向量极值问题的最优性条件及二次规划问题的一种新算法
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摘要
本文主要讨论了抽象空间中向量最优化问题的一些理论，以及求解一般二次规划问题的一种新算法。文章分别在线性空间和线性拓扑空间中给出次似凸向量值映射的定义，在线性空间中给出(u，O_2，Y_+)-广义凸的定义，证明了这些凸性假设下相应的择一定理，然后应用择一定理分别给出了线性空间中两类广义凸规划的最优性条件，在线性拓扑空间中给出了无约束规划问题的最优性条件(结合G-导数)及Lagrange对偶定理。最后文章提出了含不等式约束的二次规划问题的一种新算法，该算法结合等式约束二次规划的降维算法，保证每次迭代点都是可行点，通过数值实验结果与精确解的比较表明了算法的有效性，整个算法程序用C++语言编制，并在微机上运行通过。
In this thesis, some topics on vector optimization theory in abstract spaces are discussed, and a new algorithm for the general quadratic programming is studied as well. The concept of cone-subconvexlike map is defined in linear spaces and in linear topological spaces. The concept of (u, O?; Y+)-generalized convex is defined in linear spaces. The corresponded alternative theorems are proved with those generalized convexity conditions. And then, by applying the alternative theorems, the optimality conditions of two kinds of generalized convex extremum problems are given in linear spaces. The optimality conditions (with G-differential) and Lagrange duality theorems of non-constrained programming are presented in linear topological spaces. Finally, a new algorithm for quadratic programming with inequality constraints is offered The new method, by applying the dimension-descending algorithm with equality constraints, assures that every iterative element is feasible. It is shown that the algorithm is efficient being compared with the results of numerical tests and the accurate solutions. The whole program is designed with C++ language and runs passed on the microcomputer.
In this thesis, some topics on vector optimization theory in abstract spaces are discussed, and a new algorithm for the general quadratic programming is studied as well. The concept of cone-subconvexlike map is defined in linear spaces and in linear topological spaces. The concept of (u, O?; Y+)-generalized convex is defined in linear spaces. The corresponded alternative theorems are proved with those generalized convexity conditions. And then, by applying the alternative theorems, the optimality conditions of two kinds of generalized convex extremum problems are given in linear spaces. The optimality conditions (with G-differential) and Lagrange duality theorems of non-constrained programming are presented in linear topological spaces. Finally, a new algorithm for quadratic programming with inequality constraints is offered The new method, by applying the dimension-descending algorithm with equality constraints, assures that every iterative element is feasible. It is shown that the algorithm is efficient being compared with the results of numerical tests and the accurate solutions. The whole program is designed with C++ language and runs passed on the microcomputer.
引文
[1] M. S. 巴扎拉，C. M. 希蒂著，王化春，张春柏译，非线性规划——理论与算法，贵州人民出版社，1986．
[2] 胡毓达，非线性规划，高等教育出版社，1990．
[3] 赵凤治，尉继英，约束最优化计算方法，科学出版社，1991．
[4] 李泽民，线性拓扑空间中向量极值问题的广义Kuhn-Tucker条件，系统科学与数学，Vol．10，No．1，78—83，1990．
[5] A. J. V. Brandao, MA. Rojas-Medar, and G.N. Silva, Optimality Conditions for Pareto Nonsmooth Nonconvex Programmming in Vanach Spaces, JOTA, Vol. 103, No. 1, 65-73, 1999.
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[7] Li Zemin, The Optimality Conditions for Vector Optimization of Set-Valued Maps, JMAA, Vol. 237, 413-424, 1999.
[8] Li Zemin A theorem of the Alternative and Its Application to the Optimization of Set-Valued Maps, JOTA, Vol. 100, No.2, 365-375, 1999.
[9] T. illes and Kassay, Theorems of the Alternative and Optimality Conditions for Convexlike and General Convexlike Programming, JOTA, Vol. 101, No.2, 243-257, 1999.
[10] 袁亚湘，孙文瑜，最优化理论与方法，科学出版社, 1997.
[11] Li Zemin, The Optimality Conditions of Differentiable Vector Optimization Problems, JMAA, 201, 35-43, U.S.A, 1996.
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[32] Gwinner, J., and Jeyakumar, V, Inequality Systems and Optimization, JMAA, Vol.159, 51-71, 1991．
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[38] 李泽民，乘积空间中向量最优化问题的最优性必要条件，全国决策科学／多目标决策研讨会论文集，香港卓越出版社有限公司，2000，5．
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[45] Li, Z. F., Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps, JOTA, Vol.98, No.3, 623-649, 1998.
[46] 杨新民，一类集值映射的择一定理，运筹学杂志，1997，16(1)．
[47] 黄永伟，李泽民，集值映射向量最优化的最优性条件，经济数学，2000，17(3)，59-65．
[48] 李泽民，Hilbert空间凸规划最优解的可移性，数学物理学报，Vol．4，185-196，1984．
[49] Kouada, I., Fritz John Type Conditions and Associated Duality Forms in Convex Nondifferentiable Vector Optimization, Operations Research, Vol.28, 399-412, 1994.
[50] Chen, G. Y., Rong, W. D., Characterization of the Benson Proper Efficiency for Nonconvex Vector Optimization, JOTA, Vol.98, No.2, 365-384, 1998.
[51] Mukherjee, R. N., and Mishra, S. K., Multiobjective Programming with Semi-locally Convex Functions, Journal of Mathematical Analysis Applications, Vol. 199, 409-424, 1996.
[52]Phuong, T. D., Sach, P. H., and Yen, N.D., Strict Lower Semicontinuity of the Level Sets and Invexity of a Locally Lipschitz Function, JOTA, Vol. 87, 579-594, 1995.
[53]贺素香，张力卫，求解约束优化问题的一个对偶算法，计算数学，Vol：23(3)，2001．
[54]时贞军，二次规划子空间共轭向量法，江西师范大学学报，1990，14(2)，21-26．
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[58]魏权龄，王日爽等，数学规划与优化设计，国防工业出版社，1984．
[59]管梅谷，郑汉鼎，线性规划，山东科学技术出版社，1983．
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[86] 詹茂豪，李泽民，黄永伟，线性空间中向量极值问题的最优性条件，经济数学，18(1)，38-42，2001．
[2] 胡毓达，非线性规划，高等教育出版社，1990．
[3] 赵凤治，尉继英，约束最优化计算方法，科学出版社，1991．
[4] 李泽民，线性拓扑空间中向量极值问题的广义Kuhn-Tucker条件，系统科学与数学，Vol．10，No．1，78—83，1990．
[5] A. J. V. Brandao, MA. Rojas-Medar, and G.N. Silva, Optimality Conditions for Pareto Nonsmooth Nonconvex Programmming in Vanach Spaces, JOTA, Vol. 103, No. 1, 65-73, 1999.
[6] Dax, A., Sreedharan, V. P., Theorem of the Alternative and Duality, JOTA, Vol.94, No.9, 561-590, 1997.
[7] Li Zemin, The Optimality Conditions for Vector Optimization of Set-Valued Maps, JMAA, Vol. 237, 413-424, 1999.
[8] Li Zemin A theorem of the Alternative and Its Application to the Optimization of Set-Valued Maps, JOTA, Vol. 100, No.2, 365-375, 1999.
[9] T. illes and Kassay, Theorems of the Alternative and Optimality Conditions for Convexlike and General Convexlike Programming, JOTA, Vol. 101, No.2, 243-257, 1999.
[10] 袁亚湘，孙文瑜，最优化理论与方法，科学出版社, 1997.
[11] Li Zemin, The Optimality Conditions of Differentiable Vector Optimization Problems, JMAA, 201, 35-43, U.S.A, 1996.
[12] C. Singh, Optimality Conditions in Multiobjective Differentiable Programming, JOTA Vol.53, No.1,115-123, 1987.
[13] Hayashi, M, and Komiya, H, Perfect Duality or Convexlike Programs, JOTA Vol.38, 179-189, 1982.
[14] Postein, J., Approaches to the Theory of optimization, 1980.
[15] Jan Van Tiel, Convex Analysis, An Introductory Text, 1984.
[16] Frenk, J.B.G., Kassay, G., On Classes of Generalized Convex Functions, Gordan-Farkas Type Theorems, and Lagrangian Duality, HOTA, Vol. 102, 315-343, 1999.
[17] Simons, S., Abstract Kuhn-Tucker Theorems, JOTA, Vol. 58, 148-152,1988.
[18] H.W.Corley, Optimality Conditions of Set-Valued Functions, JOTA, Vol.58, No.1, 1-10, 1988．
[19] W.Song, Duality for Vector Optimization of Set-Valued Functions, JMAA, Vol.201,212-225, 1996．
[20] 李泽民,半无穷维向量最优化问题的最优性条件，系统科学与数学 Vol.14, No.4, 375-380, 1994．
[21] L.-J.Lin, Optimization of Set-Valued Functions, JMAA, Vol.186, 30-51, 1994．
[22] Craven, B.D., Nonsmooth Multiobjective Programming, Numerical Functional Analysis and Optimization, Vol.10,49-64,1989．
[23] Craven, B.D., Gwinner, J., and Jeyakumar, V. Nonconvex Theorems of the Alternative and Minimization, Optimization, Vol.18,151-163,1987．
[24] Minami, M, Weak Pareto-Optimal Necessary Conditions in a Nondifferential Multiobjective Program on a Banach Space, Journal Optimization Theory and Applications, Vol.41, 451-461, 1983．
[25] Jeyakumar, V, and Zaffaroni, A., Asymptotic Conditions for Weak and Proper Optimality in Infinite-Dimensional Convex Vector Optimization, Numerical Functional Analysis and Optimization, Vol.17,323-343,1996．
[26] Luc, D.T., Contingent Derivative of Set-Valued Maps and Applications to Vector Optimization, Mathematical Programming, Vol.50,99-111,1991．
[27] Chen, G.Y., Jahn, J., Optimality Conditions for Set-Valued Optimization Problems, Mathematical Methods of Operations Research, Vol.48,187-200, 1998．
[28] Illes, T, and Kassay, G., Farkas Type Theorems for Generalized Convexities, Pure Mathematics and Applications, Vol.5, 225-239, 1994．
[29] Huang, Y.W, Li, Z.M, Chen, Z.D., Theorems of the Alternative for Nearly Convexlike Set-Valued Maps, 重庆建筑大学学报,Vol.22(4) , 61-64, 2000．
[30] Wang, S.S., A Separation Theorem of Convex Sets on Ordered Vector Space and Its Applications,应用数学学报(英文版), Vol.9,309-319,1986．
[31] Yu, P.L., Cone Convexity, Cone Extreme Points, and Nondomainated Solutions indecision Problems with Multiobjectives, JOTA, Vol.14, 319-377,1974．
[32] Gwinner, J., and Jeyakumar, V, Inequality Systems and Optimization, JMAA, Vol.159, 51-71, 1991．
[33] Rong, W.D., Wu,Y.N., Characterizations of Supper Efficiency In Cone-Convexlike Vector Optimization with Set-Valued Maps, Vol.48,
247-258，1998．
[34] 应枚茜，魏权龄，非线性规划及其引论，中国人民大学出版社，1994．
[35] 林锉云，董家礼，多目标优化的方法与理论，长春，吉林教育出版社，1992．
[36] 刘光中，凸分析与极值问题，高等教育出版社，1984．
[37] 李泽民，一类多目标规划问题的二阶最优性条件，华中师范大学学报，专辑(10)，98-101，1998．
[38] 李泽民，乘积空间中向量最优化问题的最优性必要条件，全国决策科学／多目标决策研讨会论文集，香港卓越出版社有限公司，2000，5．
[39] Tamminen, E. V., Sufficient Conditions for the Existence of Multipliers and Lagrangian Duality in Abstract Optimization Problems, JOTA, Vol. 82, 93-104, 1994.
[40] Jeyakumar, V., Convexlike Alternative Theorems of Fan via a Theorem of the Alternative, JOTA, Vol.48, 525-533, 1986.
[41] 吴从忻译，一般拓扑学，科学出版社，1982．
[42] 夏道行，杨亚力，线性拓扑空间引论，上海科技文献出版社，1986．
[43] Tardella, E, On the Inage of a Constrained Extremum Problem and Some Application to the Existence of a Minimum, JOTA, Vol.60, 93-104, 1989.
[44] 陈光亚，Banach空间中向量极值问题的Lagrange定理及K-T条件，系统科学与数学，Vol．3，62-70，1983．
[45] Li, Z. F., Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps, JOTA, Vol.98, No.3, 623-649, 1998.
[46] 杨新民，一类集值映射的择一定理，运筹学杂志，1997，16(1)．
[47] 黄永伟，李泽民，集值映射向量最优化的最优性条件，经济数学，2000，17(3)，59-65．
[48] 李泽民，Hilbert空间凸规划最优解的可移性，数学物理学报，Vol．4，185-196，1984．
[49] Kouada, I., Fritz John Type Conditions and Associated Duality Forms in Convex Nondifferentiable Vector Optimization, Operations Research, Vol.28, 399-412, 1994.
[50] Chen, G. Y., Rong, W. D., Characterization of the Benson Proper Efficiency for Nonconvex Vector Optimization, JOTA, Vol.98, No.2, 365-384, 1998.
[51] Mukherjee, R. N., and Mishra, S. K., Multiobjective Programming with Semi-locally Convex Functions, Journal of Mathematical Analysis Applications, Vol. 199, 409-424, 1996.
[52]Phuong, T. D., Sach, P. H., and Yen, N.D., Strict Lower Semicontinuity of the Level Sets and Invexity of a Locally Lipschitz Function, JOTA, Vol. 87, 579-594, 1995.
[53]贺素香，张力卫，求解约束优化问题的一个对偶算法，计算数学，Vol：23(3)，2001．
[54]时贞军，二次规划子空间共轭向量法，江西师范大学学报，1990，14(2)，21-26．
[55]Yang, X. M., Generalized Subconvexlike Functions and Multipleobjective Optimization, 系统科学与数学(英文版), Vol. 8(3), 254-259, 1995.
[56]胡运权，运筹学习题集，第2版，清华大学出版社，1995．
[57]席少霖，非线性优化方法，高大教育出版社，1992．
[58]魏权龄，王日爽等，数学规划与优化设计，国防工业出版社，1984．
[59]管梅谷，郑汉鼎，线性规划，山东科学技术出版社，1983．
[60]Li, Z. F., Wang, S. Y., Connectedness of Supper Efficient Sets in Vector Optimization of Set-Valued Maps, Mathematical Methods of Operations Research, Vol. 48, 207-217, 1998.
[61]路易斯·汉格曼，大卫·扬著，蔡大用，施妙根译，实用迭代法，清华出版社，1984．
[62]周汉良，范玉妹，数学规划及其应用，冶金出版社，1995．
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