How to Make Multi-Objective Evolutionary Algorithms Invariant to Monotonically Increasing Transformation of Objective Functions

Yutaro Yamada, Kento Uchida, Shinichi Shirakawa

Abstract: Invariance properties are important for evolutionary algorithms to ensure consistent behavior and performance on broader classes of objective functions. Owing to the ranking-based transformation, several evolutionary algorithms for single-objective optimization possess the invariance property to monotonically increasing transformation of objective functions, which reduces parameter tuning costs. In contrast, in multi-objective evolutionary algorithms (MOEAs), the monotonically increasing transformation of objective functions often change the behavior of existing methods and degrade their performance because they can change the shape of the Pareto front to intractable forms, such as concave and discontinuous shapes. This study proposes the ranking-based transformation (RT) framework that gives the invariance property to monotonically increasing transformations to existing MOEAs. After MOEA generates solutions following its original procedure, RT framework computes the utility values of the solutions for each objective function. These utility values are given by a non-increasing transformation of their ranking on the unbounded archive of non-dominated solutions. RT framework then updates the population so that the MOEA acquires the Pareto optimal solutions on the utility functions. The experimental results of the bi-objective benchmark problems show that RT framework can resolve the performance deterioration of MOEA/D-DE and NSGA-II under monotonically increasing transformations.