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# Ekeland’s variational principle with weighted set order relations

Ansari, Qamrul Hasan, Hamel, Andreas H. and Sharma, Pradeep Kumar (2020) Ekeland’s variational principle with weighted set order relations. Mathematical Methods of Operations Research, 91 (1), 117-136.

Record type: Article

## 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.

Accepted/In Press date: 8 April 2019
e-pub ahead of print date: 26 August 2019
Published date: 1 February 2020
Additional Information: Funding Information: The authors are grateful to the handling editor and two anonymous referees for their valuable comments and suggestions, which helped to improve the previous draft of the paper. In this paper, first author was supported by DST-SERB Project No. MTR/2017/000135.
Keywords: Caristi’s fixed point theorem, Ekeland’s variational principle, Minimal element theorem, Nonlinear scalarization function, Order intervals, Takahashi’s minimization theorem, Weighted set relation

## Identifiers

Local EPrints ID: 485512
URI: http://eprints.soton.ac.uk/id/eprint/485512
ISSN: 1432-2994
PURE UUID: 99cda0a8-dd3e-4007-a9a2-af8927d609a9
ORCID for Pradeep Kumar Sharma: orcid.org/0000-0002-5848-3004

## Catalogue record

Date deposited: 07 Dec 2023 17:40

## Contributors

Author: Qamrul Hasan Ansari
Author: Andreas H. Hamel

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