广义黏性逼近方法及其应用

doi:10.15960/j.cnki.issn.1007-6093.2020.03.010

Abstract

Abstract: 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.

Key words: generalized viscosity approximation method, nonexpansive mapping, averaged mapping, constrained convex optimization, bilevel optimization

CLC Number:

O221.2
O224

GUO Ke, WANG Tao, Zhang Youcai. Generalized viscosity approximation method and its application[J]. Operations Research Transactions, 2020, 24(3): 127-140.

References

[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.

Generalized viscosity approximation method and its application

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	XU Zi, ZHANG Huiling. Optimization algorithms and their complexity analysis for non-convex minimax problems [J]. Operations Research Transactions, 2021, 25(3): 74-86.
[2]	KE Rongzhu, ZHANG Jin. On the first order approach for bilevel programming: moral hazard case [J]. Operations Research Transactions, 2021, 25(3): 87-104.
[3]	LI Jun, CHEN Jiawei, DENG Guangju. Optimality conditions of generalized convex interval valued optimization problems [J]. Operations Research Transactions, 2020, 24(4): 25-38.
[4]	ZHAO Dan, GAO Yuelin. Non-parameter filled function method for nonlinear integer programming [J]. Operations Research Transactions, 2020, 24(4): 63-73.
[5]	TAO Jie, GAO Yan. A price oriented optimal production and allocation strategy of protective suppliers in the epidemic situation [J]. Operations Research Transactions, 2020, 24(1): 13-22.
[6]	CHEN Yong, WANG Wei, XU Yifan. An stochastic algorithm for global optimization with linear constraints based on intermittent diffusion [J]. Operations Research Transactions, 2020, 24(1): 88-100.
[7]	LI Chaoqiong, LI Feng. A partial parallel splitting LQP alternating direction method of multipliers for solving monotone variational inequalities [J]. Operations Research Transactions, 2020, 24(1): 101-114.
[8]	MIAO Xiaonan, GU Jian, XIAO Xiantao. A cubic regularization method for solving nonsmooth equations [J]. Operations Research Transactions, 2019, 23(2): 17-30.
[9]	LIANG Renli, BAI Yanqin. A splitting augmented Lagrangian method embedding in the BB method for solving the sparse logistic problem [J]. Operations Research Transactions, 2019, 23(2): 86-94.
[10]	TANG Liping, YANG Xinmin. Characterizations of two classes of cone generalized pseudoinvexity [J]. Operations Research Transactions, 2015, 19(1): 1-8.
[11]	ZENG Youfang, TANG Chunming. A non-interior-point continuous algorithm with superlinear convergence for second-order cone programming [J]. Operations Research Transactions, 2015, 19(1): 18-30.
[12]	LIU Shuixia, CHEN Guoqing. A multiplier relaxation method for solving mathematical programs with complementarity constraints [J]. Operations Research Transactions, 2014, 18(4): 119-130.
[13]	YANG Yun, PENG Zheng. An inexact parallel splitting method for separable convex optimization problem [J]. Operations Research Transactions, 2014, 18(3): 33-46.
[14]	FENG Zengzhe, ZHANG Xixue, LIU Jianbo$, FANG Liang. A new wide neighborhood infeasible interior-point algorithm for semidefinite programming [J]. Operations Research Transactions, 2014, 18(2): 49-58.
[15]	DAI Yuhong, LIU Xinwei. Advances in linear and nonlinear programming [J]. Operations Research Transactions, 2014, 18(1): 69-92.