Nadir Point Estimation Using Evolutionary Approaches: Better Accuracy and Computational Speed Through Focused Search

Estimation of the nadir objective vector representing worst objective function values in the set of Pareto-optimal solutions is an important task, particularly for multi-objective optimization problems having more than two conflicting objectives. Along with the ideal point, nadir point can be used to normalize the objectives so that multi-objective optimization algorithms can be used more reliably. The knowledge of the nadir point is also a pre-requisite to many multiple criteria decision making methodologies. Moreover, nadir point is useful for an aid in interactive methodologies and visualization softwares catered for multi-objective optimization. However, the computation of an exact nadir point for more than two objectives is not an easy matter, simply because the nadir point demands the knowledge of extreme Pareto-optimal solutions. In the past few years, researchers have proposed several nadir point estimation procedures using evolutionary optimization methodologies. In this paper, we review the past studies and reveal an interesting chronicle of events in this direction. To make the estimation procedure computationally faster and more accurate, the methodologies were refined one after the other by mainly focusing on finding smaller and still sufficient subset of Pareto-optimal solutions to facilitate estimating the nadir point. Simulation results on a number of numerical test problems demonstrate better efficacy of the approach which aims to find only the extreme Pareto-optimal points compared to other two approaches.