Lattice approach in vector optimization and duality

Vector optimization problems are optimization problems with vector valued objective function. In this thesis, lattice approach for vector optimization problems, depend on notions infimum and supremum, is discussed and results on this subject are compiled from various literature. Firstly, the concept of conlinear space, which forms the basis of the lattice approach, the properties of these spaces, infimal and supremal sets, convex functions in conlinear spaces and their properties are given. With the help of these concepts,upper-closed and self-infimal sets of a given space were obtained as a complete lattice in an appropriate order. Afterwards, a lattice extension of a given optimization problem is obtained with the help of these complete lattices, and the relationships between the solutions of the extended lattice problem and the solutions of the primal problem are examined. In the last section, two types of conjugate duality for vector optimization problems are presented using complete lattices and conjugate functions, and the optimality conditions are examined. Keywords: Set-valued optimization, Lower semi-continuity, Scalarization, Set optimization, Vector optimization.