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 Brezis–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.
- Ekeland’s variational principle with weighted set order relations
- Qamrul Hasan Ansari - King Fahd University of Petroleum and MineralsAndreas Heinrich Hamel - Free University of Bozen-BolzanoPK Sharm
- Mathematical Methods of Operations Research, Vol.91, pp.117-136
- 1432-2994
- 1432-5217
- 91
- Springer
- 19
- (UNIBZ)88306010
991006979369901241 - 2-s2.0-85071501160
- Faculty of Economics and Management
- English
- Journal article
- Ansari QH, Hamel AH, Sharm PK