基于分解的多目标花朵授粉算法

在过去几十年里，许多多目标进化算法被广泛应用于解决多目标优化问题，其中一种比较流行的多目标进化算法是基于分解的多目标进化算法(MOEA/D)。花朵授粉算法是一种启发式优化算法，但迄今为止，花朵授粉算法在基于分解的多目标进化算法领域的研究还非常少。本文在基于分解的多目标进化算法的框架下，将花朵授粉算法拓展至多目标优化领域，提出一种基于分解的多目标花朵授粉算法(MOFPA/D)。此外，为了保证非支配解的多样性，本文提出一种基于网格的目标空间分割法，该方法从找到的Pareto最优解集中筛选出一定数量且分布均匀的Pareto最优解。实验结果表明，基于分解的多目标花朵授粉算法在收敛性与多样性方面均优于基于分解的多目标进化算法。     

基于分解和差分进化的多目标粒子群优化算法

为了提高多目标优化算法解集的分布性和收敛性,提出一种基于分解和差分进化的多目标粒子群优化算法(dMOPSO-DE).该算法通过提出方向角产生一组均匀的方向向量,确保粒子分布的均匀性;引入隐式精英保持策略和差分进化修正机制选择全局最优粒子,避免种群陷入局部最优Pareto前沿;采用粒子重置策略保证群体的多样性.与非支配排序(NSGA-II)算法、多目标粒子群优化(MOPSO)算法、分解多目标粒子群优化(dMOPSO)算法和分解多目标进化-差分进化(MOEA/D-DE)算法进行比较,实验结果表明,所提出算法在求解多目标优化问题时具有良好的收敛性和多样性.     

一种基于多策略差分进化的分解多目标进化算法

为了提高多目标优化问题非支配解集合的分布性和收敛性,根据不同差分进化策略的特点,基于切比雪夫分解机制,提出一种基于多策略差分进化的分解多目标进化算法(MOEA/D-WMSDE).该算法首先采用切比雪夫分解机制,将多目标优化问题转化为一系列单目标优化子问题;然后引入小波基函数和正态分布实现差分进化算法的参数控制,探究一种...     

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。     

一种基于偶图匹配的多目标分解进化算法

针对基于分解的多目标进化(MOEA/D)算法在选择下一代解时未考虑解和子问题之间的相对距离,可能导致算法得到的最终解多样性较差的问题,提出一种基于偶图匹配的多目标分解进化(MOEA/D-BM)算法.所提算法利用偶图匹配模型对解和子问题的相互关系进行建模,在选择下一代解的同时,考虑收敛性和多样性,以提高算法性能.通过与其他3种经典的多目标分解进化算法在多个测试函数上进行实验,验证了所提出算法的有效性.     

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.     

基于改进MOEA/D算法的WSN覆盖优化方法*

为了优化无线传感器网络（WSN）的覆盖方法,针对MOEA/D中缺少对本代优质个体的保存和最优解集中的个体极少的两个问题,提出了粒子群优化的基于分解的多目标进化算法（MOEA/D-PSO）。通过保留种群本代优质个体,改进本地优化解集在进化过程中的搜索方向和搜索进度,弥补了MOEA/D不足。仿真实验证明,相对于MOEA/D和非支配排序遗传算法（NSGA-II）,MOEA/D-PSO所得非支配解更接近Pareto最优曲面,解集分布的均匀性和多样性表现更佳,WSN的覆盖范围更广,能量消耗更少。     

基于泛化反向学习的多目标约束差分进化算法

差分进化算法是一种简单有效的进化算法,基于泛化反向学习的机制在进化算法中经常可以引导种群的进化.针对多目标的约束优化问题,提出了一种基于泛化反向学习的多目标约束差分进化算法.该算法采用基于泛化反向学习的机制(generalized opposition-based learning, GOBL)产生变换种群,然后在种群初始化和代跳跃阶段,利用非支配排序、拥挤距离和约束处理技术从原始种群和其变换种群中选择更优的种群个体作为新的种群继续迭代进化;该算法通过采用基于泛化反向学习的机制,可以引导种群个体慢慢向最优的Pareto前沿逼近,以求得最优解集.最后采用多目标Benchmark问题对该算法进行了实验评估,实验结果表明：与NSGA-Ⅱ,MOEA/D及其他的多目标进化算法相比,提出的算法具有更好的收敛性,并且产生的解能够逼近最优的Pareto前沿.     

一种改进的基于分解的多目标进化算法

利用基于分解的多目标进化算法框架(MOEA/D),将混合策略的进化算法用于求解分解后的若干单目标优化子问题,提出了一种带局部搜索的基于分解的多目标混合策略进化算法(LMS-MOEA/D)。算法利用均匀设计产生子问题的聚合权重向量,混合交叉策略能够充分利用不同交叉算子的优势;同时算法针对演化过程收敛的特点,结合局部搜索策略,获得逼近Pareto前沿的最优解集。最后通过实验验证算法在多样性和收敛性方面的有效性。     

MaOEA/d2:一种基于双距离构造的高维多目标进化算法

传统的基于Pareto支配关系的多目标进化算法(MOEA)难以有效求解高维多目标优化问题(MaOP). 提出一种利用PBI效用函数的双距离构造的支配关系, 且无需引入额外的参数. 其次, 利用双距离定义了一种多样性保持方法, 该方法不仅考虑了解个体的双距离, 而且还可以根据优化问题的目标数目自适应地调整多样性占比, 以较好地平衡高维目标解群的收敛性和多样性. 最后, 将基于双距离构造的支配关系和多样性保持方法嵌入到NSGA-II算法框架中, 设计了一种基于双距离的高维多目标进化算法MaOEA/d². 该算法与其他5种代表性的高维多目标进化算法一同在5-、10-、15-和20-目标的DTLZ和WFG基准测试问题上进行了IGD和HV性能测试, 结果表明, MaOEA/d²算法具有较好的收敛性和多样性. 由此表明, MaOEA/d²算法是一种颇具前景的高维多目标进化算法.     

MOEA/D + uniform design: A new version of MOEA/D for optimization problems with many objectives

To extend multiobjective evolutionary algorithm based on decomposition (MOEA/D) in higher dimensional objective spaces, this paper proposes a new version of MOEA/D with uniform design, named the uniform design multiobjective evolutionary algorithm based on decomposition (UMOEA/D), and compares the proposed algorithm with MOEA/D and NSGA-II on some scalable test problems with three to five objectives. UMOEA/D adopts the uniform design method to set the aggregation coefficient vectors of the subproblems. Compared with MOEA/D, distribution of the coefficient vectors is more uniform over the design space, and the population size neither increases nonlinearly with the number of objectives nor considers a formulaic setting. The experimental results indicate that UMOEA/D outperforms MOEA/D and NSGA-II on almost all these many-objective test instances, especially on problems with higher dimensional objectives and complicated Pareto set shapes. Experimental results also show that UMOEA/D runs faster than NSGA-II for the problems used in this paper. In additional, the results obtained are very competitive when comparing UMOEA/D with some other algorithm on the multiobjective knapsack problems.     

Convergence Acceleration Operator for Multiobjective Optimization

A convergence acceleration operator (CAO) is described which enhances the search capability and the speed of convergence of the host multiobjective optimization algorithm. The operator acts directly in the objective space to suggest improvements to solutions obtained by a multiobjective evolutionary algorithm (MOEA). The suggested improved objective vectors are then mapped into the decision variable space and tested. This method improves upon prior work in a number of important respects, such as mapping technique and solution improvement. Further, the paper discusses implications for many-objective problems and studies the impact of the use of the CAO as the number of objectives increases. The CAO is incorporated with two leading MOEAs, the non-dominated sorting genetic algorithm and the strength Pareto evolutionary algorithm and tested. Results show that the hybridized algorithms consistently improve the speed of convergence of the original algorithm while maintaining the desired distribution of solutions. It is shown that the operator is a transferable component that can be hybridized with any MOEA.      

基于分解的演化多目标优化算法综述

基于分解的演化多目标优化算法(MOEA/D)的基本思想是将一个多目标优化问题转化成一系列子问题(单目标或者多目标)来进行优化求解.自2007年提出以来, MOEA/D受到了国内外学者的广泛关注,已经成为最具代表性的演化多目标优化算法之一.总结过去13年中关于MOEA/D的一些研究进展,具体内容包括:(1)关于MOEA/D的算法改进;(2) MOEA/D在超多目标优化问题及约束优化问题上的研究;(3) MOEA/D在一些实际问题上的应用.然后,实验对比几个具有代表性的MOEA/D改进算法.最后,指出一些MOEA/D未来的研究方向.     

一种基于混合高斯模型的多目标进化算法

目前,大多数多目标进化算法采用为单目标优化所设计的重组算子.通过证明或实验分析了几个典型的单目标优化重组算子并不适合某些多目标优化问题.提出了基于分解技术和混合高斯模型的多目标优化算法（multiobjective evolutionary algorithm based on decomposition and mixture Gaussian models,简称MOEA/D-MG）.该算法首先采用一个改进的混合高斯模型对群体建模并采样产生新个体,然后利用一个贪婪策略来更新群体.针对具有复杂Pareto前沿的多目标优化问题的测试结果表明,对给定的大多数测试题,该算法具有良好的效果.     

Large-scale parallelization of the Borg multiobjective evolutionary algorithm to enhance the management of complex environmental systems

The Borg MOEA is a self-adaptive multiobjective evolutionary algorithm capable of solving complex, many-objective environmental systems problems efficiently and reliably. Water and environmental resources problems pose significant computational challenges due to their potential for large Pareto optimal sets, the presence of disjoint Pareto-optimal regions that arise from discrete choices, multi-modal suboptimal regions, and expensive objective function calculations. This work develops two large-scale parallel implementations of the Borg MOEA, the master–slave and multi-master Borg MOEA, and applies them to a highly challenging risk-based water supply portfolio planning problem. The performance and scalability of both implementations are compared on up to 16384 processors. The multi-master Borg MOEA is shown to scale efficiently on tens of thousands of cores while dramatically improving the reliability of attaining high-quality solutions. Our results dramatically expand the scale and scope of complex environmental systems that can be addressed using many-objective evolutionary optimization.     

A New Evolutionary Algorithm for Solving Many-Objective Optimization Problems

In this paper, we focus on the study of evolutionary algorithms for solving multiobjective optimization problems with a large number of objectives. First, a comparative study of a newly developed dynamical multiobjective evolutionary algorithm (DMOEA) and some modern algorithms, such as the indicator-based evolutionary algorithm, multiple single objective Pareto sampling, and nondominated sorting genetic algorithm II, is presented by employing the convergence metric and relative hypervolume metric. For three scalable test problems (namely, DTLZ1, DTLZ2, and DTLZ6), which represent some of the most difficult problems studied in the literature, the DMOEA shows good performance in both converging to the true Pareto-optimal front and maintaining a widely distributed set of solutions. Second, a new definition of optimality (namely, L-optimality) is proposed in this paper, which not only takes into account the number of improved objective values but also considers the values of improved objective functions if all objectives have the same importance. We prove that L-optimal solutions are subsets of Pareto-optimal solutions. Finally, the new algorithm based on L-optimality (namely, MDMOEA) is developed, and simulation and comparative results indicate that well-distributed L-optimal solutions can be obtained by utilizing the MDMOEA but cannot be achieved by applying L-optimality to make a posteriori selection within the huge Pareto nondominated solutions. We can conclude that our new algorithm is suitable to tackle many-objective problems.      

Multiobjective evolutionary algorithms for electric power dispatch problem

The potential and effectiveness of the newly developed Pareto-based multiobjective evolutionary algorithms (MOEA) for solving a real-world power system multiobjective nonlinear optimization problem are comprehensively discussed and evaluated in this paper. Specifically, nondominated sorting genetic algorithm, niched Pareto genetic algorithm, and strength Pareto evolutionary algorithm (SPEA) have been developed and successfully applied to an environmental/economic electric power dispatch problem. A new procedure for quality measure is proposed in this paper in order to evaluate different techniques. A feasibility check procedure has been developed and superimposed on MOEA to restrict the search to the feasible region of the problem space. A hierarchical clustering algorithm is also imposed to provide the power system operator with a representative and manageable Pareto-optimal set. Moreover, an approach based on fuzzy set theory is developed to extract one of the Pareto-optimal solutions as the best compromise one. These multiobjective evolutionary algorithms have been individually examined and applied to the standard IEEE 30-bus six-generator test system. Several optimization runs have been carried out on different cases of problem complexity. The results of MOEA have been compared to those reported in the literature. The results confirm the potential and effectiveness of MOEA compared to the traditional multiobjective optimization techniques. In addition, the results demonstrate the superiority of the SPEA as a promising multiobjective evolutionary algorithm to solve different power system multiobjective optimization problems.     

Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II

Partly due to lack of test problems, the impact of the Pareto set (PS) shapes on the performance of evolutionary algorithms has not yet attracted much attention. This paper introduces a general class of continuous multiobjective optimization test instances with arbitrary prescribed PS shapes, which could be used for studying the ability of multiobjective evolutionary algorithms for dealing with complicated PS shapes. It also proposes a new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE, and compares the proposed algorithm with NSGA-II with the same reproduction operators on the test instances introduced in this paper. The experimental results indicate that MOEA/D could significantly outperform NSGA-II on these test instances. It suggests that decomposition based multiobjective evolutionary algorithms are very promising in dealing with complicated PS shapes.     

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.     

Performance evaluation of evolutionary multiobjective optimization algorithms for multiobjective fuzzy genetics-based machine learning

Recently, evolutionary multiobjective optimization (EMO) algorithms have been utilized for the design of accurate and interpretable fuzzy rule-based systems. This research area is often referred to as multiobjective genetic fuzzy systems (MoGFS), where EMO algorithms are used to search for non-dominated fuzzy rule-based systems with respect to their accuracy and interpretability. In this paper, we examine the ability of EMO algorithms to efficiently search for Pareto optimal or near Pareto optimal fuzzy rule-based systems for classification problems. We use NSGA-II (elitist non-dominated sorting genetic algorithm), its variants, and MOEA/D (multiobjective evolutionary algorithm based on decomposition) in our multiobjective fuzzy genetics-based machine learning (MoFGBML) algorithm. Classification performance of obtained fuzzy rule-based systems by each EMO algorithm is evaluated for training data and test data under various settings of the available computation load and the granularity of fuzzy partitions. Experimental results in this paper suggest that reported classification performance of MoGFS in the literature can be further improved using more computation load, more efficient EMO algorithms, and/or more antecedent fuzzy sets from finer fuzzy partitions.