Weak minimizers, minimizers and variational inequalities for set-valued functions. A blooming wreath?

Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex set-valued functions. Similar results have been proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved usin...

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Bibliographic Details
Published in:Optimization 2017-12, Vol.66 (12), p.1973-1989
Main Authors: Crespi, Giovanni P., Schrage, Carola
Format: Article
Language:English
Subjects:
Dini derivative
Estimates
Inequalities
Mathematical programming
Optimization
Set optimization
variational Inequalities
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Summary:Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex set-valued functions. Similar results have been proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case, we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed, this might eventually prove the usefulness of the set optimization approach to renew the study of vector optimization.
ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2016.1189550