分布估计算法研究及在动态优化问题中的应用
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摘要
分布估计算法(EDAs)通过将统计学习理论与进化算法结合,形成一种全新的进化模式,是进化计算领域的研究热点。分布估计算法从提出到现在虽然取得了一定的进展,但是还有很多问题需要深入研究,例如理论分析、算法设计、应用研究等。本文以分布估计算法为基础,根据算法的类型和特点,主要从算法性能的改进以及在动态优化问题中的应用两个方面进行了研究,主要创新成果如下:
1.研究了分布估计算法的收敛性,首先通过在期望分布基础上引入一个误差量,建立有限群体分布估计算法模型;然后在三种不同的常用选择策略下证明了EDAs的收敛性。结果显示在有限群体模型下,在本文所述误差范围内分布估计算法具有全局收敛性。
2.针对多变量相关的分布估计算法(Bayesian优化算法:BOA),从三个方面对算法进行了改进研究。首先,针对BOA计算量大的问题,提出了结合局部结构学习的Bayesian优化算法并分析了算法的复杂度;其次,讨论了一般优化问题先验知识的挖掘和利用方式,把BOA中前一代种群所提供的信息作为先验知识结合到当前代Bayesian网络的学习中,提高了所学习网络的可靠性,从而提高算法的性能;最后,讨论了BOA的多样性,设计了一个种群多样性函数,通过此函数引入变异算子,以保持种群多样性,避免算法陷入局部最优。以上算法均通过仿真实验表明了算法的有效性。
3.提出一种改进的基于群体的增量学习分布估计算法(PBIL算法),利用改进的算法求解了一类特殊的动态优化问题。首先从动态环境随时间变化的特点进行分析,归纳出一类何时变化满足一定统计分布的动态优化问题。然后针对这类问题提出自适应PBIL算法,根据何时变化这个随机变量的概率自适应的调整当前代群体的概率模型,增加种群多样性,快速适应环境的变化,最后对设计的算法进行了比较性仿真验证。
4.针对动态离散优化问题,采用多群体的思想,提出了一种多群体单变量边缘分布算法(MUMDA),利用多个概率模型(对应多个群体)将搜索空间分成几个部分,通过对不同区域的搜索或者探索并对好解进行迁移,扩大搜索空间,增加种群多样性,跟踪最优解的变化,并证明了所提算法的收敛性。比较分析表明,所提算法能快速跟踪最优解。
5.针对动态单目标优化问题,提出一种自组织策略,利用当前环境的局部信息和最优解的历史信息,自适应的增加种群多样性,将自组织策略与单变量边缘分布算法(UMDA)结合,提出一种新的自组织单变量边缘分布算法(SOUMDA),利用动态sphere函数对所提算法进行了测试。
6.针对动态多模优化问题,提出一种新的多群体及扩散单变量边缘分布算法(MDUMDA),多群体方法用来并行地寻找多个最优解,扩散模型用来有指导的增加种群多样性,使得前一环境的最优解的邻域集逐渐远离这一最优解并扩大搜索空间,使算法快速适应环境的变化。利用动态优化问题标准测试例子MPB对所提算法进行测试,仿真结果表明了算法的有效性。
7.针对动态多目标优化问题,提出了一种基于预测模型的正则分布估计算法(PREDA)。在算法设计中,首先利用Pareto最优解集的多个类中心与参考点描述Pareto最优解集,设计了一种动态多目标问题中历史数据的存储方式;其次通过惯性预测与高斯变异产生预测点集,并将预测点集结合到当前群体中,使得算法在环境变化后有指导的增加种群多样性,提高算法适应环境的能力,快速跟踪最优解,通过标准动态测试问题对所提算法进行了仿真实验,与相关算法进行比较分析结果表明本文设计的算法能快速适应环境的变化,跟踪Pareto最优解。
Estimation of distribution algorithm (EDA) is a new class of evolutionary algorithms, which combines the statistic theory with evolutionary schemes. In recent years, EDA attracts more and more attentions. Although EDA has got some progresses since it was proposed, there are many unsolved problems, such as theory analysis, algorithm designs and applications, etc. Based on characteristics and types of EDA, we mainly focus our study on the performance improvement of EDA and the applications of EDA to dynamic optimization problems. The main contributions of this dissertation are as follows:
1. Study the convergence of estimation of distribution algorithm. Firstly, a model of EDAs with finite population is built up by incorporating an error into expected distribution of parent population. Secondly the convergence of EDAs is proved with finite population under three widely used selection schemes.
2. Bayesian optimization algorithm (BOA) is improved from three aspects. Firstly, a Bayesian optimization algorithm incorporated with local structure learning is proposed to decrease the computation, and the complexity of the proposed algorithm is analyzed. Secondly, according to the characteristics of BOA, it is discussed on how to discovery and use prior knowledge in general optimization problems. Concretely, the information discovered in the previous generation is used as prior knowledge and is incorporated into the Bayesian networks learning. As a result, the reliability of the networks and the performance of the proposed algorithm are improved. Finally, we discuss the population diversity of BOA and design a diversity function. By using this function, the mutation operator is incorporated into BOA. The objective is to maintain the diversity of population and avoid the algorithm traping in the local optima. Simulations indicate that the above three algorithms are all effective.
3. We design a class of special dynamic optimization problem and propose an algorithm to solve it. Firstly, according to the characteristic of changing environment, a class of special dynamic problem in which the change time obeys some distribution is addressed. Secondly, for this type of problems, an adaptive PBIL (Population-based incremental learning) algorithm is presented, where the probability of the random variable is used to adaptively adjust the probability model of the current population, to increase population diversity and to promptly react to the environment changing. The experimental results show that adaptive PBIL can track the optimal solution more reliably and accurately compared with PBIL.
4. For dynamic discrete optimization problems, a multi-population univariate marginal distribution algorithm (MUMDA) is proposed. The main idea of the algorithm is to divide the search space into several parts by using several probability modals corresponding to several populations. Meanwhile the algorithm explores and exploits in different regions and can make the best solutions migrate. The experimental results show that the MUMDA is effective and can react to the dynamic environments promptly.
5. For dynamic simple-objective optimization problems, a self-organization scheme is proposed. This approach combines the current information and the historic information of optimal solutions to increase the population diversity. Incorporated by the scheme, a new self-organization univariate marginal distribution algorithm (SOUMDA) is proposed. The experimental study on dynamic sphere function showed that the SOUMDA is effective and can adapt to the dynamic environments rapidly.
6. For dynamic multimodal optimization problems, a new Multi-population and Diffusion UMDA (MDUMDA) is presented. The multi-population approach is used to locate multiple local optima for dynamic multimodal problems. The diffusion model is used to guide increasing the diversity, which makes the neighbor individuals of previous optimal solutions move gradually and thus can enlarge the search space. The experimental studies on the multimodal dynamic moving peaks benchmark are made to evaluate the proposed algorithm.
7. For dynamic multi-objective optimization problems, a regularity modal-based estimation of distribution algorithm with prediction is proposed. In algorithm design, we use the central points of Pareto optimal set and reference solutions of Pareto optimal set to describe Pareto optimal solutions. So it can be used to storage the history of the Pareto optimal solutions of dynamic multi-objective optimization problems. Then by using the inertia prediction and Gauss mutation the prediction set of central points and reference solutions is generated. After an environment changed, the predict set is incorporated in the current population to guide increasing the population diversity. Finally, the experiments on dynamic multiobjective optimization problems were carried out to evaluate the performance of our proposed algorithm.
1.研究了分布估计算法的收敛性,首先通过在期望分布基础上引入一个误差量,建立有限群体分布估计算法模型;然后在三种不同的常用选择策略下证明了EDAs的收敛性。结果显示在有限群体模型下,在本文所述误差范围内分布估计算法具有全局收敛性。
2.针对多变量相关的分布估计算法(Bayesian优化算法:BOA),从三个方面对算法进行了改进研究。首先,针对BOA计算量大的问题,提出了结合局部结构学习的Bayesian优化算法并分析了算法的复杂度;其次,讨论了一般优化问题先验知识的挖掘和利用方式,把BOA中前一代种群所提供的信息作为先验知识结合到当前代Bayesian网络的学习中,提高了所学习网络的可靠性,从而提高算法的性能;最后,讨论了BOA的多样性,设计了一个种群多样性函数,通过此函数引入变异算子,以保持种群多样性,避免算法陷入局部最优。以上算法均通过仿真实验表明了算法的有效性。
3.提出一种改进的基于群体的增量学习分布估计算法(PBIL算法),利用改进的算法求解了一类特殊的动态优化问题。首先从动态环境随时间变化的特点进行分析,归纳出一类何时变化满足一定统计分布的动态优化问题。然后针对这类问题提出自适应PBIL算法,根据何时变化这个随机变量的概率自适应的调整当前代群体的概率模型,增加种群多样性,快速适应环境的变化,最后对设计的算法进行了比较性仿真验证。
4.针对动态离散优化问题,采用多群体的思想,提出了一种多群体单变量边缘分布算法(MUMDA),利用多个概率模型(对应多个群体)将搜索空间分成几个部分,通过对不同区域的搜索或者探索并对好解进行迁移,扩大搜索空间,增加种群多样性,跟踪最优解的变化,并证明了所提算法的收敛性。比较分析表明,所提算法能快速跟踪最优解。
5.针对动态单目标优化问题,提出一种自组织策略,利用当前环境的局部信息和最优解的历史信息,自适应的增加种群多样性,将自组织策略与单变量边缘分布算法(UMDA)结合,提出一种新的自组织单变量边缘分布算法(SOUMDA),利用动态sphere函数对所提算法进行了测试。
6.针对动态多模优化问题,提出一种新的多群体及扩散单变量边缘分布算法(MDUMDA),多群体方法用来并行地寻找多个最优解,扩散模型用来有指导的增加种群多样性,使得前一环境的最优解的邻域集逐渐远离这一最优解并扩大搜索空间,使算法快速适应环境的变化。利用动态优化问题标准测试例子MPB对所提算法进行测试,仿真结果表明了算法的有效性。
7.针对动态多目标优化问题,提出了一种基于预测模型的正则分布估计算法(PREDA)。在算法设计中,首先利用Pareto最优解集的多个类中心与参考点描述Pareto最优解集,设计了一种动态多目标问题中历史数据的存储方式;其次通过惯性预测与高斯变异产生预测点集,并将预测点集结合到当前群体中,使得算法在环境变化后有指导的增加种群多样性,提高算法适应环境的能力,快速跟踪最优解,通过标准动态测试问题对所提算法进行了仿真实验,与相关算法进行比较分析结果表明本文设计的算法能快速适应环境的变化,跟踪Pareto最优解。
Estimation of distribution algorithm (EDA) is a new class of evolutionary algorithms, which combines the statistic theory with evolutionary schemes. In recent years, EDA attracts more and more attentions. Although EDA has got some progresses since it was proposed, there are many unsolved problems, such as theory analysis, algorithm designs and applications, etc. Based on characteristics and types of EDA, we mainly focus our study on the performance improvement of EDA and the applications of EDA to dynamic optimization problems. The main contributions of this dissertation are as follows:
1. Study the convergence of estimation of distribution algorithm. Firstly, a model of EDAs with finite population is built up by incorporating an error into expected distribution of parent population. Secondly the convergence of EDAs is proved with finite population under three widely used selection schemes.
2. Bayesian optimization algorithm (BOA) is improved from three aspects. Firstly, a Bayesian optimization algorithm incorporated with local structure learning is proposed to decrease the computation, and the complexity of the proposed algorithm is analyzed. Secondly, according to the characteristics of BOA, it is discussed on how to discovery and use prior knowledge in general optimization problems. Concretely, the information discovered in the previous generation is used as prior knowledge and is incorporated into the Bayesian networks learning. As a result, the reliability of the networks and the performance of the proposed algorithm are improved. Finally, we discuss the population diversity of BOA and design a diversity function. By using this function, the mutation operator is incorporated into BOA. The objective is to maintain the diversity of population and avoid the algorithm traping in the local optima. Simulations indicate that the above three algorithms are all effective.
3. We design a class of special dynamic optimization problem and propose an algorithm to solve it. Firstly, according to the characteristic of changing environment, a class of special dynamic problem in which the change time obeys some distribution is addressed. Secondly, for this type of problems, an adaptive PBIL (Population-based incremental learning) algorithm is presented, where the probability of the random variable is used to adaptively adjust the probability model of the current population, to increase population diversity and to promptly react to the environment changing. The experimental results show that adaptive PBIL can track the optimal solution more reliably and accurately compared with PBIL.
4. For dynamic discrete optimization problems, a multi-population univariate marginal distribution algorithm (MUMDA) is proposed. The main idea of the algorithm is to divide the search space into several parts by using several probability modals corresponding to several populations. Meanwhile the algorithm explores and exploits in different regions and can make the best solutions migrate. The experimental results show that the MUMDA is effective and can react to the dynamic environments promptly.
5. For dynamic simple-objective optimization problems, a self-organization scheme is proposed. This approach combines the current information and the historic information of optimal solutions to increase the population diversity. Incorporated by the scheme, a new self-organization univariate marginal distribution algorithm (SOUMDA) is proposed. The experimental study on dynamic sphere function showed that the SOUMDA is effective and can adapt to the dynamic environments rapidly.
6. For dynamic multimodal optimization problems, a new Multi-population and Diffusion UMDA (MDUMDA) is presented. The multi-population approach is used to locate multiple local optima for dynamic multimodal problems. The diffusion model is used to guide increasing the diversity, which makes the neighbor individuals of previous optimal solutions move gradually and thus can enlarge the search space. The experimental studies on the multimodal dynamic moving peaks benchmark are made to evaluate the proposed algorithm.
7. For dynamic multi-objective optimization problems, a regularity modal-based estimation of distribution algorithm with prediction is proposed. In algorithm design, we use the central points of Pareto optimal set and reference solutions of Pareto optimal set to describe Pareto optimal solutions. So it can be used to storage the history of the Pareto optimal solutions of dynamic multi-objective optimization problems. Then by using the inertia prediction and Gauss mutation the prediction set of central points and reference solutions is generated. After an environment changed, the predict set is incorporated in the current population to guide increasing the population diversity. Finally, the experiments on dynamic multiobjective optimization problems were carried out to evaluate the performance of our proposed algorithm.
引文
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