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

Multi-objective optimization problems arise when more than one objective function is to be minimized over a given feasible region. Unlike the traditional mathematical programming with a single-objective function, an optimal solution in the sense of one that minimizes all the objective functions simultaneously does not necessarily exist in multi-objective optimization problems, and whence, we are troubled with conflicts among objectives in decision-making problems with multiple objectives. Applications of complex programming may be found in Mathematics, engineering, and in many other areas. In earlier works in the field of complex programming problem, all the researchers have considered only the real part of the objective function of the problem as the objective function of the problem neglecting the imaginary part of the objective function, and the constraints of the problem have considered as a cone in the complex space C^n 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