Vectorization of set-valued maps with respect to total ordering cones and its applications to set-valued optimization problems

• Published in: Journal of mathematical analysis and applications Vol. 385; no. 1; pp. 285 - 292
• Main Authors: Küçük, Mahide, Soyertem, Mustafa, Küçük, Yalçın, Atasever, İlknur
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 2012
• Subjects: Scalarization; Set-valued optimization; The Successive Weighted Sum Method; The Weighted Sum Method; Total order; Vectorization; Scalarization; Set-valued optimization; Total order; The Successive Weighted Sum Method; Vectorization; The Weighted Sum Method
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2011.06.045

As a result of our previous studies on finding the minimal element of a set in
n-dimensional Euclidean space with respect to a total ordering cone, we introduced a method which we call “The Successive Weighted Sum Method” (Küçük et al., 2011
[1,2]). In this study, we compare the Weighted Sum Method to the Successive Weighted Sum Method. A vector-valued function is derived from the special type of set-valued function by using a total ordering cone, which is a process we called vectorization, and some properties of the given vector-valued function are presented. We also prove that this vector-valued function can be used instead of the set-valued map as an objective function of a set-valued optimization problem. Moreover, by giving two examples we show that there is no relationship between the continuity of set-valued map and the continuity of the vector-valued function derived from this set-valued map.