Directional derivative for vector and set-valued mappings and applications

In this work, an additional condition for a partial ordering cone to define a total ordering cone in vector spaces is given. A process for constructing a total order by using an orthogonal base of real separable Hilbert space is developed. A characterization of total ordering cones is given by using this process. Optimality conditions and solution methods for vector or set valued optimality problems with respect to a total order are presented. A new scalarization method is developed by using the optimality condition obtained with total ordering cones and this method is named as "Successive Weighted Sum Method". This new method is compared with Weighted Sum Method on an optimization problem with respect to some criteria. The process of acquiring a vector from a set is named as vectorization. It is shown that set valued optimization problems can be solved by replacing them by vector valued optimization problems. A generalization of directional derivative to set valued maps is given by using vectorization and the definition of the limit of set valued maps. A method for calculating the set valued directional derivative of some of the set valued maps is developed by using the vector valued function. Some optimality conditions are obtained in terms of the developed calculation method.