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An ill-posed nonlocal two-point problem for systems of partial differential equations

SIBERIAN MATHEMATICAL JOURNAL 2005年第1期46卷 94-102页

作者： Il'kiv, VS Ptashnyk, BI Lviv Polytechnic National University Lviv

We study conditions for existence and uniqueness of a pseudosolution in a Sobolev space to a nonlocal two-point boundary value problem for an indefinite type nonhomogeneous system of partial differential equations wit... 详细信息

关键词： partial differential equation nonlocal condition ill-posed problem pseudosolution convex minimization problem

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Weak and strong convergence theorems for maximal monotone operators in a Banach space

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SET-VALUED ANALYSIS 2004年第4期12卷 417-429页

作者： Kamimura, S Kohsaka, F Takahashi, W Hitotsubashi Univ Grad Sch Int Corp Strategy Chiyoda Ku Tokyo 1018439 Japan Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku Tokyo 1528552 Japan

In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim. 14 (1976), 877-898] and Kamimura and Takahashi [J. Approx. Theory 106 (2000), 226-2401. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.

关键词： convex minimization problem maximal monotone operator proximal point algorithm resolvent uniformly convex Banach space

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The zeros of monotone operators for the variational inclusion problem in Hilbert spaces

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2021年第1期2021卷 1-23页

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

In this paper, we introduce a regularization method for solving the 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, which allow us to obtain a short proof of another strong convergence theorem for this problem. We also apply our main result to the fixed point problem of the nonexpansive variational inequality problem, the common fixed point problem of nonexpansive strict pseudocontractions, the convex minimization problem, and the split feasibility problem. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

关键词： Common fixed point problem convex minimization problem Split feasibility problem Variational inclusion problem Maximal monotone operator Regularization method

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Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces

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FIXED POINT THEORY AND APPLICATIONS 2015年第1期2015卷 1-13页

作者： Cholamjiak, Prasit Abdou, Afrah A. N. Cho, Yeol Je Univ Phayao Sch Sci Phayao 56000 Thailand King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia Gyeongsang Natl Univ Dept Math Educ Jinju 660701 South Korea Gyeongsang Natl Univ RINS Jinju 660701 South Korea

In this paper, we introduce a new modified proximal point algorithm involving fixed point iterates of nonexpansive mappings in CAT(0) spaces and prove that the sequence generated by our iterative process converges to ... 详细信息

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

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Characterizations of the existence of solutions for variational inequality problems in Hilbert spaces

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JOURNAL OF FIXED POINT THEORY AND APPLICATIONS 2023年第3期25卷 1-16页

作者： Alberto Hernandez-Linares, Carlos Martinez-Anteo, Eduardo Muniz-Perez, Omar Univ Veracruzana Fac Matemat Paseo 122Sec 2 S-NCol Nuevo Xalapa Xalapa Mexico CONACYT Ctr Invest Matemat AC Unidad Merida Parque Cientif & Tecnol Yucatan Km 5-5 Carretera Sierra Papacal Chuburna Puerto Merida 97302 Mexico

In this work, we give necessary and sufficient conditions for the existence of solutions to the variational inequality problem: find x(0) E K such that (F(x(0)), y - x(0)) = 0, for every y ? K, where K is a nonempty closed convex subset of a real Hilbert space H and F : K? H is a monotone and continuous operator. These characterizations are given in terms of approximate fixed points sequences, as well as by Leray- Schauder condition. We apply our obtained results in a constrained convex minimization problem.

关键词： Variational inequality fixed point monotone mapping metric projection closed convex cones convex minimization problem Hilbert spaces

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Convergence rate analysis of proximal gradient methods with applications to composite minimization problems

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OPTIMIZATION 2021年第1期70卷 75-100页

作者： Sahu, D. R. Yao, J. C. Verma, M. Shukla, K. K. Banaras Hindu Univ Dept Math Varanasi Uttar Pradesh India China Med Univ China Med Univ Hosp Res Ctr Interneural Comp Taichung Taiwan Inst Res & Dev Banking Technol Hyderabad India Indian Inst Technol BHU Dept Comp Sci & Engn Varanasi Uttar Pradesh India

First-order methods such as proximal gradient, which use Forward-Backward Splitting techniques have proved to be very effective in solving nonsmooth convex minimization problem, which is useful in solving various practical problems in different fields such as machine learning and image processing. In this paper, we propose few new forward-backward splitting algorithms, which consume less number of iterations to converge to an optimum. In addition, we derive convergence rates for the proposed formulations and show that the speed of convergence of these algorithms is significantly better than the traditional forward-backward algorithm. To demonstrate the practical applicability, we apply them to two real-world problems of machine learning and image processing. The first issue deals with the regression on high-dimensional datasets, whereas the second one is the image deblurring problem. Numerical experiments have been conducted on several publicly available real datasets to verify the obtained theoretical results. Results demonstrate the superiority of our algorithms in terms of accuracy, the number of iterations required to converge and the rate of convergence against the classical first-order methods.

关键词： Accretive operator convex minimization problem forward-backward splitting algorithm Mann iteration method S-iteration 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 1-20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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A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2019年第1期2019卷 1-33页

作者： Izuchukwu, C. Isiogugu, F. O. Okeke, C. C. Univ Kwazulu Natal Sch Math Stat & Comp Sci Durban South Africa DST NRF Ctr Excellence Math & Stat Sci CoEMass Johannesburg South Africa Univ Nigeria Dept Math Nsukka Nigeria

In this paper, we introduce a new viscosity-type iteration process for approximating a common solution of a finite family of split variational inclusion problem and fixed point problem. We prove that the proposed algorithm converges strongly to a common solution of a finite family of split variational inclusion problems and fixed point problem for a finite family of type-one demicontractive mappings between a Hilbert space and a Banach space. Furthermore, we applied our results to study a finite family of split convex minimization problems, and also considered a numerical experiment of our results to further illustrate its applicability. Our results extend and improve the results of Byrne et al. (J. Nonlinear convex Anal. 13:759-775, 2012), Kazmi and Rizvi (Optim. Lett. 8(3):1113-1124, 2014), Moudafi (J. Optim. Theory Appl. 150:275-283, 2011), Shehu and Ogbuisi (Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat. 110(2):503-518, 2016), Takahashi and Yao (Fixed Point Theory Appl. 2015:87, 2015), Chidume and Ezeora (Fixed Point Theory Appl. 2014:111, 2014), and a host of other important results in this direction.

关键词： Variational inclusion problem Viscosity iterative method convex minimization problem Strictly pseudocontractions Maximal monotone mapping Resolvent operators

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Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping

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FIXED POINT THEORY AND APPLICATIONS 2015年第1期2015卷 1-12页

作者： Dong, Qiao-Li Yuan, Han-bo Civil Aviat Univ China Coll Sci Tianjin 300300 Peoples R China Civil Aviat Univ China Tianjin Key Lab Adv Signal Proc Tianjin 300300 Peoples R China

The purpose of this paper is to present accelerations of the Mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective.

关键词： nonexpansive mapping convex minimization problem Picard algorithm Mann algorithm CQ algorithm conjugate gradient method steepest descent method

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A modified regularization method for finding zeros of monotone operators in Hilbert spaces

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2015年第1期2015卷 1-10页

作者： Cholamjiak, Prasit Cholamjiak, Watcharaporn Suantai, Suthep Univ Phayao Sch Sci Phayao 56000 Thailand Chiang Mai Univ Fac Sci Dept Math Chiang Mai 50200 Thailand

We study the regularization method for solving the variational inclusion problem of the sum of two monotone operators in Hilbert spaces. The strong convergence theorem is then established under some relaxed conditions which mainly improves and recovers that of Qin et al. (Fixed Point Theory Appl. 2014:75, 2014). We also apply our main result to the convex minimization problem, the fixed point problem and the variational inequality problem. Finally we provide numerical examples for supporting the main result.

关键词： convex minimization problem maximal monotone operator inverse strongly monotone operator regularization method variational inequality problem

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