Variational and Quasi-Variational Inequalities Under Local Reproducibility: Solution Concept and Applications

Journal article

Authors/Editors

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Publication Details

Author list: Aussel D.; Chaipunya P.

Publisher: Springer

Publication year: 2024

Journal: Journal of Optimization Theory and Applications (0022-3239)

Volume number: 203

Start page: 1531

End page: 1563

Number of pages: 33

ISSN: 0022-3239

eISSN: 1573-2878

URL: https://www.scopus.com/inward/record.uri?eid=2-s2.0-85194479964&doi=10.1007%2fs10957-024-02493-4&partnerID=40&md5=a4ab3d877de34cc94df2700f860f4e76

Languages: English-Great Britain (EN-GB)

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Abstract

Local solutions for variational and quasi-variational inequalities are usually the best type of solutions that could practically be obtained when in lack of convexity or else when available numerical techniques are too limited for global solutions. Nevertheless, the analysis of such problems found in the literature seems to be very restricted to the global treatment. Motivated by this fact, in this work, we propose local solution concepts, study their interrelations and relations with global concepts and prove existence results as well as stability of local solution map of parametric variational inequalities. The key ingredient of our results is the new concept of local reproducibility of a set-valued map, which we introduce to explore such local solutions to quasi-variational inequality problems. As a by-product, we obtain local solutions to quasi-optimization problems, bilevel quasi-optimization problems and Single-Leader-Multi-Follower games. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

Keywords

49J00, 49J53, 49K40, Local reproducibility, Local solution, Qualitative stability, Single-Leader-Multi-Follower game