An extension of the min-max method for approximate solutions of multi-objective optimization problems

It is a common characteristic of many multiobjective optimization problems that the efficient solution set can only be identified approximately. This study addresses scalarization techniques for solving multiobjective optimization problems. The min-max scalarization technique is considered, and efforts are made to overcome its weaknesses in studying approximate efficient solutions. To this end, two modifications of the min-max scalarization technique are proposed. First, an alternative form of the min-max method is introduced. Additionally, by using slack and surplus variables in the constraints and penalizing violations in the objective function, we obtain easy-to-check conditions for approximate efficiency. The established theorems clarify the relationship between \varepsilon-(weakly and properly) efficient solutions of the multiobjective optimization problem and \epsilon-optimal solutions of the proposed scalarized problems, without requiring any assumptions of convexity.