Slack-based generalized Tchebycheff norm scalarization approaches for solving multiobjective optimization problems

In this research, we propose two scalarization techniques for solving multiobjective optimization problems (MOPs). Based on the generalized Tchebycheff norm, the achieved scalarized approaches are provided by applying slack and surplus variables. We obtain results related to the presented approaches by varying the range of parameters. These results give an overview of the relationships between (weakly, properly) Pareto optimal solutions of the MOP and optimal solutions of the presented scalarized problems. We remark that all the provided theorems do not require any convexity assumption for objective functions. The main advantage of the generalized Tchebycheff norm approach is that, unlike most scalarization approaches, there is no gap between necessary and sufficient conditions for (weak, proper) Pareto optimality. Moreover, this approach, in different results, shows necessary and sufficient conditions for Pareto optimality.

Automated summary

In this research, two scalarization techniques are proposed to solve multiobjective optimization problems (MOPs). These techniques utilize the generalized Tchebycheff norm and incorporate slack and surplus variables to provide scalarized approaches. By varying the range of parameters, we obtain results that elucidate the relationship between (weakly, properly) Pareto optimal solutions of the MOP and optimal solutions of the scalarized problems. Importantly, the theorems presented in this study do not require any convexity assumption for objective functions. The main advantage of the generalized Tchebycheff norm approach is that it eliminates the gap between necessary and sufficient conditions for (weak, proper) Pareto optimality and provides necessary and sufficient conditions for Pareto optimality in different results.