集值优化问题Benson真有效解的最优性条件
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摘要
本文在赋范线性空间中讨论了集值优化问题的Benson真有效性。在近似锥次类凸等条件下,利用集值映射的切导数与凸集分离定理等得到了集值优化问题关于Benson真有效元的Fritz-John型最优性必要条件。然后,在Slater型约束规格下得到了集值优化问题Benson真有效元的Kuhn-Tucker型最优性必要条件;同时,我们给出了一个Benson真有效元的充分条件等。最后,我们讨论了集值优化问题ε-Benson真有效元的导数型最优性必要条件与充分条件。具体内容如下:
本文包括四章。
第一章对论文的背景作了简要的介绍。
第二章是预备知识,介绍了锥、锥凸集值函数、切锥与集值映射的切导数等的相关知识。
第三章讨论了集值优化问题Benson真有效解的最优性条件。
第四章研究了集值优化问题ε-Benson真有效解的最优性条件。
In this paper, the Benson proper efficiency for set-valued optimization problem is discussed in normed linear space. Under the assumption of nearly cone-subconvexlikeness, by using the tangent derivatives of set-valued mapping and the separation theory for convex sets and so on, The Fritz-John type necessary conditions for set-valued optimization problem in the sense of Benson proper efficient elements is obtained. And then, under the Slater type constraint qualification, the Kuhn-tucker type necessary conditions for set-valued optimization problem in the sense of Benson proper efficient elements is established; at the same time, we gain a sufficient conditions for Benson proper efficient elements. Finally, we discuss the derivative type necessary and sufficient conditions for set-valued optimization problem in the sense ofε-Benson proper efficient elements. The main points of this paper is as followes:
This paper includes four parts.
The first chapter is an introduction; the background of this paper is introduced.
In chapter 2, we introduce the knowledge that the paper use, introducing these conception of cone , cone convex set-valued maps , tangent cones , and tangent derivatives of set-valued mapping, etc.
In chapter 3, we discuss the optimality conditions of set-valued optimization problem in the sense of Benson proper efficient solution.
In chapter 4, we study the optimization conditions ofε- Benson proper efficient solution of set-valued optimization problem.
本文包括四章。
第一章对论文的背景作了简要的介绍。
第二章是预备知识,介绍了锥、锥凸集值函数、切锥与集值映射的切导数等的相关知识。
第三章讨论了集值优化问题Benson真有效解的最优性条件。
第四章研究了集值优化问题ε-Benson真有效解的最优性条件。
In this paper, the Benson proper efficiency for set-valued optimization problem is discussed in normed linear space. Under the assumption of nearly cone-subconvexlikeness, by using the tangent derivatives of set-valued mapping and the separation theory for convex sets and so on, The Fritz-John type necessary conditions for set-valued optimization problem in the sense of Benson proper efficient elements is obtained. And then, under the Slater type constraint qualification, the Kuhn-tucker type necessary conditions for set-valued optimization problem in the sense of Benson proper efficient elements is established; at the same time, we gain a sufficient conditions for Benson proper efficient elements. Finally, we discuss the derivative type necessary and sufficient conditions for set-valued optimization problem in the sense ofε-Benson proper efficient elements. The main points of this paper is as followes:
This paper includes four parts.
The first chapter is an introduction; the background of this paper is introduced.
In chapter 2, we introduce the knowledge that the paper use, introducing these conception of cone , cone convex set-valued maps , tangent cones , and tangent derivatives of set-valued mapping, etc.
In chapter 3, we discuss the optimality conditions of set-valued optimization problem in the sense of Benson proper efficient solution.
In chapter 4, we study the optimization conditions ofε- Benson proper efficient solution of set-valued optimization problem.
引文
[1]王其林、刘军、张建,择一定理及向量优化问题的Benson真有效性[J].重庆交通学院学报,2006,25(4):148-150.
[2]王其林、吴云,集值优化问题的ε-严有效解的最优性条件[J].西南师范大学学报,2006,31(4):40-43.
[3]王政伟、李泽民,广义锥次似凸集值映射向量优化的Benson真有效性[J].工程数学学报,2003,20(6):14-20.
[4]王明征、夏尊铨,带有广义次类凸型集值映射的集值优化中的数乘与ε-对偶[J].应用数学学报,2004,27(4):632-637.
[5]戎卫东,马毅,集值映射向量最优化问题的ε-真有效解[J].运筹学学报,2000,4(4):21-32.
[6]旷华武,集值优化问题的Benson真有效解的广义导数型最优性条件[J].应用数学学报,2006,29(5):778-788.
[7]旷华武,Benson真有效解意义下集值优化问题的最优性条件[J].运筹学学报,2006,10(4):106-114.
[8]旷华武,集值优化问题的Benson真有效解的广义最优性条件[J].高校应用数学学报,2004,19(2):233-240.
[9]李仲飞,集值映射向量优化的Benson真有效性[J].应用数学学报,1998,21(1):123-134。
[10]张从军,周光辉,关于切锥与切导数的等价命题[J],纯粹数学与应用数学,2004,20(1):14-17.
[11]胡毓达,多目标规划有效性理论[M].上海:上海科学技术出版社,1994:111-256。
[12]胡毓达、孟志青,凸分析与非光滑分析[M].上海:上海科学技术出版社,2000.
[13]侯震梅、刘三阳、周勇,超有效意义下集值优化的最优性条件[J].数学物理学报,2007,27(1):131-137.
[14]侯震梅,集值优化最优性条件与稳定性问题的研究[D].西安电子科技大学,博士学位论文,2005.
[15]徐义红,集值优化问题强有效解的Kuhn Tucker最优性条件[J].数学研究与评论,2006,26(2):354-360.
[16]徐义红,朱传喜,集值优化问题超有效解的Lagrange最优性条件[J].应用泛函分析学报,2005,7(4):339-343.
[17]徐义红,集值优化问题的最优性条件[D].西安电子科技大学,博士学位论文,2003.
[18]盛宝怀、刘三阳,Benson真有效意义下向量集值优化的广义Fritz John条件[J].应用数学和力学,2002,23(12):1289-1295.
[19]盛宝怀、刘三阳,用广义梯度刻画集值优化Benson真有效解[J].应用数学学报,2002,25(1):22-28.
[20]盛宝怀、刘三阳,Benson真有效意义下的集值优化的广义最优性条件[J].数学学报,2003,46(3):611-620.
[21]彭建文,集值分析和集值最优化[J].重庆师范学院学报,2003,20(3):5-7.
[22]Aubin J P.,Frankowska H.,Set-valued Analysis[M].Boston:Birkhauser,1990,38-197.
[23]Aubin JP.,Contingent Derivatives of Set-valued Maps and Existence of Solutions to Nonlinear and Differential Inclusion[J].Mathematical Analysis and Applications,1981,160-229.
[24]Benson X.B.An Improved Definition of Proper Efficiency for Vector Minimization with Respect to Cones[J].Mathematical Analysis and Applications,1979,71(1):232-241.
[25]Chen Guangya,Jahn J,Optimality Conditions for Set-valued Optimization Problems[J].Mathematical Methods of Operations Research,1998,48(2):187-200.
[26]Chen Ling,ε-Super Efficient Solutions of Vector Optimization Problems with Set-valued Maps[J].Or Tranctions,2001,5(3):51-56.
[27]Corley HW.,Optimality Conditions for Maximization of Set-valued Functions[J].Journal of Optimization Theory and Applications,1988,58(1):1-10.
[28]G.Y.Chen and W.D.Rong,Characterizations of the Benson Proper Efficiency for Nonconvex Vector Optimization[J].Journal of Optimization Theory and Applications,1998,1(2):365-384.
[29]Jahn J.,Rauh R,Contingent Epiderivatives and Set-valued Optimization[J].Mathematical Methods of Operations Research,1997,46(2):193-211.
[30]Johannes Jahn and Akhtar A.khan,Generalized Contingent Epiderivatives in Set-valued Optimization: Optimality conditions[J]. Numerical Functional Analysis and Optimization, 2002, 23(7-8): 807-831.
[31]Li zemin, The Optimality Conditions for Vector Optimization of Set-valued Maps[J]. Journal of Mathematical Analysis and Applications, 1999, 237(2): 413-424.
[32]Li Zhongfei. Benson proper efficiency in the vector optimization of set-valued maps[J]. Journal of Mathematical Analysis and Applications , 1998, 98(3): 623-649.
[33]Li Hong-tao, Li Cai-ping, The generalized Optimality condition of vector optimization under Benson proper efficiency[J]. Pure and Applied Mathematics, 2001, 17(3):271-278.
[34]Luis Rodriguez-Marin, Miguel Sama, About Contingent Epiderivatives[J]. Journal of Mathematical Analysis and Applications, 2007 327(2):745-762.
[35]Qiu-sheng Qiu, Hening Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions[J]. Acta Mathematice Applicatae Sinica English Series, 2007,23(2):319-328.
[36]Wei Liu, Xuhua Gong, Proper Efficiency for Set-valued Vector Optimization Problems and Vector Variational Inequalities[J]. Mathematical Methods of Operations Research ,2000, 51:443-457.
[37]X. M. Yang, D.Li, AND S. Y. Wang, Near-subconvexlikeness in Vector Optimization with Set-valued Functions[J]. Journal Of Optimization Theory and Applications, 2001, 110 (2): 413-427.
[38]Xu Yihong, Liu Sanyang, Benson proper efficiency in the nearly cone-subconvexlike vector optimization with set-value functions[J]. Applied Mathematics A Journal of Chinese Universities ,2003,18(1):95-102.
[2]王其林、吴云,集值优化问题的ε-严有效解的最优性条件[J].西南师范大学学报,2006,31(4):40-43.
[3]王政伟、李泽民,广义锥次似凸集值映射向量优化的Benson真有效性[J].工程数学学报,2003,20(6):14-20.
[4]王明征、夏尊铨,带有广义次类凸型集值映射的集值优化中的数乘与ε-对偶[J].应用数学学报,2004,27(4):632-637.
[5]戎卫东,马毅,集值映射向量最优化问题的ε-真有效解[J].运筹学学报,2000,4(4):21-32.
[6]旷华武,集值优化问题的Benson真有效解的广义导数型最优性条件[J].应用数学学报,2006,29(5):778-788.
[7]旷华武,Benson真有效解意义下集值优化问题的最优性条件[J].运筹学学报,2006,10(4):106-114.
[8]旷华武,集值优化问题的Benson真有效解的广义最优性条件[J].高校应用数学学报,2004,19(2):233-240.
[9]李仲飞,集值映射向量优化的Benson真有效性[J].应用数学学报,1998,21(1):123-134。
[10]张从军,周光辉,关于切锥与切导数的等价命题[J],纯粹数学与应用数学,2004,20(1):14-17.
[11]胡毓达,多目标规划有效性理论[M].上海:上海科学技术出版社,1994:111-256。
[12]胡毓达、孟志青,凸分析与非光滑分析[M].上海:上海科学技术出版社,2000.
[13]侯震梅、刘三阳、周勇,超有效意义下集值优化的最优性条件[J].数学物理学报,2007,27(1):131-137.
[14]侯震梅,集值优化最优性条件与稳定性问题的研究[D].西安电子科技大学,博士学位论文,2005.
[15]徐义红,集值优化问题强有效解的Kuhn Tucker最优性条件[J].数学研究与评论,2006,26(2):354-360.
[16]徐义红,朱传喜,集值优化问题超有效解的Lagrange最优性条件[J].应用泛函分析学报,2005,7(4):339-343.
[17]徐义红,集值优化问题的最优性条件[D].西安电子科技大学,博士学位论文,2003.
[18]盛宝怀、刘三阳,Benson真有效意义下向量集值优化的广义Fritz John条件[J].应用数学和力学,2002,23(12):1289-1295.
[19]盛宝怀、刘三阳,用广义梯度刻画集值优化Benson真有效解[J].应用数学学报,2002,25(1):22-28.
[20]盛宝怀、刘三阳,Benson真有效意义下的集值优化的广义最优性条件[J].数学学报,2003,46(3):611-620.
[21]彭建文,集值分析和集值最优化[J].重庆师范学院学报,2003,20(3):5-7.
[22]Aubin J P.,Frankowska H.,Set-valued Analysis[M].Boston:Birkhauser,1990,38-197.
[23]Aubin JP.,Contingent Derivatives of Set-valued Maps and Existence of Solutions to Nonlinear and Differential Inclusion[J].Mathematical Analysis and Applications,1981,160-229.
[24]Benson X.B.An Improved Definition of Proper Efficiency for Vector Minimization with Respect to Cones[J].Mathematical Analysis and Applications,1979,71(1):232-241.
[25]Chen Guangya,Jahn J,Optimality Conditions for Set-valued Optimization Problems[J].Mathematical Methods of Operations Research,1998,48(2):187-200.
[26]Chen Ling,ε-Super Efficient Solutions of Vector Optimization Problems with Set-valued Maps[J].Or Tranctions,2001,5(3):51-56.
[27]Corley HW.,Optimality Conditions for Maximization of Set-valued Functions[J].Journal of Optimization Theory and Applications,1988,58(1):1-10.
[28]G.Y.Chen and W.D.Rong,Characterizations of the Benson Proper Efficiency for Nonconvex Vector Optimization[J].Journal of Optimization Theory and Applications,1998,1(2):365-384.
[29]Jahn J.,Rauh R,Contingent Epiderivatives and Set-valued Optimization[J].Mathematical Methods of Operations Research,1997,46(2):193-211.
[30]Johannes Jahn and Akhtar A.khan,Generalized Contingent Epiderivatives in Set-valued Optimization: Optimality conditions[J]. Numerical Functional Analysis and Optimization, 2002, 23(7-8): 807-831.
[31]Li zemin, The Optimality Conditions for Vector Optimization of Set-valued Maps[J]. Journal of Mathematical Analysis and Applications, 1999, 237(2): 413-424.
[32]Li Zhongfei. Benson proper efficiency in the vector optimization of set-valued maps[J]. Journal of Mathematical Analysis and Applications , 1998, 98(3): 623-649.
[33]Li Hong-tao, Li Cai-ping, The generalized Optimality condition of vector optimization under Benson proper efficiency[J]. Pure and Applied Mathematics, 2001, 17(3):271-278.
[34]Luis Rodriguez-Marin, Miguel Sama, About Contingent Epiderivatives[J]. Journal of Mathematical Analysis and Applications, 2007 327(2):745-762.
[35]Qiu-sheng Qiu, Hening Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions[J]. Acta Mathematice Applicatae Sinica English Series, 2007,23(2):319-328.
[36]Wei Liu, Xuhua Gong, Proper Efficiency for Set-valued Vector Optimization Problems and Vector Variational Inequalities[J]. Mathematical Methods of Operations Research ,2000, 51:443-457.
[37]X. M. Yang, D.Li, AND S. Y. Wang, Near-subconvexlikeness in Vector Optimization with Set-valued Functions[J]. Journal Of Optimization Theory and Applications, 2001, 110 (2): 413-427.
[38]Xu Yihong, Liu Sanyang, Benson proper efficiency in the nearly cone-subconvexlike vector optimization with set-value functions[J]. Applied Mathematics A Journal of Chinese Universities ,2003,18(1):95-102.