A novel two-archive strategy for evolutionary many-objective optimization algorithm based on reference points

Current evolutionary many-objective optimization algorithms face two challenges: one is to ensure population diversity for searching the entire solution space. The other is to ensure quick convergence to the optimal solution set. In this paper, we propose a novel two-archive strategy for evolutionary many-objective optimization algorithm. The uniform archive strategy, based on reference points, is used to keep population diversity in the evolutionary process, and to ensure that an evolutionary algorithm is able to search the entire solution space. The single elite archive strategy is used to ensure that individuals with the best single objective value are able to evolve into the next generation and have more opportunities to generate offspring. This strategy aims to improve the convergence rate. Then this novel two-archive strategy is applied to improving the Non-dominated Sorting Genetic Algorithm (NSGA-III). Simulation experiments are conducted on benchmark test sets and experimental results show that our proposed algorithm with the two-archive strategy has a better performance than other state-of-art algorithms.     

A clustering-ranking method for many-objective optimization

In evolutionary multi-objective optimization, balancing convergence and diversity remains a challenge and especially for many-objective (three or more objectives) optimization problems (MaOPs). To improve convergence and diversity for MaOPs, we propose a new approach: clustering-ranking evolutionary algorithm (crEA), where the two procedures (clustering and ranking) are implemented sequentially. Clustering incorporates the recently proposed non-dominated sorting genetic algorithm III (NSGA-III), using a series of reference lines as the cluster centroid. The solutions are ranked according to the fitness value, which is considered to be the degree of closeness to the true Pareto front. An environmental selection operation is performed on every cluster to promote both convergence and diversity. The proposed algorithm has been tested extensively on nine widely used benchmark problems from the walking fish group (WFG) as well as combinatorial travelling salesman problem (TSP). An extensive comparison with six state-of-the-art algorithms indicates that the proposed crEA is capable of finding a better approximated and distributed solution set.     

An improved evolutionary algorithm for handling many-objective optimization problems

It has been shown that the multi-objective evolutionary algorithms (MOEAs) act poorly in solving many-objective optimization problems which include more than three objectives. The research emphasis, in recent years, has been put into improving the MOEAs to enable them to solve many-objective optimization problems efficiently. In this paper, we propose a new composite fitness evaluation function, in a novel way, to select quality solutions from the objective space of a many-objective optimization problem. Using this composite function, we develop a new algorithm on a well-known NSGA-II and call it FR-NSGA-II, a fast reference point based NSGA-II. The algorithm is evaluated for producing quality solutions measured in terms of proximity, diversity and computational time. The working logic of the algorithm is explained using a bi-objective linear programming problem. Then we test the algorithm using experiments with benchmark problems from DTLZ family. We also compare FR-NSGA-II with four competitive algorithms from the extant literature to show that FR-NSGA-II will produce quality solutions even if the number of objectives is as high as 20.     

A many-objective evolutionary algorithm based on a projection-assisted intra-family election

In recent years, many researchers have put emphasis on the study of how to keep a good balance between convergence and diversity in many-objective optimization. This paper proposes a new many-objective evolutionary algorithm based on a projection-assisted intra-family election. In the proposed algorithm, basic evolution directions are adaptively generated according to the current population and potential evolution directions are excavated in each individual's family. Based on these evolution directions, a strategy of intra-family election is performed in every family and elite individuals are elected as representatives of the specific family to join the next stage, which can enhance the convergence of the algorithm. Moreover, a selection procedure based on angles is used to maintain the diversity. The performance of the proposed algorithm is verified and compared with several state-of-the-art many-objective evolutionary algorithms on a variety of well-known benchmark problems ranging from 5 to 20 objectives. Empirical results demonstrate that the proposed algorithm outperforms other peer algorithms in terms of both the diversity and the convergence of the final solutions set on most of the test instances. In particular, our proposed algorithm shows obvious superiority when handling the problems with larger number of objectives.     

A hybridized angle-encouragement-based decomposition approach for many-objective optimization problems

Due to the large objective space when handling many-objective optimization problems (MaOPs), it is a challenging work for multi-objective evolutionary algorithms (MOEAs) to balance convergence and diversity during the search process. Although a number of decomposition-based MOEAs have been designed for the above purpose, some difficulties are still encountered for tackling some difficult MaOPs. As inspired by the existing decomposition approaches, a new Hybridized Angle-Encouragement-based (HAE) decomposition approach is proposed in this paper, which is embedded into a general framework of decomposition-based MOEAs, called MOEA/D-HAE. Two classes of decomposition approaches, i.e., the angle-based decomposition and the proposed encouragement-based boundary intersection decomposition, are sequentially used in HAE. The first one selects appropriate solutions for association in the feasible region of each subproblem, which is expected to well maintain the diversity of associated solutions. The second one acts as a supplement for the angle-based one under the case that no solution is located in the feasible region of subproblem, which aims to ensure the convergence and explore the boundaries. By this way, HAE can effectively combine their advantages, which helps to appropriately balance convergence and diversity in evolutionary search. To study the effectiveness of HAE, two series of well-known test MaOPs (WFG and DTLZ) are used. The experimental results validate the advantages of HAE when compared to other classic decomposition approaches and also confirm the superiority of MOEA/D-HAE over seven recently proposed many-objective evolutionary algorithms.     

基于最小距离和聚合策略的分解多目标进化算法

针对基于帕累托（Pareto）支配的多目标进化算法在解决高维问题时选择压力降低,以及基于分解的多目标进化算法在提高收敛性和分布性的同时降低了种群多样性的问题,提出了一种基于最小距离和聚合策略的分解多目标进化算法。首先,使用基于角度分解的技术将目标空间分解为指定个数的子空间来提高种群的多样性;然后,在生成新解的过程中加入基于聚合的交叉邻域方法,使生成的新解更接近于父代解;最后,分两阶段在每个子空间内基于最小距离和聚合策略来选择解以提高收敛性和分布性。为了验证所提算法的可行性,采用标准测试函数ZDT和DTLZ进行仿真实验,结果表明所提算法的总体性能均优于经典的基于分解的多目标进化算法（MOEA/D）、MOEA/D-DE、NSGA-Ⅲ和GrEA。可见,所提算法在提高多样性的同时可以有效平衡收敛性和多样性。     

A novel metric based on changes in pareto domination ratio for objective reduction of many-objective optimization problems

Abstract

Although many-objective optimisation can be simplified through reduction of redundant objectives, algorithms that perform this reduction still lack a convenient method of evaluation. In this paper, we address this deficiency by proposing a new method of evaluation, on the basis of changes in the Pareto-domination ratio after a reduction has occurred. Experimental results have shown that the proposed method can perform non- redundant objective set evaluation more accurately than existing evaluation methods, and also does not need the true Pareto front beforehand.     

A new decomposition based evolutionary algorithm with uniform designs for many-objective optimization

For many-objective optimization problems, how to get a set of solutions with good convergence and diversity is a difficult and challenging work. In this paper, a new decomposition based evolutionary algorithm with uniform designs is proposed to achieve the goal. The proposed algorithm adopts the uniform design method to set the weight vectors which are uniformly distributed over the design space, and the size of the weight vectors neither increases nonlinearly with the number of objectives nor considers a formulaic setting. A crossover operator based on the uniform design method is constructed to enhance the search capacity of the proposed algorithm. Moreover, in order to improve the convergence performance of the algorithm, a sub-population strategy is used to optimize each sub-problem. Comparing with some efficient state-of-the-art algorithms, e.g., NSGAII-CE, MOEA/D and HypE, on six benchmark functions, the proposed algorithm is able to find a set of solutions with better diversity and convergence.     

Ensemble mating selection in evolutionary many-objective search

Traditional multi-objective evolutionary algorithms have encountered difficulties when handling many-objective problems. This is due to the loss of selection pressure incurred by the growing size of objective space. A variety of environmental selection operators have been proposed to address the issue, each has its distinct benefits and drawbacks. We develop a novel ensemble framework to enhance the effectiveness and robustness of many-objective optimization. The framework incorporates multiple environmental selection operators to guide the search, which are then viewed as voters to construct a mating pool. We design an ensemble mating selection strategy that makes decisions based on the preference information provided by the voters: individuals elected by more voters will be assigned larger possibilities to enter the mating pool. By doing so, high quality offspring can be reproduced from the elected promising candidates. To accommodate the multiple selection operators for voting, the framework maintains multiple parallel populations, where each population is updated by one of the selection operators. An instantiation of the framework with three popular operators is presented as a prime example. Extensive experiments have been conducted on a number of many-objective problems to examine the effectiveness of the proposed approach. Experimental results show that the mating selection strategy is capable of improving the quality of the obtained solution set.     

Binary search based boundary elimination selection in many-objective evolutionary optimization

In many-objective optimization, the balance between convergence and diversity is hard to maintain, while the dominance resistant solutions (DRSs) could further harm the balance particularly in high-dimensional objective space. Thus, this paper proposes a novel selection strategy – boundary elimination selection based on binary search (called BESBS), trying to avoid the impact of DRSs during the optimization and achieve a good balance between the convergence and diversity simultaneously. During the environmental selection, the binary search (BS) is used to adaptively adjust the ϵ value in the ϵ-dominance relationship and assist in detecting the well-distributed neighbors for the elite solutions. Then the ϵ value obtained by BS is used for serving the boundary elimination selection (BES) to guarantee the stability of the elite population. To improve the convergence, BES is mainly designed to select individuals approximating to the ideal point. By modifying the fitness of solutions and choosing solutions in terms of the shuffled sequence of objective axis, the DRSs will be eliminated during the selection. Thus, BESBS could achieve a good balance between the convergence and diversity and avoid the impact from DRSs simultaneously. From a series of experiments with 35 instances, the experimental results have shown that BESBS is competitive against 8 state-of-art many-objective evolutionary algorithms.     

A radial space division based evolutionary algorithm for many-objective optimization

In evolutionary many-objective optimization, diversity maintenance plays an important role in pushing the population towards the Pareto optimal front. Existing many-objective evolutionary algorithms mainly focus on convergence enhancement, but pay less attention to diversity enhancement, which may fail to obtain uniformly distributed solutions or fall into local optima. This paper proposes a radial space division based evolutionary algorithm for many-objective optimization, where the solutions in high-dimensional objective space are projected into the grid divided 2-dimensional radial space for diversity maintenance and convergence enhancement. Specifically, the diversity of the population is emphasized by selecting solutions from different grids, where an adaptive penalty based approach is proposed to select a better converged solution from the grid with multiple solutions for convergence enhancement. The proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on a variety of benchmark test problems. Experimental results demonstrate the competitiveness of the proposed algorithm in terms of both convergence enhancement and diversity maintenance.     

Many-objective evolutionary optimization based on reference points

Many-objective optimization problems are common in real-world applications, few evolutionary optimization methods, however, are suitable for solving them up to date due to their difficulties. A reference points-based evolutionary algorithm (RPEA) was proposed in this paper to solve many-objective optimization problems. The aim of this study is to exploit the potential of the reference points-based approach to strengthen the selection pressure towards the Pareto front while maintaining an extensive and uniform distribution among solutions. In RPEA, a series of reference points with good performances in convergence and distribution are continuously generated according to the current population to guide the evolution. Furthermore, superior individuals are selected based on the evaluation of each individual by calculating the distances between the reference points and the individual in the objective space. The proposed algorithm was applied to seven benchmark optimization problems and compared with ɛ-MOEA, HypE, MOEA/D and NSGA-III. The results empirically show that the proposed algorithm has a good adaptability to problems with irregular or degenerate Pareto fronts, whereas the other reference points-based algorithms do not. Moreover, it outperforms the other four in 8 out of 21 test instances, demonstrating that it has an advantage in obtaining a Pareto optimal set with good performances.     

Newtonian clustering: An approach based on molecular dynamics and global optimization

Given a data set, a dynamical procedure is applied to the data points in order to shrink and separate, possibly overlapping clusters. Namely, Newton's equations of motion are employed to concentrate the data points around their cluster centers, using an attractive potential, constructed specially for this purpose. During this process, important information is gathered concerning the spread of each cluster. In succession this information is used to create an objective function that maps each cluster to a local maximum. Global optimization is then used to retrieve the positions of the maxima that correspond to the locations of the cluster centers. Further refinement is achieved by applying the EM-algorithm to a Gaussian mixture model whose construction and initialization is based on the acquired information. To assess the effectiveness of our method, we have conducted experiments on a plethora of benchmark data sets. In addition we have compared its performance against four clustering techniques that are well established in the literature.     

基于聚类分析的增强型蚁群算法

针对蚁群算法存在的早熟收敛、搜索时间长等不足,提出一种增强型蚁群算法.该算法构建了一优解池,保存到当前迭代为止获得的若干优解,并提出一种基于邻域的聚类算法,通过对优解池中的元素聚类,捕获不同的优解分布区域.该算法交替使用不同簇中的优解更新信息素,兼顾考虑了搜索的强化性和分散性.针对典型的旅行商问题进行仿真实验,结果表明该算法获得的解质量高于已有的蚁群算法.     

A new approach to clustering

Estimation theory is used to derive a new approach to the clustering problem. The new method is a unification of centroid and mode estimation, achieved by considering the effect of spatial scale on the estimator. The result is a multiresolution method which spans a range of spatial scales, giving enhanced robustness both to noise in the data and to changes of scale in the data, by using comparison between scales as a test of cluster validity. Iterative and non-iterative algorithms based on the new estimator are presented and are shown to be more accurate than simple scale-space filtering in identifying and locating the cluster centres from noisy test data. Results from a wide range of applications are used to illustrate the power and versatility of the new method.     

一类基于物种迁移优化的进化算法

借鉴自然界中的物种迁移机制,提出一类基于物种迁移优化的进化算法．该算法是根据生态系统中物种分布的迁移模型而提出的一种优化算法．参考其他智能算法的思想,通过物种迁移实现信息交换和共享,从而完成进化过程．讨论了物种迁移优化算法的基本原理和实现过程,同时进行一些基准函数的性能测试．实验结果表明所提出的算法是有效的,具有一定的参考和应用价值．     

Many-objective evolutionary algorithm based on adaptive weighted decomposition

Decomposition is a representative method for handling many-objective optimization problems with evolutionary algorithms. Classical decomposition scheme relies on a set of uniformly distributed reference vectors to divide the objective space into multiple subregions. This scheme often works poorly when the problem has an irregular Pareto front due to the inconsistency between the distribution of reference vectors and the shape of Pareto fronts. We propose in this paper an adaptive weighted decomposition based many-objective evolutionary algorithm to tackle complicated many-objective problems whose Pareto fronts may or may not be regular. Unlike traditional decomposition based algorithms that use a pre-defined set of reference vectors, the reference vectors in the proposed algorithm are produced from the population during the search. The experiments show that the performance of the proposed algorithm is competitive with other state-of-the-art algorithms and is less-sensitive to the irregularity of the Pareto fronts.     

A hybrid approach for data clustering based on modified cohort intelligence and K-means

Clustering is an important and popular technique in data mining. It partitions a set of objects in such a manner that objects in the same clusters are more similar to each another than objects in the different cluster according to certain predefined criteria. K-means is simple yet an efficient method used in data clustering. However, K-means has a tendency to converge to local optima and depends on initial value of cluster centers. In the past, many heuristic algorithms have been introduced to overcome this local optima problem. Nevertheless, these algorithms too suffer several short-comings. In this paper, we present an efficient hybrid evolutionary data clustering algorithm referred to as K-MCI, whereby, we combine K-means with modified cohort intelligence. Our proposed algorithm is tested on several standard data sets from UCI Machine Learning Repository and its performance is compared with other well-known algorithms such as K-means, K-means++, cohort intelligence (CI), modified cohort intelligence (MCI), genetic algorithm (GA), simulated annealing (SA), tabu search (TS), ant colony optimization (ACO), honey bee mating optimization (HBMO) and particle swarm optimization (PSO). The simulation results are very promising in the terms of quality of solution and convergence speed of algorithm.     

直觉模糊集相似度遗传算法求解多目标车间调度问题

为了提高高维多目标置换流水车间调度问题的求解质量,提出基于直觉模糊集相似度的遗传算法(similarity of intuitionistic fuzzy sets GA,SIFS_GA).算法中分别将参考解和Pareto解映射为参考解直觉模糊集和Pareto解直觉模糊集.计算两个集合之间的直觉模糊相似度,用以判断Pareto解的优劣.以直觉模糊集相似度值引导多目标遗传算法进化.对6个CEC标准测试集与10个流水车间调度测试实例进行仿真实验,结果表明SIFS_GA算法性能优于常用的多目标优化算法,且可以有效解决多目标置换流水车间调度问题,尤其在解决规模较大的问题上是一种有效方法.     

A new clustering method based on the bio-inspired cuttlefish optimization algorithm

Most of the well-known clustering methods based on distance measures, distance metrics and similarity functions have the main problem of getting stuck in the local optima and their performance strongly depends on the initial values of the cluster centers. This paper presents a new approach to enhance the clustering problems with the bio-inspired Cuttlefish Algorithm (CFA) by searching the best cluster centers that can minimize the clustering metrics. Various UCI Machine Learning Repository datasets are used to test and evaluate the performance of the proposed method. For the sake of comparison, we have also analysed several algorithms such as K-means, Genetic Algorithm and the Particle Swarm Optimization (PSO) Algorithm. The simulations and obtained results demonstrate that the performance of the proposed CFA-Clustering method is superior to the other counterpart algorithms in most cases. Therefore, the CFA can be considered as an alternative stochastic method to solve clustering problems.