یک الگوریتم نقطه مبدایی ترکیبی برای عملگر حلال در فضای باناخ
محورهای موضوعی : آمار
1 - گروه ریاضی، دانشکده علوم پایه، واحد ساری، دانشگاه آزاد اسلامی، ساری، ایران
2 - گروه ریاضی، دانشکده علوم پایه، واحد ساری، دانشگاه آزاد اسلامی، ساری، ایران
کلید واژه: maximal monotone operator, Equilibrium problem, proximal point algorithm, the resolvent operator in Banach space,
چکیده مقاله :
مسئلههای تعادل کاربردهای فراوانی در نظریه بهینه سازی و آنالیز محدب دارند و به همین دلیل است که روش‌های متفاوتی برای حل مسئلههای 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 می‌باشد. در واقع، یک الگوریتم نقطه مبدایی ترکیبی با استفاده از حلال یک عملگر یکنوای ماکسیمال در فضای باناخ را در نظر میگیریم. تحت شرایطی مناسب، همگرایی قوی دنباله تولید شده توسط الگوریتم به ریشه عملگر یکنوای ماکسیمال را ثابت میکنیم. به عنوان کاربردی از نتیجه اصلی و با استفاده از قضایای ثابت شده، برای هر دوتابع یکنوا میتوانیم 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 میبخشد.
Equilibrium 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.
[1] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Mathematics Student. 63(1), 123-145 (1994)
[2] Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers (2000)
[3] Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Springer Science Business Media 62 (2012)
[4] Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6(1), 117-136 (2005)
[5] Moudafi, A.: On finite and strong convergence of a proximal method for equilibrium problems. Numer. Funct. Anal. Optim. 28(11-12), 1347-1354 (2007)
[6] Moudafi, A.: Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 15(1-2), 91-100 (1999)
[7] Tada, A., Takahashi, W.: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. Nonlinear Convex Anal. 8,609-617 (2007)
[8] Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331(1), 506-515 (2007)
[9] Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70(1), 45-57 (2009)
[10] Hadjisavvas N.I., Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization. 59(2), 147-160, (2010)
[11] Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877-898 (1976)
[12] Rockafellar, R.: On the maximal monotonicity of subdifferential mappings. Pacific J. Math. 33(1), 209-216 (1970)
[13] Boikanyo, O.A., Morosanu, G.: Modified Rockafellar’s algorithms. Math. Sci. Res. J. 13(5), 101-122 (2009)
[14] Khatibzadeh, H.: Some remarks on the proximal point algorithm. J. Optim. Theory Appl. 153(3), 769-778 (2012)
[15] Khatibzadeh, H., Ranjbar, S.: On the strong convergence of Halpern type proximal point algorithm. J. Optim. Theory Appl. 158(2), 385-396 (2013)
[16] Rouhani, B.D., Khatibzadeh, H.: On the proximal point algorithm. J. Optim. Theory Appl. 137(2), 411-417 (2008)
[17] Solodov, M.V., Svaiter, B.F.: A hybrid approximate extragradient proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Anal. 7(4), 323-345 (1999)
[18] Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6(1), 59-70 (1999)
[19] Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66(1), 240-256 (2002)
[20] Aoyama, K., Kohsaka, F., Takahashi, W.: Shrinking projection methods for firmly nonexpansive mappings. Nonlinear Anal. 71(12), 1626-1632 (2009)
[21] Aoyama, K., Kohsaka, F., Takahashi, W.: Three generalizations of firmly nonexpansive mappings: their relations and continuity properties. J. Nonlinear Convex Anal. 10(1), 131-147 (2009)
[22] Aoyama, K.O., Kohsaka, F,U., Takahashi, W.A.: Strong convergence theorems for a family of mappings of type (P) and applications. in Nonlinear Analysis and Optimization, pp. 1-17, Yokohama Publishers, Yokohama (2009)
[23] Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13(3), 938-945 (2002)
[24] Ohsawa, S., Takahashi, W.: Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces. Arch. Math. 81(4), 439-445 (2003)
[25] Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87(1), 189-202 (2000)
[26] Li, L., Song, W.: Modified proximal-point algorithm for maximal monotone operators in Banach spaces. J. Optim. Theory Appl. 138(1), 45-64 (2008)
[27] Matsushita, S.Y., Xu, L.: Finite termination of the proximal point algorithm in Banach spaces. J. Math. Anal. Appl. 387(2), 765-769 (2012)
[28] Matsushita, S.Y., Xu, L.: On convergence of the proximal point algorithm in Banach spaces. Proc. Amer. Math. Soc. 139(11), 4087-4095 (2011)
[29] Nakajo, K., Shimoji, K., Takahashi, W.: Strong convergence theorems by the hybrid method for families of mappings in Banach spaces. Nonlinear Anal. 71(3), 812-818 (2009)
[30] Dadashi,V., Khatibzadeh, H.: On the Weak and Strong Convergence of the Proximal Point Algorithm in Reflexive Banach Spaces. Optimization, 66(9), 1487-1494 (2017)
[31] Dadashi,V., Postolache, M.: Hybrid Proximal Point Algorithm and Applications to Equilibrium Problems and Convex Programming. J. Optim. Theory Appl. 174, 518-529 (2017).
[32] Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3(4), 510-585 (1969)
[33] Tsukada, M.: Convergence of best approximations in a smooth Banach space. J. Approx. Theory. 40(4), 301-309, (1984)
[34] Barbu, V., Precupanu, Th.: convexity and optimization in Banach spaces. Dordrecht: Springer monographs in mathematics. Springer, 2012
