A Review of Nadir Point Estimation Procedures Using Evolutionary Approaches: A Tale of Dimensionality Reduction

Estimation of the nadir objective vector 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 exact nadir point for more than two objectives is not an easy matter, simply because nadir point demands the knowledge of extreme Paretooptimal 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 increasingly lower dimensional subset of Pareto-optimal solutions. 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 its higher-dimensional counterparts. Kalyanmoy Deb Department of Mechanical Engineering, Indian Institute of Technology Kanpur, PIN 208016, India, e-mail: [email protected] Also Finland Distinguished Professor, Helsinki School of Economics, Finland Kaisa Miettinen Department of Mathematical Information Technology, P.O. Box 35 (Agora), FI-40014 University of Jyväskylä, Finland, e-mail: [email protected]