Preference-Based Nonlinear Normalization for Multiobjective Optimization

Abstract

Normalization is commonly used in multiobjective evolutionary algorithms (MOEAs) in order to handle multiobjective optimization problems with differently-scaled objectives. The goal of normalization is to obtain uniformly-distributed solutions over the entire Pareto front. However, in practice, such a uniform solution set may not be a well-distributed solution set for decision making when the desired distribution of solutions is not uniform. To obtain a well-distributed solution set that meets the desired distribution, in this paper, we propose a preference-based nonlinear normalization method that transforms the objective space based on the probability integral transform theorem. As a result, the use of a standard MOEA to search for uniformly-distributed solutions in the transformed objective space leads to a desired well-distributed solution set. The proposed method is incorporated in three different MOEAs (i.e., a Pareto dominance-based MOEA, a decomposition-based MOEA, and an indicator-based MOEA). Experimental results demonstrate the flexibility and effectiveness of the proposed method. Our code is available at https://github.com/linjunhe/moea-pn .