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

Kucuk M., Soyertem M., KÜÇÜK Y., ATASEVER GÜVENÇ İ.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.385, no.1, pp.285-292, 2012 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 385 Issue: 1
  • Publication Date: 2012
  • Doi Number: 10.1016/j.jmaa.2011.06.045
  • Journal Name: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.285-292
  • Keywords: The Successive Weighted Sum Method, The Weighted Sum Method, Vectorization, Scalarization, Set-valued optimization, Total order
  • Anadolu University Affiliated: Yes

Abstract

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" (Kucuk 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 setvalued 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. (C) 2011 Elsevier Inc. All rights reserved.