Weak minimizers, minimizers and variational inequalities for set-valued functions. A blooming wreath?. (2nd December 2017)

• Record Type: Journal Article
• Authors: Crespi, Giovanni P.; Schrage, Carola
• Is Part Of: Optimization. Volume 66:Number 12(2017)
• Page Start: 1973
• Page End: 1989
• Publication Date: 2017-12-02
• Subjects: Set optimization -- variational Inequalities -- Dini derivative; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1189550
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : 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.