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

本文将多目标优化问题的解定义为多维扩展实空间中一个集合的最小值，并构造了多目标优化问题的数学理论。首先给出了扩展实空间中集合的上极值和上极值的新定义，并研究了它们的性质。特别地，将取最小值作为一个算子，并明确了该算子与集论运算如并和、代数和的关系。引入了扩展实空间中集合的除法和遍历性质，并证明了任意集合的极小值具有这些性质。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集值映射的共轭映射和子梯度的概念，并在此基础上建立了共轭对偶理论。给出了次梯度与共轭映射的关系，研究了集值映射次可微的充分条件。将一个原始多目标优化问题嵌入到摄动问题族中，用共轭映射定义其对偶问题。在原始问题和对偶问题之间建立了一些对偶结果。在此基础上，定义了参数化多目标优化问题的摄动映射，并分析了摄动映射的行为。从定性的角度考虑了摄动映射的连续性。另一方面，从定量的角度研究了摄动映射的图形导数。所得结果有助于构建一种新的多目标优化理论。

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.