Reference direction based immune clone algorithm for many-objective optimization

In this paper, a new preference multi-objective optimization algorithm called immune clone algorithm based on reference direction method (RD-ICA) is proposed for solving many-objective optimization problems. First, an intelligent recombination operator, which performs well on the functions comprising many parameters, is introduced into an immune clone algorithm so as to explore the potentially excellent gene segments of all individuals in the antibody population. Second, a reference direction method, a very strict ranking based on the desire of decision makers (DMs), is used to guide selection and clone of the active population. Then a light beam search (LBS) is borrowed to pick out a small set of individuals filling the external population. The proposed method has been extensively compared with other recently proposed evolutionary multi-objective optimization (EMO) approaches over DTLZ problems with from 4 to 100 objectives. Experimental results indicate RD-ICA can achieve competitive results.     

基于R2指标和参考向量的高维多目标进化算法

在高维多目标优化中, 不同的优化问题存在不同形状的Pareto前沿(PF), 而研究表明大多数多目标进化算法(Multi-objective evolutionary algorithms, MOEAs) 在处理不同的优化问题时普适性较差. 为了解决这个问题, 本文提出了一个基于R2指标和参考向量的高维多目标进化算法(An R2 indicator and reference vector based many-objective optimization evolutionary algorithm, R2-RVEA). R2-RVEA基于Pareto支配选取非支配解来指导种群进化, 仅当非支配解的数量超过种群规模时, 算法进一步采用种群分解策略和R2指标选择策略进行多样性管理. 通过大量的实验证明, 本文提出的算法在处理不同形状的PF时具有良好的性能.     

进化多目标优化算法研究

进化多目标优化主要研究如何利用进化计算方法求解多目标优化问题,已经成为进化计算领域的研究热点之一.在简要总结2003年以前的主要算法后,着重对进化多目标优化的最新进展进行了详细讨论.归纳出当前多目标优化的研究趋势,一方面,粒子群优化、人工免疫系统、分布估计算法等越来越多的进化范例被引入多目标优化领域,一些新颖的受自然系统启发的多目标优化算法相继提出;另一方面,为了更有效的求解高维多目标优化问题,一些区别于传统Pareto占优的新型占优机制相继涌现;同时,对多目标优化问题本身性质的研究也在逐步深入.对公认的代表性算法进行了实验对比.最后,对进化多目标优化的进一步发展提出了自己的看法.     

采用放松支配关系的高维多目标微分进化算法

为了提高进化算法在求解高维多目标优化问题时的收敛性和多样性,提出了采用放松支配关系的高维多目标微分进化算法。该算法采用放松的Pareto支配关系,以增加个体的选择压力;采用群体和外部存储器协同进化的方案,并通过混合微分变异算子,生成子代群体;采用基于指标的方法计算个体的适应度并对群体进行更新;采用基于Lp范数（0相似文献   

MCEDA: A novel many-objective optimization approach based on model and clustering

To solve many-objective optimization problems (MaOPs) by evolutionary algorithms (EAs), the maintenance of convergence and diversity is essential and difficult. Improved multi-objective optimization evolutionary algorithms (MOEAs), usually based on the genetic algorithm (GA), have been applied to MaOPs, which use the crossover and mutation operators of GAs to generate new solutions. In this paper, a new approach, based on decomposition and the MOEA/D framework, is proposed: model and clustering based estimation of distribution algorithm (MCEDA). MOEA/D means the multi-objective evolutionary algorithm based on decomposition. The proposed MCEDA is a new estimation of distribution algorithm (EDA) framework, which is intended to extend the application of estimation of distribution algorithm to MaOPs. MCEDA was implemented by two similar algorithm, MCEDA/B (based on bits model) and MCEDA/RM (based on regular model) to deal with MaOPs. In MCEDA, the problem is decomposed into several subproblems. For each subproblem, clustering algorithm is applied to divide the population into several subgroups. On each subgroup, an estimation model is created to generate the new population. In this work, two kinds of models are adopted, the new proposed bits model and the regular model used in RM-MEDA (a regularity model based multi-objective estimation of distribution algorithm). The non-dominated selection operator is applied to improve convergence. The proposed algorithms have been tested on the benchmark test suite for evolutionary algorithms (DTLZ). The comparison with several state-of-the-art algorithms indicates that the proposed MCEDA is a competitive and promising approach.     

A many-objective evolutionary algorithm based on hyperplane projection and penalty distance selection

For many-objective optimization problems, due to the low selection pressure of the Pareto-dominance relation and the ineffectivity of diversity maintenance scheme in the environmental selection, the current Pareto-dominance based multi-objective evolutionary algorithms (MOEAs) fail to balance between convergence and diversity. This paper proposes a many-objective evolutionary algorithm based on hyperplane projection and penalty distance selection (we call it MaOEA-HP). Firstly, the normalization method is used to construct an unit hyperplane and the population is projected onto the unit hyperplane. Then, a harmonic average distance is applied to calculate the crowding density of the projected points on the unit hyperplane. Finally, the perpendicular distance from the individual to the hyperplane as convergence information is added into the diversity maintenance phase, and a penalty distance selection scheme is designed to balance between convergence and diversity of solutions. Compared with six state-of-the-art many-objective evolutionary algorithms, the experimental results on two well-known many-objective optimization test suites show that MaOEA-HP has more advantage than the other algorithms, could improve the convergence and ensure the uniform distribution.     

Towards faster convergence of evolutionary multi-criterion optimization algorithms using Karush Kuhn Tucker optimality based local search

Evolutionary multi-criterion optimization (EMO) algorithms emphasize non-dominated and less crowded solutions in a population iteratively until the population converges close to the Pareto optimal set. During the search process, non-dominated solutions are differentiated only by their local crowding or contribution to hypervolume or using a similar other metric. Thus, during evolution and even at the final iteration, the true convergence behavior of each non-dominated solutions from the Pareto optimal set is unknown. Recent studies have used Karush Kuhn Tucker (KKT) optimality conditions to develop a KKT Proximity Measure (KKTPM) for estimating proximity of a solution from Pareto optimal set for a multi-objective optimization problem. In this paper, we integrate KKTPM with a recently proposed EMO algorithm to enhance its convergence properties towards the true Pareto optimal front. Specifically, we use KKTPM to identify poorly converged non-dominated solutions in every generation and apply an achievement scalarizing function based local search procedure to improve their convergence. Assisted by the KKTPM, the modified algorithm is designed in a way that maintains the total number of function evaluations as low as possible while making use of local search where it is most needed. Simulations on both constrained and unconstrained multi- and many objectives optimization problems demonstrate that the hybrid algorithm significantly improves the overall convergence properties. This study brings evolutionary optimization closer to mainstream optimization field and should motivate researchers to utilize KKTPM measure further within EMO and other numerical optimization algorithms.     

基于ε占优的正交多目标差分演化算法研究

演化多目标优化是目前演化计算中热门研究方向之一.但是,要设计一种高效、鲁棒的演化多目标优化算法,使其找到接近最优和完整的非劣解集是一项很困难的任务.为了能有效求解多目标优化问题,提出了一种新的多目标差分演化算法.新算法具有如下特征:1)利用正交实验设计和连续空间量化的方法产生初始群体,使得初始群体中的个体可以均匀分布于搜索空间,并且可以使好的个体在演化过程中得到利用;2)采用Archive群体保存非劣解,并利用ε占优方法更新Archive群体,从而可以使算法较快获得分布很好的Pareto解集;3)为了加快算法收敛,提出一种基于随机选择和精英选择的混合选择机制.通过8个标准测试函数对新算法进行测试,并与其他一些多目标演化算法进行比较,其结果表明新算法可以有效逼近真实Pareto前沿且分布均匀,并且在收敛性和多样性的求解精度和稳     

A niche-elimination operation based NSGA-III algorithm for many-objective optimization

Decomposition-based multi-objective evolutionary algorithms have been found to be very promising for many-objective optimization. The recently presented non-dominated sorting genetic algorithm III (NSGA-III) employs the decomposition idea to efficiently promote the population diversity. However, due to the low selection pressure of the Pareto-dominance relation the convergence of NSGA-III could still be improved. For this purpose, an improved NSGA-III algorithm based on niche-elimination operation (we call it NSGA-III-NE) is proposed. In the proposed algorithm, an adaptive penalty distance (APD) function is presented to consider the importance of convergence and diversity in the different stages of the evolutionary process. Moreover, the niche-elimination operation is designed by exploiting the niching technique and the worse-elimination strategy. The niching technique identifies the most crowded subregion, and the worse-elimination strategy finds and further eliminates the worst individual. The proposed NSGA-III-NE is tested on a number of well-known benchmark problems with up to fifteen objectives and shows the competitive performance compared with five state-of-the-art decomposition-based algorithms. Additionally, a vector angle based selection strategy is also proposed for handling irregular Pareto fronts.     

Preference representation using Gaussian functions on a hyperplane in evolutionary multi-objective optimization

Many-objective optimization has attracted much attention in evolutionary multi-objective optimization (EMO). This is because EMO algorithms developed so far often degrade their search ability for optimization problems with four or more objectives, which are frequently referred to as many-objective problems. One of promising approaches to handle many objectives is to incorporate the preference of a decision maker (DM) into EMO algorithms. With the preference, EMO algorithms can focus the search on regions preferred by the DM, resulting in solutions close to the Pareto front around the preferred regions. Although a number of preference-based EMO algorithms have been proposed, it is not trivial for the DM to reflect his/her actual preference in the search. We previously proposed to represent the preference of the DM using Gaussian functions on a hyperplane. The DM specifies the center and spread vectors of the Gaussian functions so as to represent his/her preference. The preference handling is integrated into the framework of NSGA-II. This paper extends our previous work so that obtained solutions follow the distribution of Gaussian functions specified. The performance of our proposed method is demonstrated mainly for benchmark problems and real-world applications with a few objectives in this paper. We also show the applicability of our method to many-objective problems.     

基于指标和自适应边界选择的高维多目标优化算法

多目标优化算法的主要目标是实现好的多样性和收敛性.传统的高维多目标优化算法,当目标维数增加时,选择方式难以平衡种群的收敛性与多样性.对此,提出一个基于指标和自适应边界选择的高维多目标优化算法.在环境选择中,首先计算种群中两两个体的指标Iε(x,y)作为第一选择标准;其次,提出一种自适应边界选择策略,利用种群进化信息对超...     

基于全局排序的高维多目标优化研究

目标数超过4的高维多目标优化是目前进化多目标优化领域求解难度最大的问题之一,现有的多目标进化算法求解该类问题时,存在收敛性和解集分布性上的缺陷,难以满足实际工程优化需求.提出一种基于全局排序的高维多目标进化算法GR-MODE,首先,采用一种新的全局排序策略增强选择压力,无需用户偏好及目标主次信息,且避免宽松Pareto支配在排序结果合理性与可信性上的损失;其次,采用Harmonic平均拥挤距离对个体进行全局密度估计,提高现有局部密度估计方法的精确性;最后,针对高维多目标复杂空间搜索需求,设计新的精英选择策略及适应度值评价函数.将该算法与国内外现有的5种高性能多目标进化算法在标准测试函数集DTLZ{1,2, 4,5}上进行对比实验,结果表明,该算法具有明显的性能优势,大幅提升了4~30维高维多目标优化的收敛性和分布性.     

An objective reduction algorithm using representative Pareto solution search for many-objective optimization problems

In recent years, many-objective optimization problems (i.e. more than three objectives) have attracted the interests of many researchers. The main difficulties of many-objective optimization problems lie in high computational cost, stagnation in search process, etc. It is almost impossible to design an algorithm effective for all problems. However, for some problems, especially for problems with redundant objectives, it is possible to design effective algorithms by removing the redundant objectives and keeping the non-redundant objectives so that the original problem becomes the one with much fewer objectives. To do so, first, a multi-objective evolutionary algorithm-based decomposition is adopted to generate a smaller number of representative non-dominated solutions widely distributed on the Pareto front. Then the conflicting objective pairs are identified through these non-dominated solutions, and the redundant objectives are determined by these pairs and then removed. Based on these, a fast non-redundant objectives generation algorithm is proposed in this paper. Finally, the experiments are conducted on a set of benchmark test problems and the results indicate the effectiveness and efficiency of the proposed algorithm.     

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 angle based evolutionary algorithm with infeasibility information for constrained many-objective optimization

Recently, angle-based approaches have shown promising for unconstrained many-objective optimization problems (MaOPs), but few of them are extended to solve constrained MaOPs (CMaOPs). Moreover, due to the difficulty in searching for feasible solutions in high-dimensional objective space, the use of infeasible solutions comes to be more important in solving CMaOPs. In this paper, an angle based evolutionary algorithm with infeasibility information is proposed for constrained many-objective optimization, where different kinds of infeasible solutions are utilized in environmental selection and mating selection. To be specific, an angle-based constrained dominance relation is proposed for non-dominated sorting, which gives infeasible solutions with good diversity the same priority to feasible solutions for escaping from the locally feasible regions. As for diversity maintenance, an angle-based density estimation is developed to give the infeasible solutions with good convergence a chance to survive for next generation, which is helpful to get across the large infeasible barrier. In addition, in order to utilize the potential of infeasible solutions in creating high-quality offspring, a modified mating selection is designed by considering the convergence, diversity and feasibility of solutions simultaneously. Experimental results on two constrained many-objective optimization test suites demonstrate the competitiveness of the proposed algorithm in comparison with five existing constrained many-objective evolutionary algorithms for CMaOPs. Moreover, the effectiveness of the proposed algorithm on a real-world problem is showcased.     

修正免疫克隆约束多目标优化算法

针对约束多目标优化问题,提出修正免疫克隆约束多目标优化算法.该算法通过引进一个约束处理策略,用一个修正算法对个体的目标函数值进行修正,并对修正后的目标函数值采用免疫克隆算法进行优化,用一个精英种群对可行非支配解进行存储.该算法在优化过程中,既保留了非支配可行解,也充分利用了约束偏离值小的非可行解,同时引进整体克隆策略来提高解分布的多样性.通过对约束多目标问题的各项性能指标的测试以及和对比算法的比较可以看出:该算法在处理约束多目标优化测试问题时,所得解的多样性得到了一定的提高.同时,解的收敛性和均匀性也得到了一定的改进.     

一种改进的多目标约束优化差分进化算法

提出一种新的多目标优化差分进化算法用于求解约束优化问题.该算法利用佳点集方法初始化个体以维持种群的多样性.将约束优化问题转化为两个目标的多目标优化问题.基于Pareto支配关系,将种群分为Pareto子集和Non-Pareto子集,结合差分进化算法两种不同变异策略的特点,对Non-Pareto子集和Pareto子集分别采用DE/best/1变异策略和DE/rand/1变异策略.数值实验结果表明该算法具有较好的寻优效果.     

基于目标分解的高维多目标并行进化优化方法

高维多目标优化问题普遍存在且难以解决, 到目前为止, 尚缺乏有效解决该问题的进化优化方法. 本文提出一种基于目标分解的高维多目标并行进化优化方法, 首先, 将高维多目标优化问题分解为若干子优化问题, 每一子优化问题除了包含原优化问题的少数目标函数之外, 还具有由其他目标函数聚合成的一个目标函数, 以降低问题求解的难度; 其次, 采用多种群并行进化算法, 求解分解后的每一子优化问题, 并在求解过程中, 充分利用其他子种群的信息, 以提高Pareto非被占优解的选择压力; 最后, 基于各子种群的非被占优解形成外部保存集, 从而得到高维多目标优化问题的Pareto 最优解集. 性能分析表明, 本文提出的方法具有较小的计算复杂度. 将所提方法应用于多个基准优化问题, 并与NSGA-II、PPD-MOEA、ε-MOEA、HypE和MSOPS等方法比较, 实验结果表明, 所提方法能够产生收敛性、分布性, 以及延展性优越的Pareto最优解集.     

自适应偏好半径划分区域的多目标进化方法

偏好多目标进化算法是一类帮助决策者找到感兴趣的Pareto最优解的算法.目前,在以参考点位置作为偏好信息载体的偏好多目标进化算法中,不合适的参考点位置往往会严重影响算法的收敛性能,偏好区域的大小难以控制,在高维问题上效果较差.针对以上问题,通过计算基于种群的自适应偏好半径,利用自适应偏好半径构造一种新的偏好关系模型,通过对偏好区域进行划分,提出基于偏好区域划分的偏好多目标进化算法.将所提算法与4种常用的以参考点为偏好信息载体的多目标进化算法g-NSGA-II、r-NSGA-II、角度偏好算法、MOEA/D-PRE进行对比实验,结果表明,所提算法具有较好的收敛性能和分布性能,决策者可以控制偏好区域大小,在高维问题上也具有较好的收敛效果.     

Handling many-objective optimisation problems with R2 indicator and decomposition-based particle swarm optimiser

An R2 indicator-based multi-objective particle swarm optimiser (R2-MOPSO) can obtain well-convergence and well-distributed solutions while solving two and three objectives optimisation problems. However, R2-MOPSO faces difficulty to tackle many-objective optimisation problems because balancing convergence and diversity is a key issue in high-dimensional objective space. In order to address this issue, this paper proposes a novel algorithm, named R2-MaPSO, which combines the R2 indicator and decomposition-based archiving pruning strategy into particle swarm optimiser for many-objective optimisation problems. The innovations of the proposed algorithm mainly contains three crucial factors: (1) A bi-level archiving maintenance approach based on the R2 indicator and objective space decomposition strategy is designed to balance convergence and diversity. (2) The global-best leader selection is based on the R2 indicator and the personal-best leader selection is based on the Pareto dominance. Meanwhile, the objective space decomposition leader selection adopts the feedback information from the bi-level archive. (3) A new velocity updated method is modified to enhance the exploration and exploitation ability. In addition, an elitist learning strategy and a smart Gaussian learning strategy are embedded into R2-MaPSO to help the algorithm jump out of the local optimal front. The performance of the proposed algorithm is validated and compared with some algorithms on a number of unconstraint benchmark problems, i.e. DTLZ1-DTLZ4, WFG test suites from 3 to 15 objectives. Experimental results have demonstrated a better performance of the proposed algorithm compared with several multi-objective particle swarm optimisers and multi-objective evolutionary algorithms for many-objective optimisation problems.