent behavior of this new algorithm, then compare it with various algorithms mentioned in the literature. It is found that our algorithm performs better than the others.

关键词： convex minimization problems Machine learning Forward-backward algorithm Linesearch Accelerated algorithm Data classification

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An accelerated viscosity forward-backward splitting algorithm with the linesearch process for convex minimization problems

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2021年第1期2021卷 1-19页

作者： Suantai, Suthep Jailoka, Pachara Hanjing, Adisak Chiang Mai Univ Fac Sci Dept Math Data Sci Res Ctr Chiang Mai 50200 Thailand

In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function;however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209-1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352-1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

关键词： convex minimization problems Forward-backward splitting Linesearch Inertial techniques Viscosity approximation Strong convergence 65K05 90C25 90C30

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On some accelerated optimization algorithms based on fixed point and linesearch techniques for convex minimization problems with applications

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ADVANCES IN CONTINUOUS AND DISCRETE MODELS 2022年第1期2022卷 25-25页

作者： Yatakoat, Pornsak Suantai, Suthep Hanjing, Adisak Nakhon Phanom Univ Fac Sci Dept Math Nakhon Phanom Thailand Chiang Mai Univ Fac Sci Data Sci Res Ctr Dept Math Chiang Mai Thailand Chiang Mai Univ Fac Sci Res Ctr Math & Appl Math Dept Math Chiang Mai Thailand Rajamangala Univ Technol Dept Sci & Math Isan Surin Campus Surin Thailand

In this paper, we introduce and study a new accelerated algorithm based on forward-backward and SP-algorithm for solving a convex minimization problem of the sum of two convex and lower semicontinuous functions in a Hilbert space. Under some suitable control conditions, a weak convergence theorem of the proposed algorithm based on a fixed point is established. Moreover, we choose the stepsize of our algorithm which is independent on the Lipschitz constant of the gradient of the objective function by using a linesearch technique, and then a weak convergence result of the proposed algorithm is analyzed. As applications, we apply the proposed algorithm for solving the image restoration problems and compare its convergence behavior with other well-known algorithms in the literature. By our experiment, the algorithms have a higher efficiency than the others.

关键词： convex minimization problems Fixed points Forward-backward algorithms Image restoration problems 47H10 47J25 65K05 90C30

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An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings

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Annali dell'Universita di Ferrara 2021年第1期67卷 1-31页

作者： Alakoya, Timilehin Opeyemi Taiwo, Adeolu Mewomo, Oluwatosin Temitope Cho, Yeol Je School of Mathematics Statistics and Computer Science University of KwaZulu-Natal Durban South Africa School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu 611731 Sichuan China Department of Mathematics Education Gyeongsang National University Jinju 52828 South Korea

In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction. © 2021, Università degli Studi di Ferrara.

关键词： convex minimization problems Equilibrium problems Fixed point problems Inertial algorithm Nonexpansive mappings Subgradient extragradient method Variational inequality problems Zeros problems

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Superiorization and bounded perturbation resilience of a gradient projection algorithm solving the convex minimization problem

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OPTIMIZATION LETTERS 2023年第8期17卷 1957-1978页

作者： Erturk, Muzeyyen Salkim, Ahmet Adiyaman Univ Dept Math TR-02040 Adiyaman Turkiye

In this article, we use the superiorization methodology to investigate the bounded perturbations resilience of the gradient projection algorithm proposed in Erturk et al. (J Nonlinear convex Anal 21(4):943-951, 2020) for solving the convex minimization problem in Hilbert space setting. We obtain that the perturbed version of this gradient projection algorithm converges weakly to a solution of the convex minimization problem just like itself. We support our conclusion with an example in an infinitely dimensional Hilbert space. We also show that the superiorization methodology can be applied to the split feasibility and the inverse linear problems with the help of the perturbed gradient projection algorithm.

关键词： Bounded perturbation resilience Superiorization Gradient projection algorithm convex minimization problems Linear inverse problems Split feasibility problems

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CONVERGENCE ANALYSIS OF A NEW ITERATIVE ALGORITHM FOR SOLVING SPLIT VARIATIONAL INCLUSION problems

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2020年第2期16卷 945-964页

作者： Tang, Yan Chongqing Technol & Business Univ Coll Math & Stat Chongqing Key Lab Social Econ & Appl Stat Chongqing 400067 Peoples R China

The split variational inclusion problem (SVIP) has been extensively studied and applied in real-world problems such as intensity-modulated radiation therapy (IMRT) and in sensor networks and in computerized tomography and data compression. Inspired by the works of Lopez et al.[24], Byrne et al.[10] and Sitthithakerngkiet et al.[34], as well as of Moudafi and Thukur[29], we propose a self-adaptive step size algorithm for solving split variational inclusion problem (SVIP) without the prior knowledge of the operator norms. Under more mild conditions we obtain weak convergence of the proposed algorithm. We also construct a self-adaptive step size two-step iterative algorithm which converges strongly to the minimum-norm element of the solution of the SVIP. Finally, the performances and computational examples are presented and a comparison with related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.

关键词： Split variational inclusion problem convex minimization problems self-adaptive step size set-valued strong convergence

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Bounded perturbation resilience of projected scaled gradient methods

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2016年第2期63卷 365-392页

作者： Jin, Wenma Censor, Yair Jiang, Ming Peking Univ Sch Math Sci LMAM Beijing 100871 Peoples R China Univ Haifa Dept Math IL-3498838 Haifa Israel Peking Univ Beijing Int Ctr Math Res Beijing 100871 Peoples R China Shanghai Jiao Tong Univ Cooperat Medianet Innovat Ctr Shanghai 200240 Peoples R China

We investigate projected scaled gradient (PSG) methods for convex minimization problems. These methods perform a descent step along a diagonally scaled gradient direction followed by a feasibility regaining step via orthogonal projection onto the constraint set. This constitutes a generalized algorithmic structure that encompasses as special cases the gradient projection method, the projected Newton method, the projected Landweber-type methods and the generalized expectation-maximization (EM)-type methods. We prove the convergence of the PSG methods in the presence of bounded perturbations. This resilience to bounded perturbations is relevant to the ability to apply the recently developed superiorization methodology to PSG methods, in particular to the EM algorithm.

关键词： convex minimization problems Proximity function Projected scaled gradient Superiorization Bounded perturbation resilience

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Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems

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NUMERICAL ALGORITHMS 2018年第3期79卷 801-824页

作者： Sitthithakerngkiet, Kanokwan Deepho, Jitsupa Martinez-Moreno, Juan Kumam, Poom King Mongkuts Univ Technol North Bangkok Nonlinear Dynam Anal Res Ctr Fac Sci Appl Dept Math Bangkok 10800 Thailand King Mongkuts Univ Technol North Bangkok Fac Sci Energy & Environm Rayong Campus19 Moo 11 Amphur Bankhai 21120 Rayong Thailand Univ Jaen Fac Sci Dept Math Campus Las Lagunillas S-N Jaen 23071 Spain King Mongkuts Univ Technol Thonburi KMUTTFixed Point Res Lab Fac Sci Dept Math Room SCL 802 Fixed Point LabSci Lab Bldg Bangkok 10140 Thailand King Mongkuts Univ Technol Thonburi KMUTT Fixed Point Theory & Applicat Res Grp KMUTT Theoret & Computat Sci Ctr TaCS Fac Sci Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand

The purpose of this paper is to introduce a general iterative method for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of split variational inclusion problem in the framework Hilbert spaces. Strong convergence theorem of the sequences generated by the purpose iterative scheme is obtained. In the last section, we present some computational examples to illustrate the assumptions of the proposed algorithms.

关键词： Split variational inclusion problem Fixed-point problem convex minimization problems Nonexpansive mapping Iterative method 47H09 47H10

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COMPUTATIONAL EXPERIENCE WITH THE CHOW-YORKE ALGORITHM

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MATHEMATICAL PROGRAMMING 1980年第1期19卷 92-101页

作者： WATSON, LT Department of Computer Science Virginia Polytechnic Institute and State University 24061 Blacksburg VA USA

The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent. It is efficient and accurate for fixed point problems. L.T. Watson, T.Y. Li, and C.Y. Wang have adapted the method for zero finding problems, the nonlinear complementarity problem, and nonlinear two-point boundary value problems. Here theoretical justification is given for applying the method to some mathematical programming problems, and computational results are presented.

关键词： Chow—Yorke Algorithm Homotopy Algorithm Globally Convergent Fixed Point Computation Nonlinear Complementarity Problem Computational Results convex minimization problems Nonlinear Equations

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