proximal point algorithm
در نشریات گروه ریاضی-
International Journal Of Nonlinear Analysis And Applications, Volume:13 Issue: 2, Summer-Autumn 2022, PP 1023 -1032
In this article, we suggest and analyze a viscosity approximation method to a zero of a monotone operator in the setting of Hadamard spaces. We derive the convergence of sequences generated by the proposed viscosity methods under some suitable assumptions. Also, some applications to solve the variational inequality, optimization and fixed point problems are given on Hadamard spaces.
Keywords: Hadamard space, proximal point algorithm, Viscosity approximation method, Monotone operators, Convex minimization -
International Journal Of Nonlinear Analysis And Applications, Volume:12 Issue: 2, Summer-Autumn 2021, PP 511 -526
In this paper, we investigate the problem of finding a common element of the solution set of convex minimization problem, the solution set of variational inequality problem and the solution set of fixed point problem with an infinite family of quasi-nonexpansive mappings in real Hilbert spaces. Based on the well-known proximal point algorithm and viscosity approximation method, we propose and analyze a new iterative algorithm for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for the sequence generated by the proposed method. Application to convex minimization and variational inequality problems coupled with inclusion problem is provided to support our main results.,Our proposed method is quite general and includes the iterative methods considered in the earlier and recent literature as special cases.
Keywords: Convex minimization problem, Proximal point algorithm, Common fixed points, Quasi-nonexpansive mappings, Variational inequality problem -
International Journal Of Nonlinear Analysis And Applications, Volume:11 Issue: 2, winter spring 2020, PP 469 -481
This paper is devoted to finding a zero point of a weighted resolvent average of a finite family of monotone operators. A new proximal point algorithm and its convergence analysis is given. It is shown that the sequence generated by this new algorithm, for a finite family of monotone operators converges strongly to the zero point of their weighted resolvent average. Finally, our results are illustrated by some numerical examples.
Keywords: Weighted resolvent average, proximal point algorithm, projection algorithm, monotone operators -
مسئله های تعادل کاربردهای فراوانی در نظریه بهینه سازی و آنالیز محدب دارند و به همین دلیل است که روش های متفاوتی برای حل مسئله های تعادل درفضاهای مختلف از جمله فضاهای هیلبرت و فضاهای باناخ ارائه شده است. هدف این مقاله، ارائه روشی برای به دست آوردن جواب مسئله تعادل در فضاهای باناخ می باشد. در واقع، یک الگوریتم نقطه مبدایی ترکیبی با استفاده از حلال یک عملگر یکنوای ماکسیمال در فضای باناخ را در نظر می گیریم. تحت شرایطی مناسب، همگرایی قوی دنباله تولید شده توسط الگوریتم به ریشه عملگر یکنوای ماکسیمال را ثابت می کنیم. به عنوان کاربردی از نتیجه اصلی و با استفاده از قضایای ثابت شده، برای هر دوتابع یکنوا می توانیم یک عملگر یکنوای ماکسیمال ارائه کنیم به طوریکه، ریشه عملگر یکنوای ماکسیمال همان جواب مسئله تعادل باشد. نتایج این مقاله، تعدادی از نتایج حاصل شده در مقالات مختلف را تعمیم داده یا بهبود می بخشد.
کلید واژگان: مسئله تعادل، عملگر یکنوای ماکسیمال، عملگر حلال در فضای باناخ، الگوریتم نقطه مبداییEquilibrium problems have many uses in optimization theory and convex analysis and which is why different methods are presented for solving equilibrium problems in different spaces, such as Hilbert spaces and Banach spaces. The purpose of this paper is to provide a method for obtaining a solution to the equilibrium problem in Banach spaces. In fact, we consider a hybrid proximal point algorithm using the resolvent of a maximal monotone operator in Banach space. Under appropriate conditions, we prove the strong convergence of the generated sequence by the algorithm to the zero of the maximal monotone operator. As an application of the main result, and using proved theorems, we can provide a maximal monotone operator for any monotone bifunction so that the zero of the maximal monotone operator is the solution to the equilibrium problem. The results of this paper generalize or improve the obtained results in the various papers.
Keywords: Equilibrium problem, maximal monotone operator, the resolvent operator in Banach space, proximal point algorithm -
In this paper, we consider a proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in real Hilbert spaces. Also, we give a necessary and sufficient condition for the common zero set of finite operators to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the set of common zeros of operators.Keywords: Maximal monotone operator, Proximal point algorithm, Nonexpansive map, Resolvent operator
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In this paperý, ýwe generalize the proximal point algorithm to complete CAT(0) spaces and showý ýthat the sequence generated by the proximal point algorithmý w-converges to a zero of the maximalý ýmonotone operatorý. ýAlsoý, ýwe prove that if fý:ýX→ýý]−∞ý,ý∞] is a properý, ýconvex and lower semicontinuousý ýfunction on the complete CAT(0) space Xý, ýthen the proximal point algorithm w-converges to a zero of the subdifferential of fý, ýi.e., a minimizer of fý. ýSome strong convergence results (convergence in metric) are also presented with additional assumptions on the monotone operator andý ýthe convex function f.Keywords: Hadamard space, maximal monotone operator, Proximal point algorithm, w-convergence, subdifferential
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