Construction of multiobjective optimization theory based on new definitions of supremum and infimum in the multi-dimensional extended real space

多维扩展实空间上上界和下确界新定义的多目标优化理论构建

• 批准号: 14550400
• 负责人: TANINO Tetsuzo
• 金额: $ 1.34万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14550400/
• 关键词: multiobjective optimization; supremum; infimum; convexity; stability; sensitivity analysis; conjugate mapping; duality

In this paper, solutions for a multiobjective optimization problem are defined as the infimum of a set in the multi-dimensional extended real space and mathematical theory of multiobjective optimization is constructed.First, we provided new definitions of supremum and infimum of a set in the extended real space, and investigated their properties. Particularly, taking infimum is regarded as an operator and some relationships between this operator and set theoretic operations such as the union and the algebraic sum are made clear. We introduced dividing and traversing properties of sets in the extended real space and proved that the infimum of an arbitrary set has these properties. This enabled us to characterize optimal solutions for a multiobjective optimization problem, particularly in the convex case.Secondly, concepts of conjugate mappings and subgradients for set-valued mappings in the extended real space were introduced and conjugate duality theory was developed based on those concepts. A relationship between subgradients and conjugate mappings was provided and sufficient conditions for subdifferentiability of set-valued mappings were studied. A primal multiobjective optimization problem was imbedded into a family of perturbed problems and its dual problem was defined in terms of the conjugate mapping. Some duality results were established between the primal problem and the dual problem.Furthermore, the perturbation mapping was defined for a parameterized multiobjective optimization problem, and its behavior was analyzed. From a qualitative viewpoint, continuity of the perturbation mapping was considered. On the other hand, from a quantitative viewpoint, graphical derivatives of the perturbation mapping were studied.The obtained results contribute to construct a new theory of multiobjective optimization.

本文将多目标优化问题的解定义为多维扩展实空间中集合的下确界，构建了多目标优化的数学理论。首先，给出了扩展实空间中集合的上界和下确界的新定义，并研究了它们的性质。特别地，将下确界视为一个算子，并明确了该算子与并集、代数和等集合论运算之间的一些关系。我们引入了扩展实空间中集合的划分和遍历性质，并证明了任意集合的下确界具有这些性质。这使我们能够描述多目标优化问题的最优解，特别是在凸情况下。其次，引入了扩展实空间中的共轭映射和集值映射的次梯度的概念，并基于这些概念发展了共轭对偶理论。提供了次梯度和共轭映射之间的关系，并研究了集值映射的次微分性的充分条件。将原始多目标优化问题嵌入到扰动问题族中，并根据共轭映射定义其对偶问题。在原问题和对偶问题之间建立了一些对偶结果。此外，定义了参数化多目标优化问题的摄动映射，并分析了其行为。从定性的角度来看，考虑了扰动映射的连续性。另一方面，从定量的角度研究了微扰映射的图导数。所得结果有助于构建新的多目标优化理论。

项目成果

Tetsuzo Tanino: "Multiobjective conjugate duality in the multi-dimensional extended real space"Proceedings of 2nd International Conference on Nonlinear Analysis and Convex Analysis. 489-499 (2003)

Tetsuzo Tanino：“多维扩展实空间中的多目标共轭对偶性”第二届非线性分析和凸分析国际会议论文集。

Tetsuzo Tanino: "Some fundamental results for the infimum of a set in the multi-dimensional extended real space"Journal of Nonlinear and Convex Analysis. 5(in press). (2004)

Tetsuzo Tanino：“多维扩展实空间中集合的下确界的一些基本结果”非线性与凸分析杂志。

Tetsuzo Tanino: "Multiobjective optimization based on a new definition of infimum"Proceedings of the First Korea-Japan Joint Symposium on Nonlinear Functional Analysis and Convex Analysis. (2003)

谷野哲三：《基于新的下确界定义的多目标优化》首届韩日非线性泛函分析与凸分析联合研讨会论文集。

Tetsuzo Tanino: "Some fundamental results for the infimum of a set in the multi-dimensional extended real space"Journal of Nonlinear and Convex Analysis. Vol.5 (in press). (2004)

Tetsuzo Tanino：“多维扩展实空间中集合的下确界的一些基本结果”非线性与凸分析杂志。

Tetsuzo Tanino: "Multiobjective conjugate duality in the multi-dimensional extended real space"Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis 2001. 489-499 (2003)

Tetsuzo Tanino：“多维扩展实空间中的多目标共轭对偶性”非线性分析和凸分析国际会议论文集 2001. 489-499 (2003)