关于广义凸的分析
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摘要
E -凸集、E -凸函数、半E -凸函数和拟E -凸函数是凸集和凸函数的推广.这些集合和函数近年来被广泛应用于许多规划问题.
借助于这些集合和函数的定义思想,本文定义了E -凸锥、伪拟E -凸函数、E -凸函数方向导数等几种更广义的概念.首先讨论了E -凸集、E -凸锥的性质.随后,讨论了定义在实线性空间中的E -凸集上的实值E -凸函数、半E -凸函数、伪拟E -凸函数、E -凸函数方向导数的性质.这是第二章、第三章、第四章的内容.在第五章中,给出了E -次微分的存在性定理,在此基础上,本文研究了E -次微分共轭性,连续性,以及单调性.第六章中,讨论了E -凸规划问题.
E -convex set and E -convex function are the promotion of convex set and convex function. Those kinds of sets and functions are applied to many mathematical programming problems in rencent years.
In this paper, with the aid of idea of defining those sets and functions, the definitions of E -convex cone, pseudo-quasi- E -convex functions, E -direction derivative are discussed. Firstly, we consider the properties of E -convex set and E -convex cone.The properties of real E -convex function, semi- E -convex functions, pseudo-quasi- E -convex functions, E -direction derivative defined on a E -convex set in the real liner spaces are discussed. Those are prime in Chapter two, Chapter three and Chapter four.In Chapter five, the Existence theorem of E -direction derivative is given .Under the assumption, conjugation, continuity, monotonicity of E -Subdifferential are studied .In Chapter six , we discuss E - convex-programming problems.
借助于这些集合和函数的定义思想,本文定义了E -凸锥、伪拟E -凸函数、E -凸函数方向导数等几种更广义的概念.首先讨论了E -凸集、E -凸锥的性质.随后,讨论了定义在实线性空间中的E -凸集上的实值E -凸函数、半E -凸函数、伪拟E -凸函数、E -凸函数方向导数的性质.这是第二章、第三章、第四章的内容.在第五章中,给出了E -次微分的存在性定理,在此基础上,本文研究了E -次微分共轭性,连续性,以及单调性.第六章中,讨论了E -凸规划问题.
E -convex set and E -convex function are the promotion of convex set and convex function. Those kinds of sets and functions are applied to many mathematical programming problems in rencent years.
In this paper, with the aid of idea of defining those sets and functions, the definitions of E -convex cone, pseudo-quasi- E -convex functions, E -direction derivative are discussed. Firstly, we consider the properties of E -convex set and E -convex cone.The properties of real E -convex function, semi- E -convex functions, pseudo-quasi- E -convex functions, E -direction derivative defined on a E -convex set in the real liner spaces are discussed. Those are prime in Chapter two, Chapter three and Chapter four.In Chapter five, the Existence theorem of E -direction derivative is given .Under the assumption, conjugation, continuity, monotonicity of E -Subdifferential are studied .In Chapter six , we discuss E - convex-programming problems.
引文
[1] Youness E A.E-convex sets,E-convex functions,and E-convex programming[J]. Journal of Optimization Theory and Application,1999,(102):439-450.
[2] Yang X.M.,Teo Kok lay,Yang Xiao Qi. A Characterization of Convex Function [J].Applied Mathematics Letter,2000,13:27-30
[3]Chen X S. Some propertises of semi- E-convex functions[J].J Math Appl,2002,(275):251-262
[4] Bector,C.R, and Singh,C.,B-vex Functions,Journal of Optimization Theory and Application,Vol:71,pp,237-253,1991.
[5] Hanson,M.A.,and Mond, B.,Convex Transformable Problems and Invexity,Journal of information and Optimization Science,Vol. 8,pp.201-207,1987.
[6] Kaul,R.N.,and Kaur,S.,Optimality Criteria in Nonlinear Programming Involving nonconvex Functions,Journal of Mathematical Analysis and Applications,Vol. 105, pp.104-112,1985.
[7] Mangasarian,O.L., Nonlinear Programming, Mcgraw-Hill,New York,1979.
[8] Yang X.M.,Teo Kok lay,Yang Xiao Qi. A Characterization of Convex Function [J].Applied Mathematics Letter,2000,13:27-30
[9]胡毓达.凸分析与非光滑分析.上海科学技术出版社,2000
[10] Hafnson M A. On Sufficiency of the Kuhn-Tucker Conditions[J].J Math Anal ppl. 1988, 80
[11] CLARKE H,LEDYAEV Y S,STEM R J,et al. Nonsmooth analysis and control theory[M]. New York: Springer-Verlag,1998.
[12]IOFFE A D.Approximate subdifferentials and applications.I: the finite dimensional theory[J].Transactions of the American Mathematical society, 1984,281(2):389-416.
[13]IOFFE A D.Proximal analysis and approximate subdifferentials[J].Jounal of London Math Socity,1990,41(1):175-192.
[14]DVILLE R,HADDAD E M E. The subdifferential of the sum of two functions in banach spaces[J].I First Order Case Convex Analysis,1996,3(3):295-308.
[15]FABIAN M.Gateaux differentiability of convex functions and topology weak asplund spaces [M].New York:J Wiley and Sons,Intersience,1997.
[16]IOFFE A D.Nonsmooth analysis: differential calculus of nondifferentiable mappings[J].Transactions of the American Mathematical Society, 1981, 266(1): 1-56.
[17]Clarke F H.Necessary conditions for nonsmooth problems in optimal control and the calculus of variationa. Ph. D thesis. Univ. of Washington,1973
[18]Jan Van Tiel. Convex Analysis,An Introductory Text,John Wiley &Sons,1984;
[19]TOINT P. A nonmonotone trust region algorithm for nonlinear optimization subject to convex constraints. Math Prog, 1997, 77(1):69-94.
[20]NOCEDAL J, YUAN Y. Combining trust region and linear search techniques// Yuan Y. Advances in nonlinear programming. Dordrecht: Kliwer Academic Publishers. 1998:153-175.
[21]Dimitri P.Bertsekas with Angelia Nedic and Asuman E.Ozdaglar.Convex Analysia and Optimization. Tsinghua University press.2006
[22]ZHANG H C, HAGER W W.A nonmonotone line search technique and its application to unconstrained optimization. SIAM J Optim.2004, 14(4): 1043-1056.
[23]莫降涛,颜世翠,刘春燕.无约束优化中带线搜索的非单调信赖域算法.广西科学. 2006 ,13(2):96-101,108.
[24]高成修,王芳华.约束优化的曲线搜索信赖域算法及其全局收敛性.数学杂志.1999, 19(2):223-236.
[25]胡毓达.实用多目标最优化.上海科学技术出版社.
[26]高峰,侯亚军.关于多目标规划信赖域算法收敛的判别准则研究.沈阳工业学院学报.1997,16(3):51-56.
[27]高峰,侯亚军.一类多目标规划问题的信赖域算法及全局收敛性.沈阳工业学院学报.1998,17(1):93-101.
[28]习会,施保昌.无约束多目标规划的信赖域方法.应用数学学报.2000,13(3):67-69.
[29]姚升保,彭叶辉,施保昌.无约束多目标规划的非单调信赖域算法.运筹与管理.2002, 11(4):52-55.
[30]姚升保,施保昌,彭叶辉.线性约束多目标规划的非单调信赖域算法.应用数学学报.2002 ,15(增) :55-59.
[31]彭叶辉,施保昌,姚升保.线性约束多目标规划的非单调信赖域算法.华中科技大学学报.2003,31(7):113-114.
[32]徐成贤.近代优化方法.科学出版社. 2002
[33]王建勇.E-拟凸函数[J].聊城大学学报,2003,16(3):34-35.
[34]杨新民.上半连续函数的拟凸性[J].运筹学学报,1999,3(1):48-51
[35] Noikodem K. On some Class of Midconvex Functions[J].Ann.Pol. Math,1989, 50:145-151.
[36]史树中.凸分析.上海:上海科学技术出版社,1990
[37]袁亚湘,孙文瑜.最优化理论与方法.科学出版社,1997.
[38]徐成贤,陈志平,李乃成.近代优化方法.科学出版社,2001.
[39] YUAN Y. Conditions for convergence of trust region algorithms for nonsmooth optimization. Math Prog, 1985, 31:220-228.
[40] YUAN Y. An example of only linearly convergence of trust region algorithms for nonsmooth optimization.IMA j Numer Anal, 1984, 47:327-335.
[41] VARDI A. A trust region algorithm for equality constrained minimization: convergence properties and implementation. SIAM J Numer Anal.1985, 22(3):575-591.
[42] BYRD R H, SCHNABEL R B, SHULTZ A.A trust region algorithm for nonlinearly constrained optimization. SIAM J Numer Anal, 1987, 24(3):1153-1170.
[43]李成林.E凸函数的次微分[J].云南大学学报:自然科学版,2006,28(5):369-37
[2] Yang X.M.,Teo Kok lay,Yang Xiao Qi. A Characterization of Convex Function [J].Applied Mathematics Letter,2000,13:27-30
[3]Chen X S. Some propertises of semi- E-convex functions[J].J Math Appl,2002,(275):251-262
[4] Bector,C.R, and Singh,C.,B-vex Functions,Journal of Optimization Theory and Application,Vol:71,pp,237-253,1991.
[5] Hanson,M.A.,and Mond, B.,Convex Transformable Problems and Invexity,Journal of information and Optimization Science,Vol. 8,pp.201-207,1987.
[6] Kaul,R.N.,and Kaur,S.,Optimality Criteria in Nonlinear Programming Involving nonconvex Functions,Journal of Mathematical Analysis and Applications,Vol. 105, pp.104-112,1985.
[7] Mangasarian,O.L., Nonlinear Programming, Mcgraw-Hill,New York,1979.
[8] Yang X.M.,Teo Kok lay,Yang Xiao Qi. A Characterization of Convex Function [J].Applied Mathematics Letter,2000,13:27-30
[9]胡毓达.凸分析与非光滑分析.上海科学技术出版社,2000
[10] Hafnson M A. On Sufficiency of the Kuhn-Tucker Conditions[J].J Math Anal ppl. 1988, 80
[11] CLARKE H,LEDYAEV Y S,STEM R J,et al. Nonsmooth analysis and control theory[M]. New York: Springer-Verlag,1998.
[12]IOFFE A D.Approximate subdifferentials and applications.I: the finite dimensional theory[J].Transactions of the American Mathematical society, 1984,281(2):389-416.
[13]IOFFE A D.Proximal analysis and approximate subdifferentials[J].Jounal of London Math Socity,1990,41(1):175-192.
[14]DVILLE R,HADDAD E M E. The subdifferential of the sum of two functions in banach spaces[J].I First Order Case Convex Analysis,1996,3(3):295-308.
[15]FABIAN M.Gateaux differentiability of convex functions and topology weak asplund spaces [M].New York:J Wiley and Sons,Intersience,1997.
[16]IOFFE A D.Nonsmooth analysis: differential calculus of nondifferentiable mappings[J].Transactions of the American Mathematical Society, 1981, 266(1): 1-56.
[17]Clarke F H.Necessary conditions for nonsmooth problems in optimal control and the calculus of variationa. Ph. D thesis. Univ. of Washington,1973
[18]Jan Van Tiel. Convex Analysis,An Introductory Text,John Wiley &Sons,1984;
[19]TOINT P. A nonmonotone trust region algorithm for nonlinear optimization subject to convex constraints. Math Prog, 1997, 77(1):69-94.
[20]NOCEDAL J, YUAN Y. Combining trust region and linear search techniques// Yuan Y. Advances in nonlinear programming. Dordrecht: Kliwer Academic Publishers. 1998:153-175.
[21]Dimitri P.Bertsekas with Angelia Nedic and Asuman E.Ozdaglar.Convex Analysia and Optimization. Tsinghua University press.2006
[22]ZHANG H C, HAGER W W.A nonmonotone line search technique and its application to unconstrained optimization. SIAM J Optim.2004, 14(4): 1043-1056.
[23]莫降涛,颜世翠,刘春燕.无约束优化中带线搜索的非单调信赖域算法.广西科学. 2006 ,13(2):96-101,108.
[24]高成修,王芳华.约束优化的曲线搜索信赖域算法及其全局收敛性.数学杂志.1999, 19(2):223-236.
[25]胡毓达.实用多目标最优化.上海科学技术出版社.
[26]高峰,侯亚军.关于多目标规划信赖域算法收敛的判别准则研究.沈阳工业学院学报.1997,16(3):51-56.
[27]高峰,侯亚军.一类多目标规划问题的信赖域算法及全局收敛性.沈阳工业学院学报.1998,17(1):93-101.
[28]习会,施保昌.无约束多目标规划的信赖域方法.应用数学学报.2000,13(3):67-69.
[29]姚升保,彭叶辉,施保昌.无约束多目标规划的非单调信赖域算法.运筹与管理.2002, 11(4):52-55.
[30]姚升保,施保昌,彭叶辉.线性约束多目标规划的非单调信赖域算法.应用数学学报.2002 ,15(增) :55-59.
[31]彭叶辉,施保昌,姚升保.线性约束多目标规划的非单调信赖域算法.华中科技大学学报.2003,31(7):113-114.
[32]徐成贤.近代优化方法.科学出版社. 2002
[33]王建勇.E-拟凸函数[J].聊城大学学报,2003,16(3):34-35.
[34]杨新民.上半连续函数的拟凸性[J].运筹学学报,1999,3(1):48-51
[35] Noikodem K. On some Class of Midconvex Functions[J].Ann.Pol. Math,1989, 50:145-151.
[36]史树中.凸分析.上海:上海科学技术出版社,1990
[37]袁亚湘,孙文瑜.最优化理论与方法.科学出版社,1997.
[38]徐成贤,陈志平,李乃成.近代优化方法.科学出版社,2001.
[39] YUAN Y. Conditions for convergence of trust region algorithms for nonsmooth optimization. Math Prog, 1985, 31:220-228.
[40] YUAN Y. An example of only linearly convergence of trust region algorithms for nonsmooth optimization.IMA j Numer Anal, 1984, 47:327-335.
[41] VARDI A. A trust region algorithm for equality constrained minimization: convergence properties and implementation. SIAM J Numer Anal.1985, 22(3):575-591.
[42] BYRD R H, SCHNABEL R B, SHULTZ A.A trust region algorithm for nonlinearly constrained optimization. SIAM J Numer Anal, 1987, 24(3):1153-1170.
[43]李成林.E凸函数的次微分[J].云南大学学报:自然科学版,2006,28(5):369-37