Proper minimality in the set-valued optimization

In this study, the basic definitions, theorems and properties required for vector-valued and set-valued optimization problems are given. In particular, differences between optimality concepts such as minimality, proper minimality and strong minimality, which are the main purpose of the thesis, were reminded. In this study, proper minimality definition is given via an expanded cone to preserve the ordering structure for the solution of set-valued optimization problems. Some optimality conditions on m_1 - minimality and m_1-maximality are obtained, and these results are compared with the conditions in vector optimization. Additionally, it is seen that the m_1 - maximality definition is equivalent to the maximality definition defined for vectors. Strong minimality for $m_1$ order is also studied and a sufficient condition is proved. Finally, the differences between the definitions and the applications of the obtained properties illustrated with geometric examples are examined. In this version, the main points of the study focusing on the concept of proper minimality defined for set-valued optimization problems and the equivalents of this concept in vector optimization are emphasized.