Abstarct: This thesis first introduces the definitions and fundamental theorems relative to vector optimization and multiobjective optimization‎. ‎Considering the importance of proper efficeint solutions‎, ‎we define Geoffrion proper efficiency‎, ‎Benson proper efficiency‎, ‎and Henig proper efficiency and propose some methods for solving multobjective problems‎. ‎Approximate solutions play an important role in vector optimization when there is no exact solution‎. Therefor we introuduce approximate efficiency‎, ‎as named as E-efficiency is described baxsed on upper comperhensive set‎. ‎Then according to the properties of improvemet sets‎, ‎we present different kinds of E-efficiency such as E-efficiency via map φ_(q,E), ‎Benson proper E-efficiency ‎and E-efficiency via map ∆_(-K) and study their properties‎. ‎Next‎, ‎by defining set-valued maps‎, ‎we introduce optimization via these maps and propose concept of subconvexlikeness for set-valued maps and express some theorems under assumption of E-subconvexlikeness‎. ‎Then provide Lagrangian multipliers theorems of Benson proper E-efficiency and in the same process, study weak E-optimal solutions for vector optimization and some relative theorems including scalarization theorem and Lagrange multiplier theorem‎. ‎Finally, we introduce weak E-saddle points for set-valued Lagrangian maps and weak E-duality‎, ‎followed by a discussion concerning the relative theorems and their properties.