A new scalarization method for finding the efficient frontier in non-convex multi-objective problems

Abstract

One of the most important issues in multi-objective optimization problems (MOPs) is finding Pareto optimal points on the Pareto frontier. This topic is one of the oldest challenges in science and engineering. Many important problems in engineering need to solve a non-convex multi-objective optimization problem (NMOP) in order to achieve the proper results. Gradient based methods, such as Normal Boundary Intersection (NBI), for solving a MOP require solving at least one optimization problem for each solution point. This method can be computationally expensive with an increase in the number of variables and/or constraints of the optimization problem. Nevertheless, the NBI method is a technique motivated by geometrical intuition to provide a better parameterization of the Pareto set than that provided by other techniques. This parameterization is better in the sense that the points obtained by using the NBI method produce a more even coverage of the Pareto curve and this coverage does not miss the interesting middle part of the Pareto curve.This useful property, provides an incentive to create a new method. The first step in this study is using a modified convex hull of individual minimum (mCHIM) in each iteration. The second step is introducing an efficient scalarization problem in order to find the Pareto points on the Pareto front. It can be shown that the corresponding solutions of the MOP have uniform spread and also weak Pareto optimal points. It is notable that the NBI and proposed methods are independent of the relative scale of different objective functions. However, it is quite possible that obtaining a solution of the NBI method not be Pareto optimal (not even locally). Actually, this method aims at getting boundary points rather than Pareto optimal points that will lead to these points which may or may not be a Pareto optimal point. The effectiveness of this method is demonstrated with various test problems in convex and non-convex MOP cases. After that, a few test instances of the CEC 2009 (Zhang et al. 2008) using the proposed method are studied. Also, the relationship between the optimal solutions of the scalarized problem and the Pareto solutions of the multi-objective optimization problem is presented by several theorems.