Banach空间中极大单调算子零点的迭代逼近定理被引量：3

A Theorem of Iterative Approximation of Zero Point for Maximal Monotone Operator in Banach Space

下载PDF

导出

摘要令E为实光滑、一致凸Banach空间,E为其对偶空间．令A■ E x E为极大单调算子, A-10≠■．本文将引入新的迭代算法,并利用Lyapunov泛函, Qr算子与广义投影算子等技巧,证明了迭代序列弱收敛于极大单调算子A的零点的结论． Let E be a real smooth and uniformly convex Banach space, and E^＊ its duality space. Let A belong to E × E^＊ be a maximal monotone operator with A^-1 0≠φ. A new iterative scheme is introduced which is proved to be weakly convergent to zero point of maximal monotone operator A by using the techniques of Lyapunov functional, Qr operator and generalized projection operator, etc.

作者魏利周海云

机构地区河北经贸大学数学与统计学学院军械工程学院应用数学与力学研究所

出处《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2007年第1期177-184,共8页 数学研究与评论（英文版）

基金国家自然科学基金(10471003)

关键词 LYAPUNOV泛函极大单调算子一致凸BANACH空间 Reich不等式 Lyapunov functional maximal monotone operator uniformly convex Banach space Reich inequality

分类号 O177.91 [理学—基础数学]

引文网络
相关文献

参考文献13

1ROCKAFELAR R.Tyrrell Monotone operators and the proximal point algorithm[J].SIAM J.Control Optimization,1976,14:877-898.
2MARTINET B.Regularization d'inequations variationelles par approximations successive[J],Rev.Franc.Inform.Rech.Aper,1970,4:154-159.
3MARTINET B.Vetermination approehe point fix d'une application pseudo-eovtrackante[J].C.R.Acad.Sci.Paris,1972,274:163-165.
4KAMIMURA S,TAKAHASHI W.Weak and strong convergence of solutions to accretive operator inclusions and applications[J].Set-Vlalued Anal.,2000,8:361-374.
5KAMIMuRA s,TAKAHAsHI W.Strong convergence of a proximal-type algorithm in a Banach space[J].SIAM J.Optim.,2002,13:938-945
6何炳生,杨振华,廖立志.极大单调算子的一个新的近似邻近点算法[J].中国科学（A辑）,2002,32(11):1026-1032. 被引量：13
7MOSCO U.Perturbation of Variational Inequalities[M].Nonlinear Functional Analysis(Proc.Sympos.Pare Math.,Vol.ⅩⅤⅢ,Part 1,Chicago,Ⅲ.,1968),Amer.Math.Soc.,Providence,R.I.,1970,182-194.
8TAKAHASHI W.Nonlinear Functional Analysis.Fixed Point Theory and Its Applications[M].Yokohama Publishers.Yokohama.2000.
9PASCALI D,SBURLAN S.Nonlinear mappings of monotone type[M].Martinus Nijhoff Publishers,The Hague;Sijthoff & Noordhoff International Publishers.Alphen aan den Rijn,1978.
10REICH S.An iterative procedure for construcing zeros of accretive sets in Banach spaces[J].Nonlinear Anal.,1978.2:85-92.

二级参考文献12

1[1]Brezis H. Operateurs Maximaux Monotone et Semi-Groups de Contractions dans les Espaces de Hilbert.Amsterdam: North-Holland, 1973
2[2]Burachik R S, Iusem A N, Svaiter B F. Enlargement of monotone operators with applications to variational inequalities. Set-Valued Analysis, 1997, 5:159～180
3[3]Rockafellar R T. Monotone operators and the proximal point algorithm SIAM Journal on Control and Optimization, 1976, 14:877～898
4[4]Teboulle M. Convergence of proximal-like algorithms. SIAM Journal on Optimization, 1997, 7:1069～1083
5[5]Eckstein J. Approximate iterations in Bregman-function-based proximal algorithms. Mathematical Programming, 1998, 83:113～123
6[6]Chen G, Teboulle M. A proximal-based decomposition method for convex minimization problems. Mathematical Programming, 1994, 64:81～101
7[7]Han D R, He B S. A new accuracy criterion for approximate proximal point algorithms. J of Mathematical Analysis and Applications, 2001, 263:343～354
8[8]He B S. Inexact implicit methods for monotone general variational inequalities. Mathematical Programming,1999, 86:199～217
9[9]Eckstein J, Bertsekas D P. On the Douglas-Rachford splitting method and the proximal points algorithm for maximal monotone operators. Mathematical Programming, 1992, 55:293～318
10[10]Bertsekas D P, Tsitsiklis J N. Parallel and Distributed Computation, Numerical Methods. Englewood Cliffs:Prentice-Hall, 1989

共引文献12

1王玮,周海云,陈东青.关于N-APPA算法的注记[J].河北师范大学学报（自然科学版）,2005,29(1):18-20.
2曾六川.关于极大强单调算子的不精确邻近点算法的收敛性分析[J].数学物理学报（A辑）,2005,25(2):281-288.
3周叔子,孙佑兰.DC函数优化问题的非精确邻近点算法[J].经济数学,2005,22(3):312-316.
4蔡时连,范江华.自反Banach空间中极大单调集值映射变分不等式的解的存在性[J].北京建筑工程学院学报,2006,22(4):77-79.
5周叔子,孙佑兰.关于DC函数最小化邻近点算法的注记[J].应用数学学报,2007,30(2):377-381.
6唐国吉.求解极大单调算子零点的一个近似邻近点算法[J].广西科学,2007,14(4):371-373.
7孙巍,吴长亮,范江华.极大单调算子的一个投影近似邻近点算法[J].广西师范大学学报（自然科学版）,2008,26(2):37-40. 被引量：1
8唐国吉.求解单调变分不等式的一个近似邻近点算法[J].广西科学,2008,15(3):257-259.
9唐国吉,赵康生.混合变分不等式的一类迭代算法[J].广西科学,2008,15(4):371-373.
10唐国吉.求解单调变分不等式的近似邻近点算法的收敛性分析[J].纯粹数学与应用数学,2009,25(1):183-189. 被引量：2

同被引文献17

1魏利,周海云.Banach空间中极大单调算子零点的带误差项的新迭代格式[J].应用数学,2006,19(1):101-105. 被引量：13
2魏利,周海云.Banach空间中极大单调算子零点的迭代收敛定理及应用[J].数学的实践与认识,2006,36(5):235-242. 被引量：13
3魏利,周海云.Banach空间中有限个极大单调算子公共零点的迭代格式[J].系统科学与数学,2007,27(2):184-193. 被引量：4
4Rockafellar R T. Monotone operators and the proximal point algorithm. SIAM. J. Control and Optim., 1976, 14: 877-898.
5Mosco U. Perturbation of variational inequalities. Nonlinear Functional Analysis, Proc. Sympos. Pure Math., Chicago, IL, 1968.
6Wei Li. A new iterative algorithm with errors for maximal monotone operators and its applications. Proceeding of ICMLC 2005 Conference, Guangzhou, 2005, 969-975.
7Takahashi W. Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, 2000.
8Pascali D, Sburlan S. Nonlinear Mappings of Monotone Type. Sijthoff and Noordhoff International Publishers, Romania, 1978.
9Zhou H Y. Gao G L, Guo J T, Cho Y J. Some general convergence principles with applications[J]. Bull Korea Math Soc.2003,40(3):351-363.
10Kamimura S, Takahashi W. Approximating Solutions of maximal monotone operators in Hilbert spaces [J]. J Approx Theory,2000,106:226-240.