On the use of the $L_{p}$ distance in reference point-based approaches for multiobjective optimization

Abstract

Reference point-based methods are very useful techniques for solving multiobjective optimization problems. In these methods, the most commonly used achievement scalarizing functions are based on the Tchebychev distance (minmax approach), which generates every Pareto optimal solution in any multiobjective optimization problem, but does not allow compensation among the deviations to the reference values given that it minimizes the value of the highest deviation. At the same time, for any $1 \le p \le \infty$, compromise programming minimizes the $L_p$ distance to the ideal objective vector from the feasible objective region. Although the ideal objective vector can be replaced by a reference point, achievable reference points are not supported by this approach, and special care must be taken in the unachievable case. In this paper, for $1 \le p < \infty$, we propose a new scheme based on the $L_p$ distance, in which different single-objective optimization problems are designed and solved depending on the achievability of the reference point. The formulation proposed allows different compensation degrees among the deviations to the reference values. It is proven that, in the achievable case, any optimal solution obtained is efficient, and, in the unachievable one, it is at least weakly efficient, although it is assured to be efficient if an augmentation term is added to the new formulation. Besides, we suggest an interactive algorithm where the new formulation is embedded. Finally, we show the empirical advantages of the new formulation by its application to both numerical problems and a real multiobjective optimization problem, for achievable and unachievable reference points.