【作者】 王文君；

【导师】 刘红卫；

【作者基本信息】 西安电子科技大学 ， 应用数学， 2011， 硕士

【摘要】 极小极大分式规划是继多目标规划的评价函数法发展起来的规划,极小极大法是在对策论中经常用到的思想,即在最不利的情况下找出一个最有利的策略。集函数的多目标规划已有很多研究,分别从不同的凸性和可微性出发,研究规划的最优性充分条件和必要条件,根据不同的对偶模型,讨论了相应的强、弱和可逆对偶理论。由于多目标规划的强大的实用价值,并且多目标规划的研究成果对现代社会的经济、政治、科技乃至军事都产生过重要的影响,而极小极大分式规划又是多目标规划发展起来的分支,极小极大分式规划已经成为一个新的研究热点,因此研究集函数的极小极大分式规划有一定的理论意义。在极小极大规划中,定义域的选择是要点,定义域选择不同,则相应的理论研究也会发生变化。本文是基于测度空间里,在具有一定凸性的集函数条件下,探讨了极小极大分式规划的最优性条件和对偶性。本文首先主要给出已知集合的子集的σ代数n-折积的含义,定义了伪测度,并介绍了集函数可微和偏导的基本概念以及集函数多目标规划的最优性条件和对偶理论。其次,在给出极小极大分式规划的模型后,介绍广义V不变凸的函数的概念。然后在这类广义凸性条件下证明了集函数极小极大分式规划的一般的最优性充分条件和拉格朗日型的最优性充分条件。最后,给出极小极大分式规划的两种对偶模型,并给出相应的对偶定理的证明。从而丰富了极小极大规划的最优性和对偶理论,是对极小极大规划的理论研究的拓展。更多还原

【Abstract】 A minimax fractional programming problem is developed by evaluation function method of multi-objective programming, whose idea is often used in countermeasure theory, i.e. the adverse circumstances to find the best strategy. The multi-objective programming with set function has been a lot of research, separately from different convexity and differentiability based on studying planning of sufficient optimality conditions and necessary conditions, and according to different dual model, it discussed the corresponding strong, weak and reversible duality theory. Because of multi-objective programming has strong practical value, and its research results has important effects in the modern society economy, politics, science and technology and military, while minimax fractional programming is a branch, which is developed from multi-objective programming, the minimax fractional programming has been a new hotspot, so the research of minimax fractional programming with set function has certain theoretical significance.In the domain of minimax programming, if the choice of domain of definition is different, accordingly, the theory research will also change. This article is based on the new space (not the Euclid space). In a certain set of convex function condition, the optimality conditions and duality for minimax fractional programming are discussed.In this paper, firstly, we define pseudometric for n-foldproduct ofσ-algebra of subsets of a given set and give the concept of differentiability and partial derivative of set function and sufficient optimality conditions and duality models for multi-objective programming with n-set functions are given. After the model of minimax programming involving n-set functions is presented, some classes of functions with generalized V-invexity are introduced. Then, sufficient optimality conditions and duality theorems for minimax fractional subset programming are established under generalized V-invex functions. Also, sufficient optimality conditions and Lagrangian optimality conditions for the programming are considered under generalized V-invex functions. Finally, two dual models are introduced, and some duality theorems are proved. These theories enriched the optimality and duality of the minimax fractional programming, which has expanded the theories of the minimax programming.更多还原