Pareto Optimal Solution Analysis of Convex Multi-Objective Programming Problem

The main method of solving multi-objective programming is changing multi-objective programming problem into single objective programming problem, and then get Pareto optimal solution. Conversely, whether all Pareto optimal solutions can be obtained through appropriate method, generally the answer is negative. In this paper, the methods of norm ideal point and membership function are used to solve the multi-objective programming problem. In norm ideal point method, norm and ideal point are given to structure the corresponding single objective programming problem. Then prove that for any Pareto optimal solution there exist weights such that Pareto optimal solution is the optimal solution of the corresponding single objective programming problem. In membership function method, firstly construct membership function for every objective function, then establish the single objective programming problem, after then solve the single objective programming problem, finally prove that for any Pareto optimal solution there exist weights such that the Pareto optimal solution is the optimal solution of the corresponding single objective programming problem. At last, two examples are given to illustrate that the two methods are effective in getting Pareto optimal solution.