Various efficient points of a set and vektor optimization

In the first part of this work, the notions of binary relation, cone and theirproperties are recalled. Relationships between ordering relations and conesare constructed and multi criteria optimization problems and scalarization aredefined. Furthermore, some weakened convexity of vector valued functions aredefined and theorems about properties of vector-valued functions are givenwithout proofs.In the second part, various efficient points with special names are recalledand interpreted geometrically for a subset of Rn. Which of these definitionsrequire to each other is determined and some examples are given in the caseof absence of requirements. Results are summarized on two schemes.In the third part, on the base of Weierstrass Theorem, necessary and sufficientoptimality conditions of vector-valued functions defined from Rn to Rpare given with respect to the cone orders.In the last part, existence of the optimal elements of a function f : X ! Ystudied where X and Y are normed Banach spaces and some existence theoremsare given. Moreover existence of a suitable scalarization for each minimalelement of a function f : X ! Y proved by means of norms. Scalarization ofthe function f is obtained by using linear mappings which belong to dual conesof the cone and separates minimal points of the set S Y .Keywords: Efficient Point, Ordering Cone, Vector Optimization,Scalarization, Optimality Conditions