The main results of the paper are a minimal element theorem and an Ekeland-type variational principle for set-valued maps whose values are compared by means of a weighted set order relation. This relation is a mixture of a lower and an upper set relation which form the building block for modern approaches to set-valued optimization. The proofs rely on nonlinear scalarization functions which admit to apply the extended Brézis–Browder theorem. Moreover, Caristi’s fixed point theorem and Takahashi’s minimization theorem for set-valued maps based on the weighted set order relation are obtained and the equivalences among all these results is verified. An application to generalized intervals is given which leads to a clear interpretation of the weighted set order relation and versions of Ekeland’s principle which might be useful in (computational) interval mathematics.

中文翻译：

---

具有加权集序关系的Ekeland变分原理

本文的主要结果是最小集定理和集合值映射的Ekeland型变分原理，其值通过加权集合顺序关系进行比较。此关系是下层和上层集合关系的混合，构成了现代集值优化方法的基础。证明依赖于非线性标量函数，这些函数允许应用扩展的Brézis-Browder定理。此外，获得了基于加权集序关系的集值映射的Caristi不动点定理和Takahashi极小化定理，并验证了所有这些结果之间的等价性。