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

Optimality Conditions of Interval-Valued Optimization Problems by Using Subdifferantials

“…But they aren't compatible with addition. For example, although ≼ , + ⋠ + for = [3,4], = [5,6], = [−1,3]. ≼ is reflexive, transitive and antisymmetric relation, that is partial order relation.…”

“…Any two intervals may not be compared according to these order relations because order relation ≼ is not a total order relation. For example, = [3,5] and = [4,6]. Now, we give some basic properties of order relations ≼ and ≺.…”

“…Hernández and Rodríguez-Marín [1], Karaman et al [2] and Xu and Li [3] obtained some scalarization methods for set optimization. The first vectorization in the optimization theory was used by Küçük et al [4,5]. Jahn [6] and Karaman et al [7] obtained some optimality conditions for set optimization by using vectorization.…”

Approximate solutions for set optimization with an order cone that has nonempty quasirelative interiors

“…Set-valued optimization is the promotion of vector optimization that has applications in many aspects, such as engineering, cybernetics, military, finance, etc. The research on setvalued optimization has achieved fruitful results (see [6,9,13,15]). The study of set optimization has been investigated by some authors (see [11,12,14,21,23]).…”

A vectorization for nonconvex set-valued optimization

“…Vectorization based on total ordering cones was first introduced by Küçük et al [17,19]. They showed that a set-valued optimization problem can be represented as a vector-valued problem.…”