Increasing the Density of Multi-Objective Multi-Modal Solutions using Clustering and Pareto Estimation Techniques

For continuous multi-objective optimization problems there exists an infinite number of solutions on the Paretooptimal front. A multi-objective evolutionary algorithm attempts to find a representative set of the Pareto-optimal solutions. In the case of multi-objective multi-modal problems, there exist multiple decision vectors which map to identical objective vectors on Pareto front. Many multi-objective evolutionary algorithms fail to find and preserve all of the multi-modal solutions in the non-dominated solutions set. Finding more of the available multi-modal solutions would give the decision maker a greater selection when choosing between solutions. In this paper, we present an extended version of the Pareto estimation method, to increase the density of the multi-objective multi-modal solutions. The method uses clustering analysis to identify and separate different clusters in the decision variables space which correspond to the multi-modal Pareto optimal solutions. Then Pareto estimation procedure is employed for these individual clusters, there by increasing the density of available multi-modal solutions. The proposed method has been tested on experimental test functions and is shown to be successful.