线性空间中具有一致子水平集的函数与标量化
Functions with uniform sublevel sets and scalarization in linear spaces
摘要：
具有一致子水平集的函数可以表示顺序、偏好关系或其他二元关系，因此成为多目标优化、决策论、数理金融、产生式理论和算子理论中的标量化工具。不一定是凸的集合可以被具有一致子级集的函数分离。本文主要研究定义在线性空间上的具有一致子水平集的实值函数和扩展实值函数的性质。函数可以是凸函数或次线性函数。它们可以与Minkowski泛函或空间子集上的序单位范数重合。所考虑的泛函应用于向量最优化问题的标量化。这些向量最优化问题涉及任意控制集。考虑这样的集合的动机是它们与决策中的控制结构的关系。
英文摘要：
Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical finance, production theory and operator theory. Sets which are not necessarily convex can be separated by functions with uniform sublevel sets. This report focuses on properties of real-valued and extended-real-valued functions with uniform sublevel sets which are defined on a linear space without assuming topological properties. The functions may be convex or sublinear. They can coincide with a Minkowski functional or with an order unit norm on a subset of the space. The considered functionals are applied to the scalarization of vector optimization problems. These vector optimization problems refer to arbitrary domination sets. The consideration of such sets is motivated by their relationship to domination structures in decision making.
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