Generalized hyperplane methods for characterizing ℝ-extreme points and trade-off rates for multiobjective optimization problems

Recently, as a generalization of the well-known scalarizing methods for characterizing Pareto optimal solutions of multiobjective optimization problems (MOPs), a fully integrated scalarizing method, called the hyperplane method has been introduced by the authors. In this paper, they further propose the generalized hyperplane method for characterizing ℝ-extreme points which can be viewed as a generalized concept of Parteo optimality to the MOPs, and investigate the properties of the generalized hyperplane problems in detail. Moreover, a meaningful formula which relates the trade-off rates within ℝ-extreme point set to the Lagrange multipliers of the generalized hyperplane problems is derived by realizing that trade-off information is significantly useful in almost all the interactive multiobjective decision making methods. Finally, the results developed in this paper are illustrated by a numerical example.