Banach空间的完全凸函数与逼近点算法
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摘要
Bregman优化方法是当前算法理论中重要的研究课题,它的出现促进了无穷维空间中迭代算法理论的快速发展.这种方法已广泛应用于最优化问题、变分不等式和非线性算子不动点的计算等各个方面.Bregman优化方法以凸分析、优化理论为基础,而且算法与这些理论相结合也促进了凸函数理论的深入研究与发展.在本论文中,我们研究了与Bregman优化方法有关的概念及其性质,以及以这些概念和性质为基础,并将这种方法应用于Banach空间中的某些类型的算子(具有Bregman单调型性质的算子)的优化算法.
完全凸函数是Bregman优化方法的基本概念,把函数的完全凸性应用于算法的设计与收敛分析中是这类算法的一个重要支点.我们考虑完全凸函数及其它函数类(一致凸函数和一致光滑函数)的性质,且在完全凸概念下考察了算子(特别是那些它们的预解算子是条件非扩张的且有Bregman单调型性质的算子及Bregman投影算子等)的性质,并将这些函数类和算子的性质用于设计Bregman优化算法以及讨论算法的收敛分析.
首先,我们使用完全凸函数研究集值映射的连续选择的存在性,以及在较弱意义下讨论了函数的完全凸性与一致凸性的等价性.我们给出函数的在有界集上的一致光滑性的特征.我们引入凸函数的局部一致完全凸的概念,讨论它的性质并将这些性质应用到在某些优化问题中使人感兴趣的算法中.我们引入局部一致完全凸Banach空间,并给出Banach空间的局部一致完全凸性的等价刻划.而且,优化问题中的某些集值算子和单值算子的零解及不动点的存在性、变分不等式问题的解的存在性也被讨论.
其次,我们在具有特定几何结构的Banach空间中使用函数‖·‖~2的几何性质研究Bregman优化算法.具体地,为在一致凸、一致光滑Banach空间X中找问题0∈T(ν)的解,这里T∶X(?)X~*是一个最大单调算子,我们研究了最大单调算子的逼近点算法的改进.在控制参数的适当假设下,我们讨论了算法的强或弱收敛性并估计了算法的收敛速度,以及这一算法对于凸优化问题的应用.另外,考虑有约束的混合型问题,即,在上述Banach空间X中找一个ν∈T~(1)0∩F~(-1)0∩C,这里T是集值的、最大单调的,C(?)X是非空闭凸子集,并且F∶X→X~*是单值的、逆单调的或它的预解F_α是强非扩张的,我们也研究了算子F的外梯度方法与算子T的逼近点算法的一个杂合方法.在某些关于算子和参数的适当的假设下,我们证明了这种杂合迭代具有弱收敛性,即,所构造的迭代序列弱收敛到上述交集中的一点.
In the present researches of algorithms,Bregman's optimization method is an important subject,which promotes the development of the theories of algorithms.The method has been exploited in each aspect such as in the researches of optimization algorithms and computing fixed points of nonlinear mappings,etc..Bregman's optimization method is based on the theories of convex analysis and optimization,and the combination of the method with these theories promotes the deeper researches for convex functions.In the paper,we studied the notions related to Bregman's optimization method and their properties,and in the context of these knowledge applied the method to the optimization algorithms of some types of operators(the operators with Bregman's monotonicity type properties)in Banach spaces.
The notion of totally convex functions is one of bases for Bregman's optimization method,and applying total convexity of functions in designing and analyzing iterative algorithms is a vital key to the class of methods.We considered the properties of totally convex functions and other classes of functions(uniformly convex functions and uniformly smooth functions),and investigated the properties of operators(the operators satisfying certain monotonicity type properties,resolvents of which have nonexpansivity type properties conditioned by Bregman functions,and Bregman projection operator,etc.)under the notion of total convexity.We applied the properties of these functions and operators to designing Bregman's optimization algorithms and discussing theirs convergence analyses.
First,by using total convexity of functions,we studied the existence of the continuous selection of set-valued maps,and discussed the equivalence of the total convexity and uniform convexity of functions in some weaker sense.We gave the characteristics of uniform smoothness on bounded sets of functions.We introduced the notion of locally uniformly total convexity of convex functions,discussed its properties and applied those to algorithms interesting in some optimization problems.We introduced locally uniformly totally convex Banach spaces,and gave the characteristics of locally uniform total convexity of Banach spaces.Also,the existence of zeros and fixed points of some multivalued operators and single-valued operators in optimization problems and the existence of solutions of variational inequality problems were discussed.
Second,in Banach spaces with a specific geometric structure,we used the properties of the function ||·||~2 to study Bregman optimization algorithms.In a uniformly convex and uniformly smooth Banach space X,for finding solutions to O∈T(u),where T:X(?)X~* is a maximal monotone operator,we studied a modified proximal point algorithm,discussed the strong or weak convergence of the algorithm and estimated the rate of convergence of the algorithm under different conditions on data,and applied the algorithm to a convex minimization problem.Additionally,considering a hybrid problem with a constraint set, i.e.,for finding a u∈T~(-1)O∩F~(-1)O∩C in the above Banach space X,where T is set-valued maximal monotone,C(?)X is a nonempty closed convex subset,and F is single-valued inverse monotone or its resolvent F_αis strongly nonexpansive,we studied a hybrid method of the extragradient method for F and the proximal point algorithm for T.We proved the weak convergence of the hybrid iterations under some assumptions on these operators and parameters,i.e.,the iteration sequences generated by converge weakly to a point of the above intersection.
完全凸函数是Bregman优化方法的基本概念,把函数的完全凸性应用于算法的设计与收敛分析中是这类算法的一个重要支点.我们考虑完全凸函数及其它函数类(一致凸函数和一致光滑函数)的性质,且在完全凸概念下考察了算子(特别是那些它们的预解算子是条件非扩张的且有Bregman单调型性质的算子及Bregman投影算子等)的性质,并将这些函数类和算子的性质用于设计Bregman优化算法以及讨论算法的收敛分析.
首先,我们使用完全凸函数研究集值映射的连续选择的存在性,以及在较弱意义下讨论了函数的完全凸性与一致凸性的等价性.我们给出函数的在有界集上的一致光滑性的特征.我们引入凸函数的局部一致完全凸的概念,讨论它的性质并将这些性质应用到在某些优化问题中使人感兴趣的算法中.我们引入局部一致完全凸Banach空间,并给出Banach空间的局部一致完全凸性的等价刻划.而且,优化问题中的某些集值算子和单值算子的零解及不动点的存在性、变分不等式问题的解的存在性也被讨论.
其次,我们在具有特定几何结构的Banach空间中使用函数‖·‖~2的几何性质研究Bregman优化算法.具体地,为在一致凸、一致光滑Banach空间X中找问题0∈T(ν)的解,这里T∶X(?)X~*是一个最大单调算子,我们研究了最大单调算子的逼近点算法的改进.在控制参数的适当假设下,我们讨论了算法的强或弱收敛性并估计了算法的收敛速度,以及这一算法对于凸优化问题的应用.另外,考虑有约束的混合型问题,即,在上述Banach空间X中找一个ν∈T~(1)0∩F~(-1)0∩C,这里T是集值的、最大单调的,C(?)X是非空闭凸子集,并且F∶X→X~*是单值的、逆单调的或它的预解F_α是强非扩张的,我们也研究了算子F的外梯度方法与算子T的逼近点算法的一个杂合方法.在某些关于算子和参数的适当的假设下,我们证明了这种杂合迭代具有弱收敛性,即,所构造的迭代序列弱收敛到上述交集中的一点.
In the present researches of algorithms,Bregman's optimization method is an important subject,which promotes the development of the theories of algorithms.The method has been exploited in each aspect such as in the researches of optimization algorithms and computing fixed points of nonlinear mappings,etc..Bregman's optimization method is based on the theories of convex analysis and optimization,and the combination of the method with these theories promotes the deeper researches for convex functions.In the paper,we studied the notions related to Bregman's optimization method and their properties,and in the context of these knowledge applied the method to the optimization algorithms of some types of operators(the operators with Bregman's monotonicity type properties)in Banach spaces.
The notion of totally convex functions is one of bases for Bregman's optimization method,and applying total convexity of functions in designing and analyzing iterative algorithms is a vital key to the class of methods.We considered the properties of totally convex functions and other classes of functions(uniformly convex functions and uniformly smooth functions),and investigated the properties of operators(the operators satisfying certain monotonicity type properties,resolvents of which have nonexpansivity type properties conditioned by Bregman functions,and Bregman projection operator,etc.)under the notion of total convexity.We applied the properties of these functions and operators to designing Bregman's optimization algorithms and discussing theirs convergence analyses.
First,by using total convexity of functions,we studied the existence of the continuous selection of set-valued maps,and discussed the equivalence of the total convexity and uniform convexity of functions in some weaker sense.We gave the characteristics of uniform smoothness on bounded sets of functions.We introduced the notion of locally uniformly total convexity of convex functions,discussed its properties and applied those to algorithms interesting in some optimization problems.We introduced locally uniformly totally convex Banach spaces,and gave the characteristics of locally uniform total convexity of Banach spaces.Also,the existence of zeros and fixed points of some multivalued operators and single-valued operators in optimization problems and the existence of solutions of variational inequality problems were discussed.
Second,in Banach spaces with a specific geometric structure,we used the properties of the function ||·||~2 to study Bregman optimization algorithms.In a uniformly convex and uniformly smooth Banach space X,for finding solutions to O∈T(u),where T:X(?)X~* is a maximal monotone operator,we studied a modified proximal point algorithm,discussed the strong or weak convergence of the algorithm and estimated the rate of convergence of the algorithm under different conditions on data,and applied the algorithm to a convex minimization problem.Additionally,considering a hybrid problem with a constraint set, i.e.,for finding a u∈T~(-1)O∩F~(-1)O∩C in the above Banach space X,where T is set-valued maximal monotone,C(?)X is a nonempty closed convex subset,and F is single-valued inverse monotone or its resolvent F_αis strongly nonexpansive,we studied a hybrid method of the extragradient method for F and the proximal point algorithm for T.We proved the weak convergence of the hybrid iterations under some assumptions on these operators and parameters,i.e.,the iteration sequences generated by converge weakly to a point of the above intersection.
引文
[1] Bregman L M. The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming[J]. USSR Computational Mathematics and Mathematical Physics, 1967, 7: 200-217.
[2] Alber Y I. Metric and generalized projection operators in Banach spaces: properties and applications[C]. In: Kartsatos A G, eds. Theory and Applications of Nonlinear Operators of Monotone and Accretive Type. New York: Marcel Dekker, New York, 1996. 178 of Lecture Notes in Pure and Applied Mathematics: 15-50.
[3] Alber Y I, Butnariu D. Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces [J]. J Optim Theory Appl, 1997, 92: 33-61.
[4] Bregman L M, Censor Y, Reich S. Dykstra's algorithm as the nonlinear extension of Breg-man's optimization method[J]. Journal of Convex Analysis, 1999, 6: 319-333.
[5] Butnariu D, Iusem A N, Zalinescu C. On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces[J]. Journal of Convex Analysis, 2003, 10: 35-61.
[6] Censor Y, Herman G T. Block-iterative algorithms with underrelaxed Bregman projections[J]. SIAM Journal on Optimizations, 2002, 13: 283-297.
[7] Alber Y I, Burachik R S, Iusem A N. A proximal point method for nonsmooth convex optimization problems in Banach spaces[J]. Abstract and Applied Analysis, 1997, 2(1-2): 97-120.
[8] Alber Y I, Iusem A N, Solodov M V. Minimization of nonsmooth convex functionals in Banach spaces[J]. Journal of Convex Analysis, 1997, 4: 235-255.
[9] Butnariu D, Iusem A N. On a proximal point method for convex optimization in Banach spaces[J]. Numer Funct Anal Optim, 1998, 18: 723-744.
[10] Drummond L, Iusem A. A projected gradient method for vector optimization problems[J]. Computational Optimization and Applications, 2004, 28(1): 5-29.
[11] Brack R E. An iterative solution of a variational inequality for certain monotone operators in Hilbert spaoes[J]. Bull Amer Math Soc, 1975, 81: 890-892.
[12] Burachik R S, Scheimberg S. A proximal point method for the variational inequality problem in Banach spaces[J]. SIAM Journal of Control and Optimization, 2000, 39: 1633-1649.
[13] Butnariu D, Iusem A N. Local moduli of convexity and their applications to finding almost common fixed points of mesurable families of operators[C]. In: Censor Y, Reich S, eds. "Recent Developments in Optimization and Nonlinear Analysis", Contemporary Mathematics. Rhode Island: American Mathematical Society, Providence, 1997. 204: 33-61.
[14] Butnariu D, Iusem A N. Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization[M]. Dordrecht, Holland: Kluwer Academic Publishers, 2000.
[15] Butnariu D, Resmerita E. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces[J]. Abstr Appl Anal, 2006, 84919. 39 pp.
[16] Iusem A N, Otero R G. Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces[J]. Numerical Functional Analysis and Optimization, 2001, 22: 609-640.
[17] Kamimura S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space[J]. SIAM Journal of Optimization, 2003, 13: 938-945.
[18] Reich S. Strong convergence theorems for resolvents of accretive operators in Banach spaces[J]. J Math Anal Appl, 1980, 75: 287-292.
[19] Butnariu D, Censor Y, Reich S. Iterative averaging of entropic projections for solving stochastic convex feasibility problems[J]. Comput Optim Appl, 1997, 8: 21-39.
[20] Bauschke H H, Borwein J M, Combettes P L. Bregman monotone optimization algorithms[J]. SIAM Journal on Control and Optimization, 2003, 42(2): 596-636.
[21] Censor Y, Zenios S. Parallel Optimization: Theory, Algorithms and Applications[M]. New York: Oxford University Press, 1997.
[22] Rockafellar R T. Maximal monotone operators and proximal point algorithm[J]. SIAM J Control Optim, 1976, 14: 877-898.
[23] Zalinescu C. Convex Analysis in General Vector Spaces[M]. River Edge, USA: World Scientific, 2002.
[24] Zalinescu C. On uniformly convex functions [J]. Journal of Mathematical Analysis and Applications, 1983, 95: 344-374.
[25] Cioranescu I. Geometry of Banach Spaces, Duality mappings and Nonlinear Problems[M]. Dordrecht, Holland: Kluwer Academic Publishers, 1990.
[26] Diestel J. Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92[M]. New York: Springer-Varlag, 1984.
[27] Barbu V, Precupanu Th. Convexity and Optimization in Banach Spaces[M]. Bucuresti, Romania: Romania International Publishers, 1978.
[28] Bauschke H H, Borwein J M, Combettes P L. Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces [J]. Communications in Contemporary Mathematics, 2001, 3(4): 615-647.
[29] Tan K K, Xu H K. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process[J]. Journal of Mathematical Analysis and Applications, 1993, 178: 301-308.
[30] Aubin J P, Frankowska J. Set Valued Analysis[M]. Boston: Birkhaser, 1990.
[31] Aubin J P. Optima and Equilibria[M]. Berlin Heidelberg: Springer-Verlag, 1993.
[32] Lim T C, Xu H K. Fixed point theorems for asymptotically nonexpansive mappings[J]. Nonlinear Analysis, 1994, 22(11): 1345-1355.
[33] Kamimura S, Takahashi W. Approximation solutions of maximal monotone operators in Hilbert spaces[J]. Journal of Approximation Theory, 2000, 106: 226-240.
[34] Kamimura S, Kohsaka F, Takahashi W. Weak and strong convergence theorem for maximal monotone operators in a Banach space[J]. Set-Valued Analysis, 2004, 12: 417-429.
[35] Kohsaka F, Takahashi W. Strong convergence of an iterative sequence for maximal monotone operators in a Banach space[J]. Abstract and Applied Analysis, 2004, 3: 239-249.
[36] Li L, Song W. Modified proximal point algorithm for maximal monotone operators in Banach spaces [J]. J Optim Theory Appl, 2008, 137(2).
[37] Otero R G, Svaiter B F. A strongly convergent hybrid proximal method in Banach spaces [J], J Math Anal Appl, 2004, 289: 700-711.
[38] Solodov M V, Svaiter B F. Forcing strong convergence of proximal point iterations in a Hilbert space[J]. Mathmatical Programming, 2000, 87: 189-202.
[39] Iusem A N, Pennanen T, Svaiter B F. Inexact variants of the proximal point algorithm without monotonicity[J]. SIAM Journal on Optimization, 2003, 13: 1080-1097.
[40] Guler O. On the convergence of the proxiaml point algorithm for convex minimization[J]. SIAM Journal of Control and Optimization, 1991, 29: 403-419.
[41] Xu H K. Iterative algorithms for nonlinear operators[J]. Journal of the London Mathematical Society, 2002, 66: 240-256.
[42] Kim T H, Xu H K. Strong convergence of modified Mann iterations[J]. Nonlinear Analysis, 2005, 61: 51-60.
[43] Takahashi W. Nonlinear Functional Analysis[M]. Yokohama, Japan: Yokohama Publishers, 2000.
[44] Diestel J. Geometry of Banach Spaces, Lecture Notes in Math[M]. New York: Springer-Verlag, New York, 1975.
[45] Xu H K. Inequalities in Banach spaces with applications[J]. Nonlinear Analysis, 1991, 16: 1127-1138.
[46] Liu L S. Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces[J]. Journal of Mathematical Analysis and Applications, 1995, 194: 114-125.
[47] Li L, Song W. A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces[J]. Nonlinear Analysis: Hybrid Systems, 2007, 1: 398-413.
[48] Iiduka H, Takahashi W. Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings[J]. Nonlinear Anal, 2005, 61: 341-350.
[49] Nadezhkina N, Takahashi W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings[J]. J Optim Theory Appl, 2006, 128: 191-201.
[50] Rockafellar R T. On the maximality of sums of nonlinear monotone operators[J]. Trans Amer Math Soc, 1970, 149: 75-88.
[51] Butnariu D, Iusem A N, Burachik R S. Iterative methods for solving stochastic convex feasibility problems and applications [J]. Comput Optim Appl, 2000, 15: 269-307.
[52] Butnariu D, Resmerita E. The outer Bregman projection method for stochastic feasibility problems in Banach spaces[C]. In: Butnariu D, Censor Y, Reich S, eds. Inherently Parallel Algorithms for Feasibility and Optimization and their Applications. Amsterdam: Elsevier, 2001.
[53] Eckstein J. Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming [J]. Math Oper Res, 1993, 18(1): 202-226.
[2] Alber Y I. Metric and generalized projection operators in Banach spaces: properties and applications[C]. In: Kartsatos A G, eds. Theory and Applications of Nonlinear Operators of Monotone and Accretive Type. New York: Marcel Dekker, New York, 1996. 178 of Lecture Notes in Pure and Applied Mathematics: 15-50.
[3] Alber Y I, Butnariu D. Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces [J]. J Optim Theory Appl, 1997, 92: 33-61.
[4] Bregman L M, Censor Y, Reich S. Dykstra's algorithm as the nonlinear extension of Breg-man's optimization method[J]. Journal of Convex Analysis, 1999, 6: 319-333.
[5] Butnariu D, Iusem A N, Zalinescu C. On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces[J]. Journal of Convex Analysis, 2003, 10: 35-61.
[6] Censor Y, Herman G T. Block-iterative algorithms with underrelaxed Bregman projections[J]. SIAM Journal on Optimizations, 2002, 13: 283-297.
[7] Alber Y I, Burachik R S, Iusem A N. A proximal point method for nonsmooth convex optimization problems in Banach spaces[J]. Abstract and Applied Analysis, 1997, 2(1-2): 97-120.
[8] Alber Y I, Iusem A N, Solodov M V. Minimization of nonsmooth convex functionals in Banach spaces[J]. Journal of Convex Analysis, 1997, 4: 235-255.
[9] Butnariu D, Iusem A N. On a proximal point method for convex optimization in Banach spaces[J]. Numer Funct Anal Optim, 1998, 18: 723-744.
[10] Drummond L, Iusem A. A projected gradient method for vector optimization problems[J]. Computational Optimization and Applications, 2004, 28(1): 5-29.
[11] Brack R E. An iterative solution of a variational inequality for certain monotone operators in Hilbert spaoes[J]. Bull Amer Math Soc, 1975, 81: 890-892.
[12] Burachik R S, Scheimberg S. A proximal point method for the variational inequality problem in Banach spaces[J]. SIAM Journal of Control and Optimization, 2000, 39: 1633-1649.
[13] Butnariu D, Iusem A N. Local moduli of convexity and their applications to finding almost common fixed points of mesurable families of operators[C]. In: Censor Y, Reich S, eds. "Recent Developments in Optimization and Nonlinear Analysis", Contemporary Mathematics. Rhode Island: American Mathematical Society, Providence, 1997. 204: 33-61.
[14] Butnariu D, Iusem A N. Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization[M]. Dordrecht, Holland: Kluwer Academic Publishers, 2000.
[15] Butnariu D, Resmerita E. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces[J]. Abstr Appl Anal, 2006, 84919. 39 pp.
[16] Iusem A N, Otero R G. Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces[J]. Numerical Functional Analysis and Optimization, 2001, 22: 609-640.
[17] Kamimura S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space[J]. SIAM Journal of Optimization, 2003, 13: 938-945.
[18] Reich S. Strong convergence theorems for resolvents of accretive operators in Banach spaces[J]. J Math Anal Appl, 1980, 75: 287-292.
[19] Butnariu D, Censor Y, Reich S. Iterative averaging of entropic projections for solving stochastic convex feasibility problems[J]. Comput Optim Appl, 1997, 8: 21-39.
[20] Bauschke H H, Borwein J M, Combettes P L. Bregman monotone optimization algorithms[J]. SIAM Journal on Control and Optimization, 2003, 42(2): 596-636.
[21] Censor Y, Zenios S. Parallel Optimization: Theory, Algorithms and Applications[M]. New York: Oxford University Press, 1997.
[22] Rockafellar R T. Maximal monotone operators and proximal point algorithm[J]. SIAM J Control Optim, 1976, 14: 877-898.
[23] Zalinescu C. Convex Analysis in General Vector Spaces[M]. River Edge, USA: World Scientific, 2002.
[24] Zalinescu C. On uniformly convex functions [J]. Journal of Mathematical Analysis and Applications, 1983, 95: 344-374.
[25] Cioranescu I. Geometry of Banach Spaces, Duality mappings and Nonlinear Problems[M]. Dordrecht, Holland: Kluwer Academic Publishers, 1990.
[26] Diestel J. Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92[M]. New York: Springer-Varlag, 1984.
[27] Barbu V, Precupanu Th. Convexity and Optimization in Banach Spaces[M]. Bucuresti, Romania: Romania International Publishers, 1978.
[28] Bauschke H H, Borwein J M, Combettes P L. Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces [J]. Communications in Contemporary Mathematics, 2001, 3(4): 615-647.
[29] Tan K K, Xu H K. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process[J]. Journal of Mathematical Analysis and Applications, 1993, 178: 301-308.
[30] Aubin J P, Frankowska J. Set Valued Analysis[M]. Boston: Birkhaser, 1990.
[31] Aubin J P. Optima and Equilibria[M]. Berlin Heidelberg: Springer-Verlag, 1993.
[32] Lim T C, Xu H K. Fixed point theorems for asymptotically nonexpansive mappings[J]. Nonlinear Analysis, 1994, 22(11): 1345-1355.
[33] Kamimura S, Takahashi W. Approximation solutions of maximal monotone operators in Hilbert spaces[J]. Journal of Approximation Theory, 2000, 106: 226-240.
[34] Kamimura S, Kohsaka F, Takahashi W. Weak and strong convergence theorem for maximal monotone operators in a Banach space[J]. Set-Valued Analysis, 2004, 12: 417-429.
[35] Kohsaka F, Takahashi W. Strong convergence of an iterative sequence for maximal monotone operators in a Banach space[J]. Abstract and Applied Analysis, 2004, 3: 239-249.
[36] Li L, Song W. Modified proximal point algorithm for maximal monotone operators in Banach spaces [J]. J Optim Theory Appl, 2008, 137(2).
[37] Otero R G, Svaiter B F. A strongly convergent hybrid proximal method in Banach spaces [J], J Math Anal Appl, 2004, 289: 700-711.
[38] Solodov M V, Svaiter B F. Forcing strong convergence of proximal point iterations in a Hilbert space[J]. Mathmatical Programming, 2000, 87: 189-202.
[39] Iusem A N, Pennanen T, Svaiter B F. Inexact variants of the proximal point algorithm without monotonicity[J]. SIAM Journal on Optimization, 2003, 13: 1080-1097.
[40] Guler O. On the convergence of the proxiaml point algorithm for convex minimization[J]. SIAM Journal of Control and Optimization, 1991, 29: 403-419.
[41] Xu H K. Iterative algorithms for nonlinear operators[J]. Journal of the London Mathematical Society, 2002, 66: 240-256.
[42] Kim T H, Xu H K. Strong convergence of modified Mann iterations[J]. Nonlinear Analysis, 2005, 61: 51-60.
[43] Takahashi W. Nonlinear Functional Analysis[M]. Yokohama, Japan: Yokohama Publishers, 2000.
[44] Diestel J. Geometry of Banach Spaces, Lecture Notes in Math[M]. New York: Springer-Verlag, New York, 1975.
[45] Xu H K. Inequalities in Banach spaces with applications[J]. Nonlinear Analysis, 1991, 16: 1127-1138.
[46] Liu L S. Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces[J]. Journal of Mathematical Analysis and Applications, 1995, 194: 114-125.
[47] Li L, Song W. A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces[J]. Nonlinear Analysis: Hybrid Systems, 2007, 1: 398-413.
[48] Iiduka H, Takahashi W. Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings[J]. Nonlinear Anal, 2005, 61: 341-350.
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