The viscosity approximation method plays an important role in the study of the fixed point problem of nonexpansive mappings. In this paper, we proposed a kind of generalized viscosity approximation method. Under certain conditions, we proved the convergence of the algorithm. As applications, we applied the obtained convergence results to solve the constrained convex optimization problems and the bilevel optimization problems.
[1] 张恭庆, 林源渠. 泛函分析讲义上册[M]. 北京:北京大学出版社, 2005:1-9.
[2] Browder F E. Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces[J]. Archive for Rational Mechanics Analysis, 1967, 24:82-90.
[3] Halpern B. Fixed points of nonexpanding maps[J]. Bulletin of the American Mathematical Society, 1967, 73(73):957-961.
[4] Lions P L. Approximation de points fixes de contractions[J]. C. R. Acad. Sci. Ser. A-B Paris, 1977, 248:1357-1359.
[5] Wittmann R. Approximation of fixed points of nonexpansive mappings[J]. Archiv Der Mathematik, 1992, 58(5):486-491.
[6] Reich S. Approximating fixed points of nonexpansive mappings[J]. Panamerican Mathematical Journal, 1994, 4(2):224-238.
[7] Wongchan K, Saejung S. Strong convergence of Browder's and Halpern's type iterations in Hilbert spaces[J]. Positivity, 2018:1-14.
[8] Xu H K. Viscosity approximation methods for nonexpansive mappings[J]. Journal of Mathematical Analysis & Applications, 2004, 298(1):279-291.
[9] Krasnosell M A. Two remarks on the method of successive approximations[C]//Uspehi Mat. Nauk N.S., Russian, October, 1955:123-127.
[10] Kanzow C, Shehu Y. Generalized Krasnoselskii-Mann-type iterations for nonexpansive mappings in Hilbert spaces[J]. Computational Optimization & Applications, 2017, 67(3):1-26.
[11] Yosida K. Functional Analysis[M]. New York:Springer-Verlag, 1978:126-127.
[12] Browder F E. Convergence theorems for sequences of nonlinear operators in Banach spaces[J]. Mathematische Zeitschrift, 1967, 100(3):201-225.
[13] Bauschke H H, Combettes P L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces[M]. New York:Springer, 2011.
[14] Xu H K. Averaged mappings and the Gradient-Projection algorithm[J]. Journal of Optimization Theory & Applications, 2011, 150(2):360-378.
[15] Baillon J B, Haddad G. Quelques propriWtWs des opWrateurs angle-bornWs et n-cycliquement monotones[J]. Israel Journal of Mathematics, 1977, 26(2):137-150.
[16] Masao Fukushima. 非线性最优化基础[M]. 北京:科学出版社, 2011.
[17] Xu H K. Iterative algorithms for nonlinear operators[J]. Journal of the London Mathematical Society, 2002, 66(1):240-256.
[18] Levitin E S, Polyak B T. Constrained minimization methods[J]. Ussr Computational Mathematics & Mathematical Physics, 1966, 6(5):1-50.
[19] Bruck R E, Jr. On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space[J]. Journal of Mathematical Analysis & Applications, 1977, 61(1):159-164.
[20] Passty G B. Ergodic convergence to a zero of the sum of monotone operators in Hilbert space[J]. Journal of Mathematical Analysis & Applications, 1979, 72(2):383-390.
[21] Beck A, Teboulle M. A fast iterative Shrinkage-Thresholding algorithm for linear inverse problems[J]. Siam Journal on Imaging Sciences, 2009, 2(1):183-202.
[22] Beck A, Sabach S. A first order method for finding minimal norm-like solutions of convex optimization problems[J]. Mathematical Programming, 2014, 147(1-2):25-46.
[23] Sabach S, Shtern S. A first order method for solving convex bi-level optimization problems[J]. Siam Journal on Optimization, 2017, 27(2):640-660.
