A Proportion-Based Selection Scheme for Multi-objective Optimization Liuwei Fu School of Information Engineering Xiangtan University Xiangtan 411105, China Email: 844871556@qq.com Juan Zou (Corresponding author) School of Information Engineering Xiangtan University Xiangtan 411105, China Email: zoujuan@xtu.edu.cn Shengxiang Yang School of Information Engineering Xiangtan University Xiangtan 411105, China School of Computer Science and Informatics De Montfort University Leicester LE1 9BH, U.K. Email: syang@dmu.ac.uk Gan Ruan School of Information Engineering Xiangtan University Xiangtan 411105, China Email: ruangan199332@gmail.com Zhongwei Ma School of Information Science and Engineering Central South University Changsha 410083, China Email: mzw cemo@csu.edu.cn Jinhua Zheng School of Information Engineering Xiangtan University Xiangtan 411105, China Email: jhzheng@xtu.edu.cn Abstract—Classical multi-objective evolutionary algorithms (MOEAs) have been proven to be inefficient for solving multi- objective optimizations problems when the number of objectives increases due to the lack of sufficient selection pressure towards the Pareto front (PF). This poses a great challenge to the design of MOEAs. To cope with this problem, researchers have developed reference-point based methods, where some well- distributed points are produced to assist in maintaining good diversity in the optimization process. However, the convergence speed of the population may be severely affected during the searching procedure. This paper proposes a proportion-based selection scheme (denoted as PSS) to strengthen the convergence to the PF as well as maintain a good diversity of the population. Computational experiments have demonstrated that PSS is sig- nificantly better than three peer MOEAs on most test problems in terms of diversity and convergence. I. I NTRODUCTION Many real world application problems are multi-objective optimization problems (MOPs), which include multiple con- flicting objectives that must be optimized simultaneously. In the evolutionary multi-objective optimization (EMO) com- munity, multi-objective optimization evolutionary algorithms (MOEAs) have benn demonstrated to be effective in solving these problems [1]-[4]. So far, many different EMO algorithms have been developed, such as Pareto-based methods, e.g., NSGA-II [5], SPEA2 [6], decomposition-based approaches, e.g., MOEA/D [7], and indicator-based approaches [8, 9], e.g., HypE [10], SMS-EMOA [11]. However, Pareto-based algorithms more or less lose selection pressure to the PF in the optimization process when solving problems having more than three objectives, i.e., many-objective optimization problems (MaOPs). As a result, the whole performance of MOEAs can be affected by the decrease of convergence. Pareto-based approaches compare solutions according to their dominance relation and density. The nondominated in- dividuals are considered as primarily selected solutions. How- ever, the Pareto-based dominance relationship has encountered great difficulties in MaOPs when the number of obectives in- creases [12, 13, 14]. Therefore, some researchers have focused on modifying the dominance relation to provide sufficient selection pressure towards the PF. Many improved methods have been proposed, such as SPEA2+SDE [15], ǫ-dominance [16, 17], and fuzzy Pareto dominance [18-24]. Indicator-based approaches use a single performance indi- cator to guide the search during the evolutionary process. The indicator-based EA (IBEA) [25] is a pioneer in this group. Recently, the hypervolume [26], Two Arch2 [27] and S metric selection evolutionary algorithm [28] have been proposed. Indicator-based approaches have been demonstrated to be effective in balancing convergence and diversity due to their good theoretical properties. Nevertheless, the computational cost of the used metrics, e.g., hypervolume, grows exponen- tially with an increase in the number of objectives [29]. Decomposition-based methods decompose a problem with multiple objectives into a set of single-objective subproblems, which are then optimized simultaneously using evolutionary algorithms. The diversity of population is maintained by a set of pre-defined well-distributed reference points. The per- pendicular distance between the individual and the reference line is usually used in reference vector-based decomposition methods. Reference lines are obtained by connecting reference points and the origin. Consequently, the convergence speed is affected by the perpendicular distance-based method to some degree, although these methods can balance convergence and diversity.