On the Normalization in Evolutionary Multi-Modal Multi-Objective Optimization Yiping Liu 1 , Hisao Ishibuchi 2 , Gary G. Yen 3 , Yusuke Nojima 1 , Naoki Masuyama 1 , and Yuyan Han 4 1 Department of Computer Science and Intelligent Systems, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan 2 Guangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation, Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China 3 School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, OK 74078, USA 4 School of Computer Science, Liaocheng University, Liaocheng, 252059, China Email: yiping0liu@gmail.com, hisao@sustech.edu.cn, gyen@okstate.edu, nojima@cs.osakafu-u.ac.jp, masuyama@cs.osakafu-u.ac.jp, hanyuyan@lcu-cs.com Abstract—Multi-modal multi-objective optimization problems may have different Pareto optimal solutions with the same objective vector. A number of evolutionary multi-modal multi- objective algorithms have been developed to solve these problems. They aim to search for a Pareto optimal solution set with good diversity in both the objective and decision spaces. Although the normalization in both the objective and decision spaces is very important for these algorithms, there are few studies on this topic. In this paper, we investigate the effect of four normalization methods on two evolutionary multi-modal multi- objective algorithms. Six distance minimization problems are chosen as test problems. The experimental results show that the effect of normalization in evolutionary multi-modal multi- objective optimization is algorithm- and problem-dependent. Index Terms—multi-modal multi-objective evolutionary opti- mization, normalization, objective space, decision space I. I NTRODUCTION An optimization problem, such as job shop scheduling [1] and financial portfolio management [2], may have multiple objectives which conflict with each other. Such a problem is regarded as a multi-objective optimization problem (MOP). There is no single optimal solution for the problem. Instead, it has a set of Pareto optimal solutions, i.e., the Pareto optimal solution set (PS). The image of PS in the objective space is called the Pareto front (PF). Without loss of generality, an MOP with box constraints can be formulated as follows: min f (x) = min(f 1 (x),...,f M (x)), s.t. x ∈ S, (1) where x is an n-dimensional decision vector in the feasible space S, f m (x) is the m-th objective to be minimized (m = This work was supported in part by the National Natural Science Foun- dation of China under Grant 61876075 and Grant 61803192, in part by the Guangdong Provincial Key Laboratory under Grant 2020B121201001, in part by the Program for Guangdong Introducing Innovative and Enterpreneurial Teams under Grant 2017ZT07X386, in part by the Shenzhen Science and Technology Program under Grant KQTD2016112514355531, in part by the Program for University Key Laboratory of Guangdong Province under Grant 2017KSYS008, and in part by Japan Society for the Promotion of Science (JSPS) KAKENHI under Grant JP19K20358. Corresponding Author: Hisao Ishibuchi 1, ..., M ), and M is the number of objectives. S = {x ∈ R n : x lower i ≤ x i ≤ x upper i ,i =1,...,n} where x upper i and x lower i are the lower and upper bounds of x i , respectively. An MOP may have multiple Pareto optimal solutions cor- responding to the same objective vector, which are called equivalent Pareto optimal solutions. In such a case, the MOP is termed as a multi-modal multi-objective optimization problem (MMOP). In recent years, a number of evolutionary multi- modal multi-objective algorithms (EMMAs) have been de- veloped to solve MMOPs. The general goal of EMMAs is to search for a set of Pareto optimal solutions with good diversity both in the objective and decision spaces. Based on the diversity evaluation mechanisms, existing EMMAs can be roughly classified into two groups. One evaluates the diversity in the objective and decision spaces simultaneously, while the other separately. The first group includes Omni-optimizer [3], Niching- covariance matrix adaptation (Niching-CMA) [4], double-niched evolutionary algorithm (DNEA) [5], [6], multi-objective particle swarm optimization algorithm using ring topology and special crowding distance (MO Ring PSO SCD) [7], and multi-modal multi-objective differential evolution optimization algorithm (MMODE) [8]. In the environmental selection, these algorithms usually use the Pareto rank (i.e., the front number according to the non-dominated sorting [9]) as the primary selection criterion. For solutions with the same Pareto rank, they use the objective and decision space density values to select solutions with good diversity. One benefit for these EMMAs is that there exist a larger number of state-of-the-art density evaluation methods to choose. For example, DNEA uses two niche-based sharing functions [10] to estimate the objective and decision space density values, respectively, and then sum them together. In Omni-optimizer, MO Ring PSO SCD, and MMODE, the objective and decision space density values are estimated by the crowding distance [9]. However, one open issue of the EMMAs in this group is how to balance the 978-1-7281-6929-3/20/$31.00 ©2020 IEEE