Differential evolution algorithm as a tool for optimal feature subset selection in motor imagery EEG

One of the challenges in developing a Brain Computer Interface (BCI) is dealing with the high dimensionality of the data when extracting features from EEG signals. Different feature selection algorithms have been proposed to overcome this problem but most of them involve complex transformed features, which require high computation and also result in increasing size of the feature set. In this paper, we present a new hybrid method to select features that involves a Differential Evolution (DE) optimization algorithm for searching the feature space to generate the optimal feature subset, with performance evaluated by a classifier. We provide a comprehensive study of the significance of evolutionary algorithm in selecting the best features for EEG signals. The BCI competition III, dataset IVa has been used to evaluate the method. Experimental results demonstrate that the proposed method performs well with Support Vector Machine (SVM) classifier, with an average classification accuracy of above 95% with a minimum of just 10 features. We also present a comparison of Differential Evolution (DE) with other evolutionary algorithms, and the results show the superiority of DE which implies that, with the selection of a good searching algorithm, a simple Common Spatial Pattern filter features can produce good results.     

Exploring dynamic self-adaptive populations in differential evolution

Although the Differential Evolution (DE) algorithm has been shown to be a simple yet powerful evolutionary algorithm for optimizing continuous functions, users are still faced with the problem of preliminary testing and hand-tuning of the evolutionary parameters prior to commencing the actual optimization process. As a solution, self-adaptation has been found to be highly beneficial in automatically and dynamically adjusting evolutionary parameters such as crossover rates and mutation rates. In this paper, we present a first attempt at self-adapting the population size parameter in addition to self-adapting crossover and mutation rates. Firstly, our main objective is to demonstrate the feasibility of self-adapting the population size parameter in DE. Using De Jong's F1–F5 benchmark test problems, we showed that DE with self-adaptive populations produced highly competitive results compared to a conventional DE algorithm with static populations. In addition to reducing the number of parameters used in DE, the proposed algorithm actually outperformed the conventional DE algorithm for one of the test problems. It was also found that that an absolute encoding methodology for self-adapting population size in DE produced results with greater optimization reliability compared to a relative encoding methodology.     

Three modified versions of differential evolution algorithm for continuous optimization

Differential evolution (DE) is one simple and effective evolutionary algorithm (EA) for global optimization. In this paper, three modified versions of the DE to improve its performance, to repair its defect in accurate converging to individual optimal point and to compensate the limited amount of search moves of original DE are proposed. In the first modified version called bidirectional differential evolution (BDE), to generate a new trial point, is used from the bidirectional optimization concept, and in the second modified version called shuffled differential evolution (SDE), population such as shuffled frog leaping (SFL) algorithm is divided in to several memeplexes and each memeplex is improved by the DE algorithm. Finally, in the third modified version of DE called shuffled bidirectional differential evolution (SBDE) to improve each memeplex is used from the proposed BDE algorithm. Three proposed modified versions are applied on two types of DE and six obtained algorithms are compared with original DE and SFL algorithms. Experiments on continuous benchmark functions and non-parametric analysis of obtained results demonstrate that applying bidirectional concept only improves one type of the DE. But the SDE and the SBDE have a better success rate and higher solution precision than original DE and SFL, whereas those are more time consuming on some functions. In a later part of the comparative experiments, a comparison of the proposed algorithms with some modern DE and the other EAs reported in the literature confirms a better or at least comparable performance of our proposed algorithms.     

Augmented grasshopper optimization algorithm by differential evolution: a power scheduling application in smart homes

With the increasing number of electricity consumers, production, distribution, and consumption problems of produced energy have appeared. This paper proposed an optimization method to reduce the peak demand using smart grid capabilities. In the proposed method, a hybrid Grasshopper Optimization Algorithm (GOA) with the self-adaptive Differential Evolution (DE) is used, called HGOA. The proposed method takes advantage of the global and local search strategies from Differential Evolution and Grasshopper Optimization Algorithm. Experimental results are applied in two scenarios; the first scenario has universal inputs and several appliances. The second scenario has an expanded number of appliances. The results showed that the proposed method (HGOA) got better power scheduling arrangements and better performance than other comparative algorithms using the classical benchmark functions. Moreover, according to the computational time, it runs in constant execution time as the population is increased. The proposed method got 0.26?% enhancement compared to the other methods. Finally, we found that the proposed HGOA always got better results than the original method in the worst cases and the best cases.

     

一种基于新型差分进化模型的MOEA/D改进算法

针对MOEA/D算法中差分进化操作收敛精度不高且速度较慢的不足,提出了一种综合基于可控支配域的向量差生成策略和基于主成分的动态缩放因子的新型差分进化模型,均衡显性与隐性搜索引导;并实现了一种基于新型差分进化模型的MOEA/D改进算法（MOEA/D-iDE）。新型差分进化是借助基于可控支配域的非支配排序对邻域进行分层,根据分层信息生成与不同进化阶段相匹配的向量差,实现对种群收敛速度的显性引导;同时对决策空间进行主成分分析,动态调整差分进化缩放因子,实现对种群收敛精度的隐性引导。实验选取ZDT、DTLZ和WFG等为测试问题,以IGD+,ER作为评价指标,将MOEA/D-iDE算法与6个同类算法进行对比实验,结果表明新算法在保证多样性的同时具有更好的收敛速度与精度,从而验证了新型差分进化模型的有效性。     

A new hybrid differential evolution algorithm with self-adaptation for function optimization

In this study, a new hybrid algorithm, hDEBSA, is proposed with the aid of two evolutionary algorithms, Differential Evolution (DE) and Backtracking Search Optimization Algorithm (BSA). The control parameters of both algorithms are simultaneously considered as a self-adaptation basis such that the values of the parameters update automatically during the optimization process to improve performance and convergence speed. To validate the proposed algorithm, twenty-eight CEC2013 test functions are considered. The performance results of hDEBSA are validated by comparing them with several state-of-the-art algorithms that are available in literature. Finally, hDEBSA is applied to solve four real-world optimization problems, and the results are compared with the other algorithms, where it was found that the hDEBSA performance is better than that of the other algorithms.     

Disturbed Exploitation compact Differential Evolution for limited memory optimization problems

This paper proposes a novel and unconventional Memetic Computing approach for solving continuous optimization problems characterized by memory limitations. The proposed algorithm, unlike employing an explorative evolutionary framework and a set of local search algorithms, employs multiple exploitative search within the main framework and performs a multiple step global search by means of a randomized perturbation of the virtual population corresponding to a periodical randomization of the search for the exploitative operators. The proposed Memetic Computing approach is based on a populationless (compact) evolutionary framework which, instead of processing a population of solutions, handles its statistical model. This evolutionary framework is based on a Differential Evolution which cooperatively employs two exploitative search operators: the first is based on a standard Differential Evolution mutation and exponential crossover, and the second is the trigonometric mutation. These two search operators have an exploitative action on the algorithmic framework and thus contribute to the rapid convergence of the virtual population towards promising candidate solutions. The action of these search operators is counterbalanced by a periodical stochastic perturbation of the virtual population, which has the role of “disturbing” the excessively exploitative action of the framework and thus inhibits its premature convergence. The proposed algorithm, namely Disturbed Exploitation compact Differential Evolution, is a simple and memory-wise cheap structure that makes use of the Memetic Computing paradigm in order to solve complex optimization problems. The proposed approach has been tested on a set of various test problems and compared with state-of-the-art compact algorithms and with some modern population based meta-heuristics. Numerical results show that Disturbed Exploitation compact Differential Evolution significantly outperforms all the other compact algorithms present in literature and reaches a competitive performance with respect to modern population algorithms, including some memetic approaches and complex modern Differential Evolution based algorithms. In order to show the potential of the proposed approach in real-world applications, Disturbed Exploitation compact Differential Evolution has been implemented for performing the control of a space robot by simulating the implementation within the robot micro-controller. Numerical results show the superiority of the proposed algorithm with respect to other modern compact algorithms present in literature.     

一种改进的基于差分进化的多目标进化算法

近年来运用进化算法（EAs）解决多目标优化问题（Multi-objective Optimization Problems MOPs）引起了各国学者们的关注。作为一种基于种群的优化方法,EAs提供了一种在一次运行后得到一组优化的解的方法。差分进化（DE）算法是EA的一个分支,最开始是用来解决连续函数空间的问题。提出了一种改进的基于差分进化的多目标进化算法（CDE）,并且将它与另外两个经典的多目标进化算法（MOEAs）NSGA-II和SPEA2进行了对比实验。     

基于非支配解排序的快速多目标微分进化算法

为解决基于帕累托(Pareto)支配解排序的多目标进化算法高时间复杂度问题,依据非支配解排序潜在特性,介绍了一种快速的非支配解排序方法,每次只处理当前种群中最高等级个体,且在分配等级的同时,能选择个体进入下一代,下一代被选足时即结束程序,减少了排序处理个体的数量,大幅度降低时间复杂度;另外,给出一种均匀的拥挤距离计算方法;最后,将快速非支配解排序和均匀拥挤距离计算与微分进化算法结合,提出基于非支配解排序的快速多目标微分进化算法(FMODE)。采用标准多目标优化问题ZDTl～ZDT4和ZDT6进行仿真实验：当种群个体较多(大于500)时,FMODE所用时间远小于NSGAⅡ;FMODE的总体性能上均优于经典的NSGAⅡ、SPEAⅡ和DEMO;在FMODE框架内,采用均匀拥挤距离在性能上也明显优于经典拥挤计算方法;并通过实验确定了FMODE算法的参数。实验结果表明FMODE能够减少计算等级时的处理时间,并在收敛性和多样性指标上明显优于对比算法。     

基于搜索空间大小的动态变异算子差分进化算法

差分进化算法（DE）是一种较新的进化计算技术,具有概念简单、易于实现、收敛速度快等优点,得到了广泛的关注和应用.为了解决经典DE计算开销大,参数设置与问题本身过于相关等缺陷,提出了一种改进的差分进化算法（IDE）,它采用了一种动态变异算子,可根据进化代数的增加,基于搜索空间大小,实时地调整变异步长,从而提高算法的求解精度.通过在MATLAB仿真环境下对著名的基准测试函数分别进行求解,将改进后的算法和已有的多种优化算法进行比较,结果表明,改进的IDE算法性能明显优于已知的算法,证明动态变异是一种有效的改进思路.     

A Differential Covariance Matrix Adaptation Evolutionary Algorithm for real parameter optimization

Hybridization in context to Evolutionary Computation (EC) aims at combining the operators and methodologies from different EC paradigms to form a single algorithm that may enjoy a statistically superior performance on a wide variety of optimization problems. In this article we propose an efficient hybrid evolutionary algorithm that embeds the difference vector-based mutation scheme, the crossover and the selection strategy of Differential Evolution (DE) into another recently developed global optimization algorithm known as Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES). CMA-ES is a stochastic method for real parameter (continuous domain) optimization of non-linear, non-convex functions. The algorithm includes adaptation of covariance matrix which is basically an alternative method of traditional Quasi-Newton method for optimization based on gradient method. The hybrid algorithm, referred by us as Differential Covariance Matrix Adaptation Evolutionary Algorithm (DCMA-EA), turns out to possess a better blending of the explorative and exploitative behaviors as compared to the original DE and original CMA-ES, through empirical simulations. Though CMA-ES has emerged itself as a very efficient global optimizer, its performance deteriorates when it comes to dealing with complicated fitness landscapes, especially landscapes associated with noisy, hybrid composition functions and many real world optimization problems. In order to improve the overall performance of CMA-ES, the mutation, crossover and selection operators of DE have been incorporated into CMA-ES to synthesize the hybrid algorithm DCMA-EA. We compare DCMA-EA with original DE and CMA-EA, two best known DE-variants: SaDE and JADE, and two state-of-the-art real optimizers: IPOP-CMA-ES (Restart Covariance Matrix Adaptation Evolution Strategy with increasing population size) and DMS-PSO (Dynamic Multi Swarm Particle Swarm Optimization) over a test-suite of 20 shifted, rotated, and compositional benchmark functions and also two engineering optimization problems. Our comparative study indicates that although the hybridization scheme does not impose any serious burden on DCMA-EA in terms of number of Function Evaluations (FEs), DCMA-EA still enjoys a statistically superior performance over most of the tested benchmarks and especially over the multi-modal, rotated, and compositional ones in comparison to the other algorithms considered here.     

混合模式搜索的分布式memetic差分进化算法

针对差分进化（DE）算法存在的早熟收敛与搜索停滞的问题,提出memetic分布式差分进化（DDE）算法。将memetic算法的思想融入到差分进化算法中,采用分布式的种群结构以及memetic算法中的混合策略,前者将初始种群分为多个子种群,子种群间根据冯·诺依曼拓扑结构周期性地实现信息交流,后者将差分进化算法作为进化的主要框架,模式搜索作为辅助手段,从而平衡算法的探索与开发能力。所提算法充分利用了模式搜索和差分进化算法的优势,建立了有效的搜索机制,增强了算法摆脱局部最优的能力,能够满足搜索过程对种群多样性及收敛速度的需求。将所提算法与几种先进的差分进化算法相比较,对标准测试函数进行优化的实验结果显示：所提算法在解的质量和收敛性能方面,均优于其他几种相比较的先进的差分进化算法。     

双种群差分进化规划算法

标准差分进化算法（SDE）具有算法简单,控制参数少,易于实现等优点。但在难优化问题中,算法存在收敛速度较慢和容易早熟等缺陷。为克服此缺点,提出一种改进算法--双种群差分进化规划算法（BGDEP）。该算法将种群划分为两个子群独立进化,分别采用DE/rand/1/bin和DE/best/2/bin版本生成变异个体。每隔δt（取5~10）代,将两个子群合并为一个种群,再应用混沌重组算子将之划分为两个子群,以实现子群间的信息交流。在双种群协同差分进化的同时,应用非均匀变异算子对其最优个体执行进化规划操作,使得算法具有较快的收敛速度和较强的全局寻优能力。为测试BGDEP的性能,给出了4个30维benchmark函数优化问题的对比数值实验。结果表明,BGDEP的求解精度、收敛速度、鲁棒性等性能优于SDE、双种群差分进化（BGDE）和非均匀变异进化规划（NUMEP）等4种算法。     

Solution of Jiles-Atherton vector hysteresis parameters estimation by modified Differential Evolution approaches

Differential Evolution (DE) is a simple and efficient stochastic global optimization algorithm of evolutionary computation field, which involves the evolution of a population of solutions using operators such as mutation, crossover, and selection. The basic idea of DE is to adapt the search during the evolutionary process. At the start of the evolution, the perturbations are large since parent populations are far away from each other. As the evolutionary process matures, the population converges to a small region and the perturbations adaptively become small. DE approaches have been successfully applied to solve a wide range of optimization problems. In this paper, the parameters set of the Jiles-Atherton vector hysteresis model is obtained with an approach based on modified Differential Evolution (MDE) approaches using generation-varying control parameters based on generation of random numbers with uniform distribution. Several evaluated MDE approaches perform better than the classical DE methods and a genetic algorithm approach in terms of the quality and stability of the final solutions in optimization of vector Jiles-Atherton vector hysteresis model from a workbench containing a rotational single sheet tester.     

基于多变异策略的自适应差分进化算法

为了克服差分进化算法寻优精度低、收敛速度慢、稳定性差等不足,提出一种基于多变异策略的自适应差分进化算法（ADE-MM）。首先,在3个变异策略的选择过程中添加2个具有学习功能的扰动阈值,以提高种群多样性,扩大搜索范围;然后,根据上次迭代的成功参数自适应调整当前参数,提高寻优精度和寻优速度;最后,利用向量粒子池法和中心粒子法产生新的向量粒子,进一步提高寻优效果。使用8个函数、5种对比算法（RMDE、OLCPDE、JADE、SaDE、MDE_pBX）进行测试,且每种例子都独立执行30次。ADE-MM算法在均值和方差的比较中取得了全胜,其中在30维的情况下取得了5个独立胜利,3个并列胜利;在50维的情况下取得了6个独立胜利,2个并列胜利;在100维的情况下全部为独立胜利。同时在Wilcoxon rank sum test、胜率和算法耗时分析中,ADE-MM算法也取得优异的表现。实验结果表明,相对于其他5种对比算法,ADE-MM算法具有更强的全局寻优能力、收敛性和稳定性。     

基于差分进化的生物地理学优化算法

针对生物地理学优化算法在实数编码时搜索能力较弱的缺点,提出一种基于差分进化的混合优化算法(BBO/DEs)。通过将差分进化的搜索性与生物地理优化算法的利用性有机结合,以解决原算法在局部搜索时容易出现早熟的问题;并构造一种基于Levy分布的变异方式,确保种群在进化过程中保持多样性;最后通过实验比较,选取了合适的试验策略。利用高维标准测试函数对相关算法进行实验,结果表明该算法能够克服搜索能力不足的缺点,并继承了原算法的快速收敛性能,可以有效兼顾精度与速度的要求。     

Colonial competitive differential evolution: An experimental study for optimal economic load dispatch

Differential evolution (DE) algorithm is a population-based algorithm designed for global optimization of the optimization problems. This paper proposes a different DE algorithm based on mathematical modeling of socio-political evolution which is called Colonial Competitive Differential Evolution (CCDE). The two typical CCDE algorithms are benchmarked on three well-known test functions, and the results are verified by a comparative study with two original DE algorithms which include DE/best/1 and DE/rand/2. Also, the effectiveness of CCDE algorithms is tested on Economic Load Dispatch (ELD) problem including 10, 15, 40, and 140-unit test systems. In this study, the constraints and operational limitations, such as valve-point loading, transmission losses, ramp rate limits, and prohibited operating zones are considered. The comparative results show that the CCDE algorithms have good performance and are reliable tools in solving ELD problem.     

Self-adaptive population sizing for a tune-free differential evolution

The study and research of evolutionary algorithms (EAs) is getting great attention in recent years. Although EAs have earned extensive acceptance through numerous successful applications in many fields, the problem of finding the best combination of evolutionary parameters especially for population size that need the manual settings by the user is still unresolved. In this paper, our system is focusing on differential evolution (DE) and its control parameters. To overcome the problem, two new systems were carried out for the self-adaptive population size to test two different methodologies (absolute encoding and relative encoding) in DE and compared their performances against the original DE. Fifty runs are conducted for every 20 well-known benchmark problems to test on every proposed algorithm in this paper to achieve the function optimization without explicit parameter tuning in DE. The empirical testing results showed that DE with self-adaptive population size using relative encoding performed well in terms of the average performance as well as stability compared to absolute encoding version as well as the original DE.     

一种新型的差分演化算法及其应用研究

提出了一种新的基于简单多样性规则的改进差分演化算法,并把它运用于约束全局最优化问题的求解中。新算法的特征是: 1)提出一种新的混合自适应交叉变异算子,以增强算法的搜索能力; 2)采用具有保持群体多样性的约束函数处理技术; 3)简化基本差分演化算法的缩放因子,尽量减少算法的控制参数,方便工程人员的使用。通过对13个标准测试函数进行测试,并与其他演化算法结果进行比较。实验结果表明,新算法在求解精度和稳定性具有很好的性能,而且其函数平均评价次数要低于所比较的其他演化算法。     

一种新的改进的混合蛙跳算法

针对混合蛙跳算法在优化过程中受初始值影响较大且容易陷入局部最优的缺陷,提出了一个改进的混合蛙跳算法,该算法利用基于对立学习的策略产生初始种群,提高了产生解的质量;在进化过程中,将差分进化有机地嵌入其中,维持了种群的多样性。数值结果表明,改进的混合蛙跳算法对复杂函数优化问题具有较强的求解能力。