向量优化中的若干对偶和稳定性研究
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摘要
本文研究了向量集值优化问题的导数型高阶Fritz John型及Kuhn-Tucker型最优性条件、高阶Mond-Weir型及Wolfe型对偶问题、共轭对偶问题,若干向量变分不等式及向量平衡问题的解集映射的上、下半连续性和连续性。全文共分九章,具体内容如下:
在第一章里,我们介绍了向量优化的高阶导数型最优性条件和对偶、共轭对偶理论和向量变分不等式及向量平衡问题的解集映射的半连续性等方面的研究概况,并且阐述了本文的选题动机和主要工作。
在第二章里,我们引入集值映射的高阶广义相依上图导数和邻接上图导数概念。基于高阶广义上图导数,研究了约束集值优化问题在Henig真有效解意义下的高阶Fritz John型必要和充分最优性条件。
在第三章里,我们引入集值映射的高阶弱相依上图导数和邻接上图导数概念。基于高阶弱上图导数和Henig有效性,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型对偶问题,分别讨论了其弱对偶、强对偶和逆对偶性质。同时建立了原问题的高阶Kuhn-Tucker型必要和充分最优性条件。
在第四章里,我们基于Tanino的共轭对偶理论,针对约束集值优化问题构造得到三个共轭对偶问题:Lagrange、Fenchel、Fenchel-Lagrange对偶问题。引入了若干新的概念——正双共轭映射、正次梯度和正次微分,并讨论了相应的性质。研究了原问题与Lagrange和Fenchel-Lagrange对偶问题间的弱对偶、强对偶和稳定性准则。同时讨论了三个对偶问题的像集间的包含关系,并给出了它们在向量平衡问题的变分原理方面的应用。
在第五章里,我们提出了参数(集值)弱向量变分不等式的参数间隙函数概念,借助参数间隙函数引入一个关键假设,从而在Banach空间中在没有单调性和凸性的假设下讨论了参数(集值)弱向量变分不等式的解集映射的Hausdorff下半连续性和连续性。
在第六章里,我们利用非线性标量化函数构造了参数广义向量拟变分不等式的一个参数间隙函数,借助参数间隙函数也引入一个重要假设,进而在局部凸Hausdorff拓扑向量空间中讨论了参数广义向量拟变分不等式的解集映射的下半连续性。同时用例子说明了在拟变分情形下,保证解集映射的Hausdorff下半连续性成立的条件未必是解集映射连续的充分条件,但进一步在紧空间假设下解集映射是连续的。
在第七章里,我们指出文献[X.H. Gong and J.C. Yao, Lower semicontinuity ofthe set of efficient solutions for generalized systems, Journal of Optimization Theory and Applications, 138 (2008) 197-205]和文献[X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, Journal of Optimization Theory and Applications, 139 (2008) 35-46]在证明一个参数广义系统(或称参数向量平衡问题)的有效解集映射和弱有效解集映射(或参数弱向量平衡问题的解集映射)的下半连续性时,对可行映射A的一致紧性的假定是多余的。同时基于标量化方法证明了在保证下半连续的假设下,解集映射是连续的。并且讨论了参数广义系统的若干真有效解集映射的连续性。
在第八章里,我们基于解集映射的一个标量刻画和有关下半连续的集值映射族的并性质,建立了一类带集值映射的参数广义向量平衡问题的解映射的下半连续性和连续性结果,改进和推广了文献[Y.H. Cheng and D.L. Zhu, Global stability results for the weak vector variational inequality, Journal of Global Optimization, 32 (2005) 543-550]和文献[X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, Journal of Optimization Theory and Applications, 139 (2008) 35-46]的对应结论。
在第九章里,我们作了一个简要的总结和讨论。
In this thesis, we study higher order Fritz John and Kuhn-Tucker type optimality conditions, higher order Mond-Weir and Wolfe type dual problems, and conjugate dual problems for constrained set-valued optimization problems, and we also study the upper, lower semicontinuity and continuity results on the solution mappings to parametric vector variational inequalities and vector equilibrium problems. This thesis is divided into nine chapters. It is organized as follows:
In Chapter 1, we describe the development and current researches on the topic of vector optimization, including higher order optimality conditions and duality, conjugate duality and the semicontinuity of the solution mappings to parametric vector variational inequalities and vector equilibrium problems. We also give the motivation and the main research work.
In Chapter 2, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued mappings are introduced. Necessary and sufficient Fritz John type optimality conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives.
In Chapter 3, the notions of higher order weak contingent epiderivative and higher order weak adjacent epiderivative for set-valued mappings are defined. By virtue of higher order weak epiderivatives and Henig efficiency, 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 (SOP) and discuss the corresponding weak duality, strong duality and converse duality properties. We also establish higher order Kuhn-Tucker type necessary and sufficient optimality conditions for (SOP).
In Chapter 4, three conjugate dual problems are proposed by considering the different perturbations to a set-valued vector optimization problem with explicit constraints. Some concepts are introduced, such as the positive biconjugate mapping, positive subgradient and positive subdifferential. The weak duality, inclusion relations between the image sets of dual problems, strong duality and stability criteria are investigated. Some applications to so-called variational principles for a generalized vector equilibrium problem are shown.
In Chapter 5, the concept of the parametric gap function is proposed and a key assumption is introduced by virtue of the parametric gap function. Then, by using the key assumption, sufficient conditions of the Hausdorff lower semicontinuity and continuity of the solution set map for a parametric (set-valued) weak vector variational inequality are obtained in Banach spaces.
In Chapter 6, the parametric gap function for a parametric generalized vector quasivariational inequality is proposed by using a nonlinear scalarization function. By virtue of the parametric gap function and a key assumption, the Hausdorff lower semicontinuity of the solution set map is established in locally convex Hausdorff topological vector spaces. Simultaneously, we use an example to explain that the theorem concerning Hausdorff lower semicontinuity may not be a sufficient condition for the upper semicontinuity of the solution set map.
In Chapter 7, we show that the uniform compactness assumptions used in proving the lower semicontinuity of the efficient solution set in [X.H. Gong and J.C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl. 138 (2008) 197-205] and the weak efficient solution set in [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl. 139 (2008) 35-46] are superfluous. Furthermore, we point out that under the assumptions of lower semicontinuity theorems, the solution set mappings are continuous actually. The upper semicontinuity of the solution set mappings are derived by scalarization methods. In addition, we also give some continuity results of various proper efficient solution sets to parametric generalized systems.
In Chapter 8, based on a scalarization representation of the solution mapping and a property involving the union of a family of lower semicontinuous set-valued mappings, we establish the lower semicontinuity and continuity of the solution mapping to a parametric generalized vector equilibrium problem with set-valued mappings. Our consequences are new and include the corresponding results in [Y.H. Cheng and D.L. Zhu, Global stability results for the weak vector variational inequality, J. Global Optim. 32 (2005) 543-550] and [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl. 139 (2008) 35-46] as special cases.
In Chapter 9, we summarize the results of this thesis and make some discussions.
在第一章里,我们介绍了向量优化的高阶导数型最优性条件和对偶、共轭对偶理论和向量变分不等式及向量平衡问题的解集映射的半连续性等方面的研究概况,并且阐述了本文的选题动机和主要工作。
在第二章里,我们引入集值映射的高阶广义相依上图导数和邻接上图导数概念。基于高阶广义上图导数,研究了约束集值优化问题在Henig真有效解意义下的高阶Fritz John型必要和充分最优性条件。
在第三章里,我们引入集值映射的高阶弱相依上图导数和邻接上图导数概念。基于高阶弱上图导数和Henig有效性,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型对偶问题,分别讨论了其弱对偶、强对偶和逆对偶性质。同时建立了原问题的高阶Kuhn-Tucker型必要和充分最优性条件。
在第四章里,我们基于Tanino的共轭对偶理论,针对约束集值优化问题构造得到三个共轭对偶问题:Lagrange、Fenchel、Fenchel-Lagrange对偶问题。引入了若干新的概念——正双共轭映射、正次梯度和正次微分,并讨论了相应的性质。研究了原问题与Lagrange和Fenchel-Lagrange对偶问题间的弱对偶、强对偶和稳定性准则。同时讨论了三个对偶问题的像集间的包含关系,并给出了它们在向量平衡问题的变分原理方面的应用。
在第五章里,我们提出了参数(集值)弱向量变分不等式的参数间隙函数概念,借助参数间隙函数引入一个关键假设,从而在Banach空间中在没有单调性和凸性的假设下讨论了参数(集值)弱向量变分不等式的解集映射的Hausdorff下半连续性和连续性。
在第六章里,我们利用非线性标量化函数构造了参数广义向量拟变分不等式的一个参数间隙函数,借助参数间隙函数也引入一个重要假设,进而在局部凸Hausdorff拓扑向量空间中讨论了参数广义向量拟变分不等式的解集映射的下半连续性。同时用例子说明了在拟变分情形下,保证解集映射的Hausdorff下半连续性成立的条件未必是解集映射连续的充分条件,但进一步在紧空间假设下解集映射是连续的。
在第七章里,我们指出文献[X.H. Gong and J.C. Yao, Lower semicontinuity ofthe set of efficient solutions for generalized systems, Journal of Optimization Theory and Applications, 138 (2008) 197-205]和文献[X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, Journal of Optimization Theory and Applications, 139 (2008) 35-46]在证明一个参数广义系统(或称参数向量平衡问题)的有效解集映射和弱有效解集映射(或参数弱向量平衡问题的解集映射)的下半连续性时,对可行映射A的一致紧性的假定是多余的。同时基于标量化方法证明了在保证下半连续的假设下,解集映射是连续的。并且讨论了参数广义系统的若干真有效解集映射的连续性。
在第八章里,我们基于解集映射的一个标量刻画和有关下半连续的集值映射族的并性质,建立了一类带集值映射的参数广义向量平衡问题的解映射的下半连续性和连续性结果,改进和推广了文献[Y.H. Cheng and D.L. Zhu, Global stability results for the weak vector variational inequality, Journal of Global Optimization, 32 (2005) 543-550]和文献[X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, Journal of Optimization Theory and Applications, 139 (2008) 35-46]的对应结论。
在第九章里,我们作了一个简要的总结和讨论。
In this thesis, we study higher order Fritz John and Kuhn-Tucker type optimality conditions, higher order Mond-Weir and Wolfe type dual problems, and conjugate dual problems for constrained set-valued optimization problems, and we also study the upper, lower semicontinuity and continuity results on the solution mappings to parametric vector variational inequalities and vector equilibrium problems. This thesis is divided into nine chapters. It is organized as follows:
In Chapter 1, we describe the development and current researches on the topic of vector optimization, including higher order optimality conditions and duality, conjugate duality and the semicontinuity of the solution mappings to parametric vector variational inequalities and vector equilibrium problems. We also give the motivation and the main research work.
In Chapter 2, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued mappings are introduced. Necessary and sufficient Fritz John type optimality conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives.
In Chapter 3, the notions of higher order weak contingent epiderivative and higher order weak adjacent epiderivative for set-valued mappings are defined. By virtue of higher order weak epiderivatives and Henig efficiency, 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 (SOP) and discuss the corresponding weak duality, strong duality and converse duality properties. We also establish higher order Kuhn-Tucker type necessary and sufficient optimality conditions for (SOP).
In Chapter 4, three conjugate dual problems are proposed by considering the different perturbations to a set-valued vector optimization problem with explicit constraints. Some concepts are introduced, such as the positive biconjugate mapping, positive subgradient and positive subdifferential. The weak duality, inclusion relations between the image sets of dual problems, strong duality and stability criteria are investigated. Some applications to so-called variational principles for a generalized vector equilibrium problem are shown.
In Chapter 5, the concept of the parametric gap function is proposed and a key assumption is introduced by virtue of the parametric gap function. Then, by using the key assumption, sufficient conditions of the Hausdorff lower semicontinuity and continuity of the solution set map for a parametric (set-valued) weak vector variational inequality are obtained in Banach spaces.
In Chapter 6, the parametric gap function for a parametric generalized vector quasivariational inequality is proposed by using a nonlinear scalarization function. By virtue of the parametric gap function and a key assumption, the Hausdorff lower semicontinuity of the solution set map is established in locally convex Hausdorff topological vector spaces. Simultaneously, we use an example to explain that the theorem concerning Hausdorff lower semicontinuity may not be a sufficient condition for the upper semicontinuity of the solution set map.
In Chapter 7, we show that the uniform compactness assumptions used in proving the lower semicontinuity of the efficient solution set in [X.H. Gong and J.C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl. 138 (2008) 197-205] and the weak efficient solution set in [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl. 139 (2008) 35-46] are superfluous. Furthermore, we point out that under the assumptions of lower semicontinuity theorems, the solution set mappings are continuous actually. The upper semicontinuity of the solution set mappings are derived by scalarization methods. In addition, we also give some continuity results of various proper efficient solution sets to parametric generalized systems.
In Chapter 8, based on a scalarization representation of the solution mapping and a property involving the union of a family of lower semicontinuous set-valued mappings, we establish the lower semicontinuity and continuity of the solution mapping to a parametric generalized vector equilibrium problem with set-valued mappings. Our consequences are new and include the corresponding results in [Y.H. Cheng and D.L. Zhu, Global stability results for the weak vector variational inequality, J. Global Optim. 32 (2005) 543-550] and [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl. 139 (2008) 35-46] as special cases.
In Chapter 9, we summarize the results of this thesis and make some discussions.
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