Ekeland’s variational principle with weighted set order relations

Qamrul Hasan Ansari (), Andreas H Hamel () and Pradeep Kumar Sharma ()
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Qamrul Hasan Ansari: Aligarh Muslim University
Andreas H Hamel: Free University of Bozen
Pradeep Kumar Sharma: Aligarh Muslim University

Mathematical Methods of Operations Research, 2020, vol. 91, issue 1, No 7, 117-136

Abstract: Abstract 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.

Keywords: Weighted set relation; Nonlinear scalarization function; Minimal element theorem; Ekeland’s variational principle; Caristi’s fixed point theorem; Takahashi’s minimization theorem; Order intervals; 49J53; 90C29; 46N10; 47J30; 47H04; 47H10 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s00186-019-00679-5

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