A modified direction approach for proper efficiency of multiobjective optimization

Abstract

In order to generate properly efficient solutions of a general multiobjective optimization problem, a unified direction approach was introduced by Ghaznavi et al. (Optim. Methods Softw., 2019). However, there are some deficiencies in this work, and a counterexample is given to show that properly efficient solutions of a general multiobjective optimization problem cannot be obtained via optimal solutions of the proposed scalarized problem. In the present paper, by adding slack variables and surplus variables, we modify the direction scheme proposed by Ghaznavi et al. (Optim. Methods Softw., 2019), and achieve necessary and sufficient conditions for (weakly and properly) efficient solutions of a general multiobjective optimization problem via optimal solutions of the corresponding scalarized problem. Further, the efficiency of our method is demonstrated by numerical examples. This work fills some gaps in the work of Ghaznavi et al. (Optim. Methods Softw., 2019).