Estimating nadir point in multi-objective optimization using mobile reference points

Nadir point represents important information to multi-objective optimization practitioners. Along with the ideal point, the nadir point: (1) provides information about the ranges of the objectives at the Pareto optimality stage, (2) helps the decision maker to easily state his/her preferences, (3) facilitates the visualization of Pareto optimal solutions for highly dimension multi-objective problems, etc. Contrary to the ideal point which can be easily computed by optimizing each objective individually over the search space, the nadir point is constructed from worst objective function values of Pareto optimal solutions which makes the accurate estimation of the nadir objective values a difficult task especially when the number of objective functions increases. In this paper, we propose a new memetic preference-based multi-objective evolutionary algorithm, termed MR-NSGA-II_N, to estimate the nadir point. The basic idea is to use extreme solutions from the best non-dominated front as mobile reference points. The mobile reference points are updated in every generation by means of a gradient-based local search procedure in order to speed up the convergence towards the Pareto optimal extreme solutions. The performance assessment of MR-NSGA-II_N is carried out on a set of three-to twenty-objective unconstrained/constrained linear/non-linear problems. The proposed approach has shown competitive and better results when compared to other recently proposed nadir point estimation approaches.