Partial order relations on family of sets and scalarizations for set optimization

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 22; no. 3; pp. 783 - 802
• Main Authors: Karaman, Emrah, Soyertem, Mustafa, Atasever Güvenç, İlknur, Tozkan, Didem, Küçük, Mahide, Küçük, Yalçın
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-07-2018; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Econometrics; Fourier Analysis; Interval arithmetic; Mathematical programming; Mathematics; Mathematics and Statistics; Operator Theory; Optimization; Potential Theory; Scalarization; Optimality conditions; 90C26; 80M50; Partial order; Set optimization
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-017-0544-3

In this study, some new order relations on family of sets are introduced by using Minkowski difference. The relations between these orders and the ordering cone of the vector space are obtained. It is shown that depending on the corresponding cone, these order relations are partial orders on the family of nonempty bounded sets. Some relationships between these order relations and upper and lower set less order relations are investigated. Also, two scalarizing functions are introduced in order to replace set optimization problems with respect to these partial order relations with scalar optimization problems. Moreover, necessary and sufficient optimality conditions are presented.