向量极值问题的最优性条件及线性不等式约束二次规划问题的一种算法
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摘要
本文主要讨论了抽象空间中向量优化问题的一些理论以及求解线性不等式约束二次规划问题的一种算法及其应用。 文章在Banach空间中界定了C-切锥的概念，并给出其有关性质，然后引入一种广义约束规格，从而得到了广义凸规划问题的最优性充分与必要条件；在线性拓扑空间中，给出集合（弱）有效点的重要性质，然后导出了约束向量极值问题像集的性质，在此基础上得到了原问题（弱）有效解存在的充分与必要条件；最后，在线性等式约束二次规划降维算法基础上，重点研究了线性不等式约束二次规划问题的一种算法，并对此算法的收敛性做出了一定分析，之后将该算法应用到求解一般的线性不等式约束非线性规划与多目标规划问题中，通过编制C++程序进行数值实验，表明此算法是实际可行﹑有效的
In this thesis, some topics on vector optimization theory in abstract spaces are discussed, and a algorithm for quadratic programming problem with linear inequalities constraints is studied as well. In Banach space, the optimal concept of Contingent Cone is defined, and then, a generalized constrained qualification is given, thereafter, the optimal conditions of the differentiable optimization problem are obtained in Banach space; In linear toplogical space, a important property of (weak)efficient points of set is given , and then, the sufficient and the neccessory conditions of the vector extremum problem with constraint are obtained; Finally, this thesis gives a algorithm for quadratic programming problem with linear inequalities constraints, and its convergence is analysized in some degree, moreover, this algorithm is efficient compared with the results of numerical tests
In this thesis, some topics on vector optimization theory in abstract spaces are discussed, and a algorithm for quadratic programming problem with linear inequalities constraints is studied as well. In Banach space, the optimal concept of Contingent Cone is defined, and then, a generalized constrained qualification is given, thereafter, the optimal conditions of the differentiable optimization problem are obtained in Banach space; In linear toplogical space, a important property of (weak)efficient points of set is given , and then, the sufficient and the neccessory conditions of the vector extremum problem with constraint are obtained; Finally, this thesis gives a algorithm for quadratic programming problem with linear inequalities constraints, and its convergence is analysized in some degree, moreover, this algorithm is efficient compared with the results of numerical tests
引文
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[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．
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[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.
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[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．
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[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, F., On the Inage of a Constrained Extremum Problem and Some Application to the Existence of a Minimum, JOTA, Vol.60, 93-104, 1989.
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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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