Search

Vectorization is a technique that replaces a set-valued optimization problem with a vector optimization problem. In this work, by using an extension of the Gerstewitz function, a vectorizing function is defined to replace a given set-valued optimization problem with respect to the set less order relation. Some properties of this function are studied. Moreover, relationships between a set-valued optimization problem and a vector optimization problem, derived via vectorization of this set-valued optimization problem, are examined. Furthermore, necessary and sufficient optimality conditions are presented without any convexity assumption. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work, it is proved that weak exhausters of a positively homogeneous function can be reduced if weak subdifferential/superdifferential can be represented as the sum of a subset of it and weak subdifferential of zero function. It is also shown that weak exhausters can be reduced if this subset is the set of minimal elements of weak subdifferential/superdifferential with respect to weak subdifferential of zero function. At the end, some optimality conditions are given using reduced weak exhausters. [ABSTRACT FROM AUTHOR]

Abstract: The characterization of total ordering cones of was given with some properties and optimality conditions in Küçük et al. (2011) . In addition, total ordering cones were used to derive a vector valued function from a special class of set valued mappings in Küçük et al. (2012) . In this study, we give a method for construction of a total ordering cone in a separable Hilbert space by using an orthogonal base. Moreover, we show that every total order can be represented by such a cone. The relationship between the notion of total ordering cone and the notion of vectorization of some set valued mappings are given and some results are obtained. [Copyright &y& Elsevier]

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” (Küçük et al., 2011 ). 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. [Copyright &y& Elsevier]

In this paper, we introduce a construction method of total ordering cone on $${\mathbb{R}^n}$$ . It is shown that any total ordering cone on $${\mathbb{R}^n}$$ is isomorphic to the cone $${\mathbb{R}^n_{lex}}$$ . Existence of a total ordering cone that contain given cone with a compact base is shown. By using this cone, a solving method of vector and set valued optimization problems is presented. [ABSTRACT FROM AUTHOR]