on characterizing efficient and properly efficient solutions for multi- objective programming problems in a complex space

In this paper, a complex non- linear programming problem with the two parts (real and imaginary) is considered. the efficient and proper efficient solutions in terms of optimal solutions of related appropriate scalar optimization problems are characterized. also, the kuhn-tuckers' conditions for efficiency and proper efficiency are derived. this paper is divided into two independently parts: the first provides the relationships between the optimal solutions of a complex single-objective optimization problem and solutions of two related real programming problems. the second part is concerned with the theory of a multi-objective optimization in complex space.