【作者】 王其林；

【导师】 李声杰；

【作者基本信息】 重庆大学 ， 计算数学， 2010， 博士

【摘要】 本文研究了集值优化问题的高阶最优性条件、非凸集值优化问题的高阶最优性条件、约束集值优化问题的高阶Mond-Weir型及Wolfe型对偶问题、向量优化问题的高阶灵敏性和二阶稳定性。全文共分八章,具体内容如下:在第一章里,我们介绍了集值映射向量优化问题的最优性条件和对偶理论研究、向量优化问题的稳定性和灵敏性理论研究的概况,并且阐述了本文选题的目的和主要研究工作。在第二章里,我们介绍了本文的一些基本假设和概念。引入一个集合的广义高阶相依集和广义高阶邻接集的概念,并讨论了它们的一些性质。在第三章里,引入集值映射的广义高阶相依上图导数和广义高阶邻接上图导数,同时讨论了这两种高阶导数的一些重要性质。基于广义高阶上图导数和Henig真有效性,获得了约束集合由一个固定集合决定的集值优化问题的高阶必要和充分最优性条件。同时还获得了约束集合由一个集值映射决定的集值优化问题的高阶Kuhn-Tucker型必要和充分最优性条件。在第四章里,引入集值映射的高阶广义相依导数和高阶广义邻接导数,同时讨论了它们的一些性质。利用这些性质和第三章中的高阶上图导数的一些性质,在没有任何凸性假设条件下,分别研究了无约束集值优化问题和约束集值优化问题在弱有效解意义下的必要和充分最优性条件。在第五章里,引入集值映射的高阶弱广义相依上图导数和高阶弱广义邻接上图导数,并讨论了它们的一些性质.基于高阶弱广义邻接上图导数,针对约束集值优化问题,提出了一个高阶Mond-Weir型对偶问题和一个高阶Wolfe型对偶,并讨论了相应的弱对偶、强对偶和逆对偶性质。在第六章里,讨论的是向量优化问题的高阶灵敏性。首先,我们分别讨论了集值映射与它的剖面映射的高阶相依导数(高阶邻接导数)之间的关系。其次,给定一簇参数向量优化问题,定义了此问题的扰动映射和弱扰动映射。最后,我们讨论了弱扰动映射的高阶邻接导数与可行集映射的高阶邻接导数的弱极小点集之间的关系,同时,也讨论了扰动映射的高阶邻接导数与目标空间中的可行集映射的高阶邻接导数的极小点集和弱极小点集之间的关系。在第七章里,讨论向量优化问题的二阶导数的稳定性。首先,我们建立了集值映射二阶相依导数和二阶邻接导数的连续性和闭性。其次,给定一簇参数向量优化问题,我们定义了此问题的弱扰动映射。最后,在恰当的假设条件下,获得了弱扰动映射的二阶邻接导数的上半连续和下半连续性。在第八章里,我们对本文作了一个简要的总结和讨论。更多还原

【Abstract】 In this thesis, we study higher-order optimality conditions for set-valued optimization problems, higher-order optimality conditions for nonconvex set-valued optimization problems, and higher-order Mond-Weir and Wolfe type duality problems for constrained set-valued optimization problems. We also study higher-order sensitivity and second-order stability for vector optimization problems. This thesis is divided into eight chapters. It is organized as follows:In Chapter 1, we describe the development and current researches on the topic of optimality conditions and duality for set-valued vector optimization problems, and stability and sensitivity for vector optimization problems. We also give the motivation and the main research works.In Chapter 2,we introduce some basic assumptions and concepts in this thesis. We introduce generalized higher-order contingent sets and generalized higher-order adjacent sets for sets, and also discuss some properties of them.In Chapter 3, we introduce generalized higher-order contingent epiderivatives and generalized higher-order adjacent epiderivatives of set-valued maps, and discuss some important properties of the two kinds of higher-order derivatives. By virtue of generalized higher-order epiderivatives and Henig efficiency, we obtain higher-order necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Simultaneously, we also obtain higher-order Kuhn-Tucker type necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.In Chapter 4, we introduce higher-order generalized contingent derivatives and higher-order generalized adjacent derivatives for set-valued maps, and discuss some of their properties. By virtue of these properties and some properties of generalized higher-order epiderivatives in Chapter 3, without any convexity assumptions, we obtain necessary and sufficient optimality conditions of weak efficient solutions for unconstrained set-valued optimization problems and constrained set-valued optimization problems, respectively.In Chapter 5, we introduce higher-order weakly generalized contingent epiderivatives and higher-order weakly generalized adjacent epiderivatives of set-valued maps,and discuss some of their properties. By virtue of higher-order weakly generalized adjacent epiderivatives,we introduce a higher-order Mond-Weir type dual problem and a higher-order Wolfe type dual problem for a constrained set-valued optimization problem, and discuss the corresponding weak duality, strong duality and converse duality properties, respectively.In Chapter 6, we study higher-order sensitivity in vector optimization problem. Firstly, we discuss relationships between higher-order contingent derivatives (higher-order adjacent derivatives) of set-valued maps and their profile maps. Secondly, given a family of parametrized vector optimization problems, we define a perturbation map and a weak perturbation map for the problems, respectively. Finally, we investigate the relationship between higher-order adjacent derivatives for the weak perturbation maps and weakly minimal point sets for higher-order adjacent derivatives of feasible set maps in the objective space. We also investigate the relationships between higher-order adjacent derivatives for the perturbation maps and two types of minimal point sets (i.e. minimal point sets and weakly minimal point sets) for higher-order adjacent derivatives of feasible set maps in the objective space.In Chapter 7, we study stability of second-order derivatives in vector optimization problems. Firstly, we establish continuity and closedness of second-order contingent derivatives and second-order adjacent derivatives for set-valued maps. Secondly, given a family of parametrized vector optimization problems, we define the weak perturbation maps for the problems. Finally, under suitable assumptions, we obtain the upper semicontinuity and the lower semicontinuity of second-order adjacent derivatives for the weak perturbation maps.In Chapter 8,we summarize the results of this thesis and make some discussions.更多还原