基于种群自适应策略的差分演化算法及其应用研究
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摘要
为解决复杂的计算问题,研究人员多年来一直在寻找以大自然为蓝本的模型和象征。优化,是许多自然过程的核心。正如达尔文的进化论,每一个物种都要经过数百万年,通过调整自身结构来适应周围的环境。我们观察到,优化和生物演化之间的基本关系是发展计算智能的一个重要范例。正是基于这样的思想,演化算法被提出用于执行非常复杂的搜索和优化。
差分演化算法(DE),一直被视为一种可靠和通用的基于种群的元启发式优化技术,并广泛的在各类问题中展现了令人瞩目的性能。在过去十年里,归功于差分演化算法的简单性、可靠性、高性能和易于实施,DE在众多研究人员中已经获得了广泛的知名度。与传统的演化算法不同,DE算法通过增加一个带权值的移动向量来执行扰动操作,并修正一些随机选择的候选粒子维度的值。正因为这样的内在机制,差分演化算法能能够在演化早期高度地探索整个搜索空间,而在优化的后期变得更加注重自身剥削和开发。然而,DE并是不总能保证收敛到全局最优解,他会偶尔陷入局部停滞或者早熟收敛,而导致优化精度的降低甚至失败。
本论文针对传统差分演化算法局部停滞和早熟收敛等问题,研究了一类自适应种群谐调框架和方法,采用马尔科夫链和信息熵的理论,提出了基于DE的改进种群自适应策略。同时,将改进算法应用于系统设计问题,解决了分数阶混沌系统的参数辨识问题,无限脉冲响应数字滤波器的设计问题以及质子交换膜燃料电池的最优化建模问题。本文主要工作概括如下：
(1)差分演化算法的自适应种群谐调控制研究
通过权衡当前的解搜索状态和需要的种群分布两个指标,提出了一种崭新的动态自适应种群谐调策略(APTS)。在APTS中,首先设计了一个基于精英的种群增量策略,他在决策空间的适当位置中生成一些新个体帮助搜索更优的可行解。其次,设计了一个基于平庸的种群缩减策略,他依据排序方法删除一些性能较差的个体以减少计算负荷,并预留一些空间给新的带有种群多样性的扰动个体。此外,上述两个种群策略都由一个状态观测器所控制。该状态观测器被建立用于监控种群的演化进程,并适时地控制APTS的灵敏度。为验证算法有效性,实现了APTS的收敛性分析,为其提供了理论保障。同时,通过一个全局性的性能比较实验,与其他6种顶尖的DE算法比较来发现最优者。实验结果表明JADE-APTS在低维问题(30维)中获得了富有竞争性的性能,在高维问题(100维)中获得了最佳性能。此外,方差分析的结果同样证实了APTS能够有效地加速收敛率和提高可行解的搜索精度。
(2)基于马尔科夫链的种群自适应改进研究及其信息熵指标判据
一个改进的种群自适应处理技术(CP)被应用于DE以解决各种优化问题。在CPDE中,实现了一个随机的策略跳变框架(MHT),依靠非均匀的马尔科夫链来选择不同的子优化控制器,更好地改进当前解搜索的状态。具体的来说,子优化控制器有两种,其一,称作改进的sigmoid函数种群增长策略。增加一些新个体进入种群,提供他们最新的信息分享给种群并帮助粒子逃脱局部困境。其二,称作基于信息熵和等级排序指标的种群减少策略。基于每个粒子的聚集熵指标和等级排序指标,删除一些过渡个体以避免不期望地计算损失和过度的搜索复杂度。其次,实现了CEC05基准函数下,CPDE与其他8种最先进演化算法(即,5种DEs和3种EAs)的性能比较实验以证明所提出方法的可行性。同时,维度可扩展性测试实验也同样证实,CP框架总能加速DE算法的搜索效能和效率,尤其是在高维问题中效果更为明显。最后,收敛速度实验和时间复杂度推导进一步证明了CP框架对迄今为止的所有差分演化变体算法不产生任何额外的计算负担。
(3)基于改进差分演化算法的分数阶混沌系统参数辨识
采用一种改进的差分演化算法(SDE),首先研究了分数阶Lorenz,Lu和Chen系统在确定性环境下的未知阶次和参数的估计问题。SDE的主要特点是有效的种群切换利用策略。他同时考虑收敛速度和计算负荷,根据适应度多样性非周期地增加和减少一些粒子。其次,研究上述3个系统在随机性环境下的未知阶次和参数的估计问题,即噪声扰动下的算法性能。五种最先进智能算法被应用于测试实验来验证SDE算法的有效性。实验结果表明我们的方法要比其他5种算法性能更优,尤其是在噪声扰动情况下。
(4)基于种群概率可控差分演化算法的无限脉冲响应数字滤波器设计
提出了一种基于马尔科夫跳变(开关切换)的种群更新DE算法用以解决限脉冲响应数字滤波器的设计问题。所提出的算法是一种带有可控概率种群大小的差分演化变体(CPDE),通过适应度多样性非周期地增加和减少一些粒子,权衡搜索广度和自身精度。进一步,6种公认优秀的演化算法被采纳用于设计上述6种典型的IIR滤波器,并和CPDE进行性能比较实验,以证明所提出方法的可行性。此外,我们还讨论了IIR数字滤波器设计的一些关键方面,如价值函数值、噪声扰动、收敛速度、成功率以及参数测量等。实验结果表明,我们提出的算法是可行且强有力的。
(5)基于混合差分演化算法的质子交换膜燃料电池最优化建模
基于极化曲线研究了一类适用于工程优化的电化学PEMFC模型。采用改进的一种改进的差分演化算法(HDE),引入动态种群谐调策略,对3个质子交换膜燃料电池模型进行参数辨识(即SR-12Modular PEM Generator, Ballard Mark V FC和BCS500-W stack模型)。在HDE中,种群的大小可以动态自适应地根据现今的搜索状态和所需的种群分布进行调节。同时,我们还测试在3%的噪声扰动下,算法对PEMFC的辨识性能。实验结果表明,即使实验过程被噪声破坏,HDE仍能获得较令人满意的辨识性能。此外,6种最顶尖的智能算法被应用于测试实验来验证HDE算法的有效性。
To tackle complex computational problems, researchers have been looking into na-ture for years for inspiration. Optimization is at the heart of many natural processes like Darwinian evolution itself. Through millions of years, every species had to adapt their physical structures to fit to the environments they were in. A keen observation of the underlying relation between optimization and biological evolution led to the development of an important paradigm of computational intelligence-the evolutionary computing techniques for performing very complex search and optimization.
Differential Evolution is a reliable and versatile function optimizer. DE, like most popular Evolutionary Algorithms (EAs), is a population-based tool. Over the past decade, the DE algorithm has gained wide-spread popularity among researchers due to its sim-plicity, reliability, high performance and easy implementation. Unlike traditional EAs, DE operates perturbation by adding a weighted moving vector and modifying the values of some randomly selected coordinates. Due to its structure, a DE scheme can be highly explorative in the prophase of the evolution and subsequently become more exploita-tive during the optimization. However, the DE does not guarantee the convergence to the global optimum. It is occasionally trapped into local stagnation or premature convergence resulting in a low optimizing precision or even failure.
In this dissertation, firstly, to remedy some of DE's pitfalls, I propose an adaptive population tuning scheme (APTS) for DE. Secondly, an enhanced adaptive population-handling technique is proposed to solve various types of optimization problems. Thirdly, I study the applications of differential evolution in identification of fractional-order sys-tems, digital IIR filters design and optimum modeling of PEM fuel cells. The main con-tribution of this dissertation are as follows.
(1) Adaptive population tuning scheme for differential evolution
An adaptive population tuning scheme (APTS) for DE is proposed to dynamically adjust the population size. More specifically, an elite-based population-incremental strat- egy is proposed to place several new individuals in appropriate areas to discover new pos-sible solutions. Meanwhile, an inferior-based population-cut strategy is also presented to remove several poor particles according to its ranking method and to reserve a place for a better reproduction. Moreover, both dynamic population strategies are controlled by a status monitor, which is used to keep track of the progress of individuals and improve the sensitivity of the proposed APTS. The experimental studies were carried out on25global numerical optimization problems used in the CEC2005special session on real-parameter optimization. An overall performance comparison between the JADE-APTS variant and other five State-of-the-Art DEs was also carried out. The experimental results illustrated that JADE-APTS achieves a competitive performance in30dimensional problems and exhibits the best performance in100dimensional problems. In addition, the ANOVA results verify that APTS can accelerate the convergence and enhance accuracy.
(2) Enhanced differential evolution with entropy-based population adapta-tion and markov chain model
An enhanced adaptive population-handling technique (CP) is proposed for DE algo-rithm to solve various types of optimization problems. In CPDE, we advocate a stochastic strategy-hopping framework in which the probability of selecting different sub-optimizers to improve the online solution-searching status is completely followed by a Markov chain. One sub-optimizer, called population increasing strategy, adds new individuals into the population to share their up-to-date information when particles are clustered together in a region and trapped into the local basin; the other sub-optimizer, namely population de-creasing strategy, removes redundant particle with its entropy and ranking metrics to save computational load. Extensive experiments have been carried out to compare it with five state-of-the-art DE variants and three other EAs on25commonly used CEC2005con-test test instances. In addition, a scalability study was implemented to show the effect of problem dimension.In the end, runtime complexity analysis and convergence rate com-parison are also validated that CP framework does not impose any serious burden on the time complexity of the existing DE variants.
(3) Identification of fractional-order systems via a switching differential evo-lution subject to noise perturbations
A switching differential evolution (SDE) algorithm is employed to identify the or-ders and parameters of incommensurate fractional-order Lorenz, Lii and Chen systems. The main feature of SDE is the switching population utilization strategy which improves the quality of the population and decreases the number of calculations by nonperiodic partial increasing or declining individuals. The results are shown in comparison with five other existing methods. The results obtained by our approach are better than other EAs especially in stochastic environment.
(4) Digital IIR filters design using differential evolution with a controllable probabilistic population size
An improved differential evolution is proposed for digital ⅡR filter design. The suggested algorithm is a kind of DE variants with a controllable probabilistic (CPDE) population size. It considers the convergence speed and the computational cost simul-taneously by nonperiodic partial increasing or declining individuals according to fitness diversities. Compared with six existing State-of-the-Art algorithms-based digital Ⅱ filter design methods obtained by numerical experiments, CPDE is relatively more promising and competitive. In addition, we discuss as well some important aspects for ⅡR filter design, such as the cost function value, the influence of (noise) perturbations, the conver-gence rate and successful percentage, the parameter measurement, etc. As to the simula-tion result, it shows that the presented algorithm is viable and comparable.
(5) Optimum modeling of PEM fuel cells by a hybrid differential evolution
A hybrid differential evolution (HDE) algorithm-based parameter identification ap-proach is proposed in terms of the voltage-current characteristics for the problem of pro-ton exchange membrane (PEM) fuel cell stack modeling. It attempts to utilize an adaptive population tuning scheme, in which the population size can be adjusted dynamically based on the solution-searching status and the desired population distribution. Thus, this active control approach realizes the optimum modeling of the SR-12Modular, the Ballard and the BCS500-W stack. The results indicate that satisfactory identification performance can be achieved by HDE even if the experimental data are corrupted by noise. Simulation results manifest that the proposed HDE reaches both better and more robust results in comparison with three versions of DEs, two versions of EAs, as well as CLPSO.
差分演化算法(DE),一直被视为一种可靠和通用的基于种群的元启发式优化技术,并广泛的在各类问题中展现了令人瞩目的性能。在过去十年里,归功于差分演化算法的简单性、可靠性、高性能和易于实施,DE在众多研究人员中已经获得了广泛的知名度。与传统的演化算法不同,DE算法通过增加一个带权值的移动向量来执行扰动操作,并修正一些随机选择的候选粒子维度的值。正因为这样的内在机制,差分演化算法能能够在演化早期高度地探索整个搜索空间,而在优化的后期变得更加注重自身剥削和开发。然而,DE并是不总能保证收敛到全局最优解,他会偶尔陷入局部停滞或者早熟收敛,而导致优化精度的降低甚至失败。
本论文针对传统差分演化算法局部停滞和早熟收敛等问题,研究了一类自适应种群谐调框架和方法,采用马尔科夫链和信息熵的理论,提出了基于DE的改进种群自适应策略。同时,将改进算法应用于系统设计问题,解决了分数阶混沌系统的参数辨识问题,无限脉冲响应数字滤波器的设计问题以及质子交换膜燃料电池的最优化建模问题。本文主要工作概括如下：
(1)差分演化算法的自适应种群谐调控制研究
通过权衡当前的解搜索状态和需要的种群分布两个指标,提出了一种崭新的动态自适应种群谐调策略(APTS)。在APTS中,首先设计了一个基于精英的种群增量策略,他在决策空间的适当位置中生成一些新个体帮助搜索更优的可行解。其次,设计了一个基于平庸的种群缩减策略,他依据排序方法删除一些性能较差的个体以减少计算负荷,并预留一些空间给新的带有种群多样性的扰动个体。此外,上述两个种群策略都由一个状态观测器所控制。该状态观测器被建立用于监控种群的演化进程,并适时地控制APTS的灵敏度。为验证算法有效性,实现了APTS的收敛性分析,为其提供了理论保障。同时,通过一个全局性的性能比较实验,与其他6种顶尖的DE算法比较来发现最优者。实验结果表明JADE-APTS在低维问题(30维)中获得了富有竞争性的性能,在高维问题(100维)中获得了最佳性能。此外,方差分析的结果同样证实了APTS能够有效地加速收敛率和提高可行解的搜索精度。
(2)基于马尔科夫链的种群自适应改进研究及其信息熵指标判据
一个改进的种群自适应处理技术(CP)被应用于DE以解决各种优化问题。在CPDE中,实现了一个随机的策略跳变框架(MHT),依靠非均匀的马尔科夫链来选择不同的子优化控制器,更好地改进当前解搜索的状态。具体的来说,子优化控制器有两种,其一,称作改进的sigmoid函数种群增长策略。增加一些新个体进入种群,提供他们最新的信息分享给种群并帮助粒子逃脱局部困境。其二,称作基于信息熵和等级排序指标的种群减少策略。基于每个粒子的聚集熵指标和等级排序指标,删除一些过渡个体以避免不期望地计算损失和过度的搜索复杂度。其次,实现了CEC05基准函数下,CPDE与其他8种最先进演化算法(即,5种DEs和3种EAs)的性能比较实验以证明所提出方法的可行性。同时,维度可扩展性测试实验也同样证实,CP框架总能加速DE算法的搜索效能和效率,尤其是在高维问题中效果更为明显。最后,收敛速度实验和时间复杂度推导进一步证明了CP框架对迄今为止的所有差分演化变体算法不产生任何额外的计算负担。
(3)基于改进差分演化算法的分数阶混沌系统参数辨识
采用一种改进的差分演化算法(SDE),首先研究了分数阶Lorenz,Lu和Chen系统在确定性环境下的未知阶次和参数的估计问题。SDE的主要特点是有效的种群切换利用策略。他同时考虑收敛速度和计算负荷,根据适应度多样性非周期地增加和减少一些粒子。其次,研究上述3个系统在随机性环境下的未知阶次和参数的估计问题,即噪声扰动下的算法性能。五种最先进智能算法被应用于测试实验来验证SDE算法的有效性。实验结果表明我们的方法要比其他5种算法性能更优,尤其是在噪声扰动情况下。
(4)基于种群概率可控差分演化算法的无限脉冲响应数字滤波器设计
提出了一种基于马尔科夫跳变(开关切换)的种群更新DE算法用以解决限脉冲响应数字滤波器的设计问题。所提出的算法是一种带有可控概率种群大小的差分演化变体(CPDE),通过适应度多样性非周期地增加和减少一些粒子,权衡搜索广度和自身精度。进一步,6种公认优秀的演化算法被采纳用于设计上述6种典型的IIR滤波器,并和CPDE进行性能比较实验,以证明所提出方法的可行性。此外,我们还讨论了IIR数字滤波器设计的一些关键方面,如价值函数值、噪声扰动、收敛速度、成功率以及参数测量等。实验结果表明,我们提出的算法是可行且强有力的。
(5)基于混合差分演化算法的质子交换膜燃料电池最优化建模
基于极化曲线研究了一类适用于工程优化的电化学PEMFC模型。采用改进的一种改进的差分演化算法(HDE),引入动态种群谐调策略,对3个质子交换膜燃料电池模型进行参数辨识(即SR-12Modular PEM Generator, Ballard Mark V FC和BCS500-W stack模型)。在HDE中,种群的大小可以动态自适应地根据现今的搜索状态和所需的种群分布进行调节。同时,我们还测试在3%的噪声扰动下,算法对PEMFC的辨识性能。实验结果表明,即使实验过程被噪声破坏,HDE仍能获得较令人满意的辨识性能。此外,6种最顶尖的智能算法被应用于测试实验来验证HDE算法的有效性。
To tackle complex computational problems, researchers have been looking into na-ture for years for inspiration. Optimization is at the heart of many natural processes like Darwinian evolution itself. Through millions of years, every species had to adapt their physical structures to fit to the environments they were in. A keen observation of the underlying relation between optimization and biological evolution led to the development of an important paradigm of computational intelligence-the evolutionary computing techniques for performing very complex search and optimization.
Differential Evolution is a reliable and versatile function optimizer. DE, like most popular Evolutionary Algorithms (EAs), is a population-based tool. Over the past decade, the DE algorithm has gained wide-spread popularity among researchers due to its sim-plicity, reliability, high performance and easy implementation. Unlike traditional EAs, DE operates perturbation by adding a weighted moving vector and modifying the values of some randomly selected coordinates. Due to its structure, a DE scheme can be highly explorative in the prophase of the evolution and subsequently become more exploita-tive during the optimization. However, the DE does not guarantee the convergence to the global optimum. It is occasionally trapped into local stagnation or premature convergence resulting in a low optimizing precision or even failure.
In this dissertation, firstly, to remedy some of DE's pitfalls, I propose an adaptive population tuning scheme (APTS) for DE. Secondly, an enhanced adaptive population-handling technique is proposed to solve various types of optimization problems. Thirdly, I study the applications of differential evolution in identification of fractional-order sys-tems, digital IIR filters design and optimum modeling of PEM fuel cells. The main con-tribution of this dissertation are as follows.
(1) Adaptive population tuning scheme for differential evolution
An adaptive population tuning scheme (APTS) for DE is proposed to dynamically adjust the population size. More specifically, an elite-based population-incremental strat- egy is proposed to place several new individuals in appropriate areas to discover new pos-sible solutions. Meanwhile, an inferior-based population-cut strategy is also presented to remove several poor particles according to its ranking method and to reserve a place for a better reproduction. Moreover, both dynamic population strategies are controlled by a status monitor, which is used to keep track of the progress of individuals and improve the sensitivity of the proposed APTS. The experimental studies were carried out on25global numerical optimization problems used in the CEC2005special session on real-parameter optimization. An overall performance comparison between the JADE-APTS variant and other five State-of-the-Art DEs was also carried out. The experimental results illustrated that JADE-APTS achieves a competitive performance in30dimensional problems and exhibits the best performance in100dimensional problems. In addition, the ANOVA results verify that APTS can accelerate the convergence and enhance accuracy.
(2) Enhanced differential evolution with entropy-based population adapta-tion and markov chain model
An enhanced adaptive population-handling technique (CP) is proposed for DE algo-rithm to solve various types of optimization problems. In CPDE, we advocate a stochastic strategy-hopping framework in which the probability of selecting different sub-optimizers to improve the online solution-searching status is completely followed by a Markov chain. One sub-optimizer, called population increasing strategy, adds new individuals into the population to share their up-to-date information when particles are clustered together in a region and trapped into the local basin; the other sub-optimizer, namely population de-creasing strategy, removes redundant particle with its entropy and ranking metrics to save computational load. Extensive experiments have been carried out to compare it with five state-of-the-art DE variants and three other EAs on25commonly used CEC2005con-test test instances. In addition, a scalability study was implemented to show the effect of problem dimension.In the end, runtime complexity analysis and convergence rate com-parison are also validated that CP framework does not impose any serious burden on the time complexity of the existing DE variants.
(3) Identification of fractional-order systems via a switching differential evo-lution subject to noise perturbations
A switching differential evolution (SDE) algorithm is employed to identify the or-ders and parameters of incommensurate fractional-order Lorenz, Lii and Chen systems. The main feature of SDE is the switching population utilization strategy which improves the quality of the population and decreases the number of calculations by nonperiodic partial increasing or declining individuals. The results are shown in comparison with five other existing methods. The results obtained by our approach are better than other EAs especially in stochastic environment.
(4) Digital IIR filters design using differential evolution with a controllable probabilistic population size
An improved differential evolution is proposed for digital ⅡR filter design. The suggested algorithm is a kind of DE variants with a controllable probabilistic (CPDE) population size. It considers the convergence speed and the computational cost simul-taneously by nonperiodic partial increasing or declining individuals according to fitness diversities. Compared with six existing State-of-the-Art algorithms-based digital Ⅱ filter design methods obtained by numerical experiments, CPDE is relatively more promising and competitive. In addition, we discuss as well some important aspects for ⅡR filter design, such as the cost function value, the influence of (noise) perturbations, the conver-gence rate and successful percentage, the parameter measurement, etc. As to the simula-tion result, it shows that the presented algorithm is viable and comparable.
(5) Optimum modeling of PEM fuel cells by a hybrid differential evolution
A hybrid differential evolution (HDE) algorithm-based parameter identification ap-proach is proposed in terms of the voltage-current characteristics for the problem of pro-ton exchange membrane (PEM) fuel cell stack modeling. It attempts to utilize an adaptive population tuning scheme, in which the population size can be adjusted dynamically based on the solution-searching status and the desired population distribution. Thus, this active control approach realizes the optimum modeling of the SR-12Modular, the Ballard and the BCS500-W stack. The results indicate that satisfactory identification performance can be achieved by HDE even if the experimental data are corrupted by noise. Simulation results manifest that the proposed HDE reaches both better and more robust results in comparison with three versions of DEs, two versions of EAs, as well as CLPSO.
引文
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