基于模型多目标原理的星座优化算法设计
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摘要
科学研究、经济领域和工程实践中的优化问题大多是多目标问题。多目标问题的最优解是一个集合,而多目标演化算法一次进化过程可以得到多个可行解,近年来演化算法逐渐成为求解多目标问题的主流方法。
卫星星座是由多颗卫星组成一个卫星网来协同工作,以满足特定空间任务所需的对地覆盖要求。卫星凭借其独特的空间位置优势,在通信、导航定位系统、防御间谍卫星、对地观测、气象预测及地质勘探等领域中都有着广泛的应用。如何设计高效、合理的卫星星座配置方案,是星座优化的关键问题。星座优化问题往往涉及多个特征点和多项优化指标,是一个复杂的高维、动态、多目标优化问题,鉴于传统演化算法忽略了Pareto解集分布规则对解集搜索的指导性且大多存在求解效率不高,算法收敛性不强等不足,随着科技的不断发展,研究和应用的不断深入,实际要优化的问题将越来越复杂,这必然会对算法用到的技术和性能提出新的挑战。
分布估计算法由Thierens、Bosman等人于2001年引入多目标优化领域,张清富、周爱民等人将被传统算法忽略的“某一类连续的多目标问题的Pareto解集的分布呈现一定的规则性”这一重要特性运用进了该算法,并在解决高维多目标优化的问题上取得了很好的效果。
分布估计算法(EDA)与遗传算法最大的不同就是它没有传统演化算法的交叉、变异操作,而是运用统计学,在每一代建立描述当前种群分布的概率模型,并从中以一定概率随机取样来产生新的子代。2007年,张清富、周爱民等人提出了基于规则模型的多目标评估分布算法(RM-MEDA),给出规则：m个目标的连续多目标优化问题的Pareto解集在决策空间的分布是一个分段连续的m-1维流形,并对符合这一规则的多目标优化问题进行求解,求得解集的多样性和收敛性比PCX-NSGA-Ⅱ、GDE3(Generalized Differential Evolution)、MIEDA等算法的要更好。
然而基于规则模型的多目标分布估计算法也存在一定的不足。基于模型的算法的聚类分析、概率建模的过程更复杂,算法运行较耗时。在某些测试问题上,在算法初期就采用概率模型产生新个体往往会使搜索方向与实际目标搜索方向相差甚远,如基于规则模型的分布估计多目标算法RM-MEDA在ZDT4、ZDT6、DTLZ1及DTLZ3上就是发散的,从而在一定程度上造成了算法的不稳定性。
本文首先对课题的研究背景、意义和国内外研究现状进行综述,深入探讨了多目标优化问题,分析了传统演化算法和分布估计算法各自的优势及不足,着重研究了基于模型原理的多目标优化算法,引入并阐述了当前存在的两类多目标优化问题：第一类连续多目标优化问题,PS和PF都是m-1维的流形；第二类连续多目标问题,PF是m-1维流形,而PS是更高维的连续流形(大于m-1维)。然后,本文针对两类多目标优化问题分别改进、设计算法,包括了：1)流形初始化种群策略,在正交基础上建立一个理想Pareto前沿,并从样本点中选取离理想Pareto前沿最近的一定量个体作为初始种群个体,从而使种群个体尽可能均匀地逼近Pareto前沿；2)在前人用GA+EDA来产生子代的工作基础上,对第一类问题快速收敛准则中的ε参数进行修正,通过不同的ε值进行试验从而选取恰当的参数值,在保证甚至提高结果质量的前提下大大提升算法采用概率建模生成子代的比例；3)第二类问题中潜在维数的确定和快速收敛准则的设计,当种群各个聚类的潜在维数相差不大时,就断定种群的分布呈现了一定规律,此时第二类问题就可以使用概率建模产生子代了。
同时,本文对当前常用的测试函数集的概念、构造及特性做了一个简要的阐述,在回顾当前使用得较多测试集的同时,着重讲述WFG测试集的构造及特性,并在原有测试函数ZDT及DTLZ系列的基础上,新构造了三个PS要复杂得多的测试函数ZDT2.3、DTLZ2.3及DTLZ2.4(前一个的PS为多峰曲线,后两个的PS为多峰曲面)。本文还在当前常用的度量指标收敛性Υ及多样性△的基础上引入了IGD度量指标,IGD度量在评估收敛性的同时兼顾了种群的均匀分布性评估,更好地补充了算法对三目标问题的性能评价。
本文将该改进后的算法n-RM-MEDA与NSGA-Ⅱ、m-NSGA-Ⅱ(加入流形初始化的NSGA-Ⅱ、基本的基于规则模型的算法RM-MEDA及基于正交的模型算法IORM-MEDA分别进行比较,通过对结果的统计和分析,发现当问题Pareto解集分布规则的复杂度很低时(如ZDT1-6),会弱化基于模型建模算法的优势,此时m-NSGA-Ⅱ优势显著；但在问题的PS分布规则变复杂之后,概率建模算法逐渐显露优势,NSGA-Ⅱ和m-NSGA-Ⅱ则显得力不从心,不仅在多样性上大失水准,连收敛性能都有所下降,这时在算法一开始运行就采用概率建模产生子代并不利于算法在整个空间的搜索,而一味采用遗传算法产生子代(如NSGA-Ⅱ和m-NSGA-Ⅱ)一样得不到好的结果,相比之下改进后的算法融合遗传算法与概率建模两种方法来产生子代则是个不错的选择,也验证了改进后的算法在收敛性和均匀分布上的有效性。
在第五章中,本文先对星座优化设计中的相关物理模型和航天方面的背景基本知识进行了简要介绍,用改进后的算法来解决低轨星座优化问题,并将改进后的算法与m-NSGA-Ⅱ、RM-MEDA、IORM-MEDA及文献结果分别进行了比较,从结果可以看到算法优化后的覆盖性能,能较好满足特定点持续覆盖的要求,对卫星星座设计的决策者来说有一定的参考价值。
Many optimization problems, in scientific research economy areas and engineering areas, belong to multi-objective optimization problems. The optimal solution of multi-objective problem is a set of solutions. As multi-objective evolutionary algorithms can get many feasible solutions, multi-objective evolutionary algorithms become hot spots and achieved many good results in multi-objective area gradually in recent decades.
The constellation composed of multi-satellites that worked synergically to satisfy the regional coverage demand required by specifically spatial mission. Satellites play a more and more important role in many fields such as communication, navigation and surveillance, weather forecast, geological prospecting by right of their unique spatial position advantage. It is a key issue for constellation optimization to design efficient and feasible constellation layout. Constellation optimization relating to multi-aim points and multi-optimized index is a complex multi-dimensional, dynamic, multi-objective problem. Since it is not high efficiency to get solutions and is not strong convergence and doesn't utilize distributing regulation of Pareto set for conventional evolvement algorithms, with the development of research and application, the complexity of resolution bring forward new challenge for performance of algorithm.
Estimation of distribution algorithms (EDAs) were introduced to the field of multi-objective optimization by Thierens, Bosman and etc in 2001. EDAs use the important speciality that the Pareto sets'distributing of certain continuous multi-objective optimization problems present definite regulation, and reach good solutions on the high dimensional multi-objective optimization problem.
EDAs don't use crossover or mutation opteration like conventional evolvement algorithms but extract globally statistical information from the selected solutions and build probability distribution models of promising solutions based on this extracted information. New solutions are sampled from the models thus built. Qingfu Zhang etc. have developed a regularity model-based multi-objective estimation of distribution (RM-MEDA) in 2007 which captures and utilizes the regularity of the Pareto set in the decision space. Systematic experiments show that, overall RM-MEDA outperforms GDE2, PCX-NSGA-Ⅱand MIDEA, on a set of test instances with variable linkages.
However, RM-MEDA has its drawbacks and shortcomings. Clustering and building probability distribution models of algorithm based on model are more complex and cost much time. Adopt probability distribution models to produce new individuals at early of algorithm on some test problems usually make the search direction far away from objective direction. For example, the regularity model-based multi-objective estimation of distribution (RM-MEDA) is not convergence on ZDT4, ZDT6, DTLZ1and DTLZ3, which make algorithm not stability on some degree.
This paper first presents the research background, significance and actuality in and out of problem, discussing multi-objective optimization problem and analyzing advantage and shortcoming of conventional evolvement algorithms and estimation of distribution algorithms. Study of model based algorithm is emphasized and two kinds of multi-objective optimization problem are brought in and expatiated. The manifold of PF and PS for the firs kind of multi-objective problem are all (m-1)-D, while the The manifold of PF for the second kind of multi-objective problem is (m-1)-D and the dimension of its PS is higher than (m-1)-D.
Then we improve and design model based algorithm for the two kinds of algorithm including:1) strategy of manifold initialization population, construct a upotian Pareto front(UPF) and choose N(size of population) individuals that are closestt to UPF from current samples as initialized samples. This strategy make the individuals of population trying their best to approach Pareto front uniformly; 2) revise parameter of convergence criterion in the first kind of optimization problem based on anterior work that producing individuals in the way of GA+EDA, which enhance the proportion of using EDA to produce individuals in the precondition of keeping even improving quality of solutions; 3) make sure of latent dimension and design convergence criterion for the second kind optimization problem, when the latent dimension of K clusters in population are not far away from each other we decide the distribution of population present certain regulation, and this time we can use EDA to creat the next generation.
In addition, we construct three new test functions ZDT2.3, DTLZ2.3 and DTLZ2.4 with more complex PS(multimodal curve or multimodal curved surface) and supply IGD measurement based on convergence measurement and diversity measurement. The IGD measurement evaluates convergence and distributing uniformity at the same time which renew performance evaluate of three-objective problem.
We compare improvement algorithm m-RM-MEDA with NSGA-Ⅱ、NSGA-Ⅱwith manifold initialization(m-NSGA-Ⅱ)、RM-MEDA and IORM-MEDA. After analyzing the result we found that the advantage of model-based algorithm will be weaken when the complexity of problems' regulation is low while the advantage of m-NSGA-Ⅱis prominence. However when the distribution regulation of PS of problems become more complex, the model-based algorithm appear advantage gradually, at the same time the convergence of NSGA-Ⅱand m-NSGA-Ⅱare down and the their diversity are off form. As the PS of problems become more complex, use EDA to creat the next generation at early make against search in all solutions space for algorithm. But it is aslo bad if we just use GA to creat child generation. So it is a good choice to merge GA and EDA to produce child. The experimental results aslo show that m-RM-MEDA is efficient on convergence and distributing uniformity.
The fifth chapter use m-RM-MEDA to resolve low earth orbit (LEO) constellation optimization problem and compared with m-NSGA-Ⅱ、RM-MEDA and IORM-MEDA. The experiment results show that the convergence percentage of our improved algorithm is better than other algorithms, and can better meet the regional durative coverage demand, providing a useful reference for decision maker dealing with constellation optimization.
卫星星座是由多颗卫星组成一个卫星网来协同工作,以满足特定空间任务所需的对地覆盖要求。卫星凭借其独特的空间位置优势,在通信、导航定位系统、防御间谍卫星、对地观测、气象预测及地质勘探等领域中都有着广泛的应用。如何设计高效、合理的卫星星座配置方案,是星座优化的关键问题。星座优化问题往往涉及多个特征点和多项优化指标,是一个复杂的高维、动态、多目标优化问题,鉴于传统演化算法忽略了Pareto解集分布规则对解集搜索的指导性且大多存在求解效率不高,算法收敛性不强等不足,随着科技的不断发展,研究和应用的不断深入,实际要优化的问题将越来越复杂,这必然会对算法用到的技术和性能提出新的挑战。
分布估计算法由Thierens、Bosman等人于2001年引入多目标优化领域,张清富、周爱民等人将被传统算法忽略的“某一类连续的多目标问题的Pareto解集的分布呈现一定的规则性”这一重要特性运用进了该算法,并在解决高维多目标优化的问题上取得了很好的效果。
分布估计算法(EDA)与遗传算法最大的不同就是它没有传统演化算法的交叉、变异操作,而是运用统计学,在每一代建立描述当前种群分布的概率模型,并从中以一定概率随机取样来产生新的子代。2007年,张清富、周爱民等人提出了基于规则模型的多目标评估分布算法(RM-MEDA),给出规则：m个目标的连续多目标优化问题的Pareto解集在决策空间的分布是一个分段连续的m-1维流形,并对符合这一规则的多目标优化问题进行求解,求得解集的多样性和收敛性比PCX-NSGA-Ⅱ、GDE3(Generalized Differential Evolution)、MIEDA等算法的要更好。
然而基于规则模型的多目标分布估计算法也存在一定的不足。基于模型的算法的聚类分析、概率建模的过程更复杂,算法运行较耗时。在某些测试问题上,在算法初期就采用概率模型产生新个体往往会使搜索方向与实际目标搜索方向相差甚远,如基于规则模型的分布估计多目标算法RM-MEDA在ZDT4、ZDT6、DTLZ1及DTLZ3上就是发散的,从而在一定程度上造成了算法的不稳定性。
本文首先对课题的研究背景、意义和国内外研究现状进行综述,深入探讨了多目标优化问题,分析了传统演化算法和分布估计算法各自的优势及不足,着重研究了基于模型原理的多目标优化算法,引入并阐述了当前存在的两类多目标优化问题：第一类连续多目标优化问题,PS和PF都是m-1维的流形；第二类连续多目标问题,PF是m-1维流形,而PS是更高维的连续流形(大于m-1维)。然后,本文针对两类多目标优化问题分别改进、设计算法,包括了：1)流形初始化种群策略,在正交基础上建立一个理想Pareto前沿,并从样本点中选取离理想Pareto前沿最近的一定量个体作为初始种群个体,从而使种群个体尽可能均匀地逼近Pareto前沿；2)在前人用GA+EDA来产生子代的工作基础上,对第一类问题快速收敛准则中的ε参数进行修正,通过不同的ε值进行试验从而选取恰当的参数值,在保证甚至提高结果质量的前提下大大提升算法采用概率建模生成子代的比例；3)第二类问题中潜在维数的确定和快速收敛准则的设计,当种群各个聚类的潜在维数相差不大时,就断定种群的分布呈现了一定规律,此时第二类问题就可以使用概率建模产生子代了。
同时,本文对当前常用的测试函数集的概念、构造及特性做了一个简要的阐述,在回顾当前使用得较多测试集的同时,着重讲述WFG测试集的构造及特性,并在原有测试函数ZDT及DTLZ系列的基础上,新构造了三个PS要复杂得多的测试函数ZDT2.3、DTLZ2.3及DTLZ2.4(前一个的PS为多峰曲线,后两个的PS为多峰曲面)。本文还在当前常用的度量指标收敛性Υ及多样性△的基础上引入了IGD度量指标,IGD度量在评估收敛性的同时兼顾了种群的均匀分布性评估,更好地补充了算法对三目标问题的性能评价。
本文将该改进后的算法n-RM-MEDA与NSGA-Ⅱ、m-NSGA-Ⅱ(加入流形初始化的NSGA-Ⅱ、基本的基于规则模型的算法RM-MEDA及基于正交的模型算法IORM-MEDA分别进行比较,通过对结果的统计和分析,发现当问题Pareto解集分布规则的复杂度很低时(如ZDT1-6),会弱化基于模型建模算法的优势,此时m-NSGA-Ⅱ优势显著；但在问题的PS分布规则变复杂之后,概率建模算法逐渐显露优势,NSGA-Ⅱ和m-NSGA-Ⅱ则显得力不从心,不仅在多样性上大失水准,连收敛性能都有所下降,这时在算法一开始运行就采用概率建模产生子代并不利于算法在整个空间的搜索,而一味采用遗传算法产生子代(如NSGA-Ⅱ和m-NSGA-Ⅱ)一样得不到好的结果,相比之下改进后的算法融合遗传算法与概率建模两种方法来产生子代则是个不错的选择,也验证了改进后的算法在收敛性和均匀分布上的有效性。
在第五章中,本文先对星座优化设计中的相关物理模型和航天方面的背景基本知识进行了简要介绍,用改进后的算法来解决低轨星座优化问题,并将改进后的算法与m-NSGA-Ⅱ、RM-MEDA、IORM-MEDA及文献结果分别进行了比较,从结果可以看到算法优化后的覆盖性能,能较好满足特定点持续覆盖的要求,对卫星星座设计的决策者来说有一定的参考价值。
Many optimization problems, in scientific research economy areas and engineering areas, belong to multi-objective optimization problems. The optimal solution of multi-objective problem is a set of solutions. As multi-objective evolutionary algorithms can get many feasible solutions, multi-objective evolutionary algorithms become hot spots and achieved many good results in multi-objective area gradually in recent decades.
The constellation composed of multi-satellites that worked synergically to satisfy the regional coverage demand required by specifically spatial mission. Satellites play a more and more important role in many fields such as communication, navigation and surveillance, weather forecast, geological prospecting by right of their unique spatial position advantage. It is a key issue for constellation optimization to design efficient and feasible constellation layout. Constellation optimization relating to multi-aim points and multi-optimized index is a complex multi-dimensional, dynamic, multi-objective problem. Since it is not high efficiency to get solutions and is not strong convergence and doesn't utilize distributing regulation of Pareto set for conventional evolvement algorithms, with the development of research and application, the complexity of resolution bring forward new challenge for performance of algorithm.
Estimation of distribution algorithms (EDAs) were introduced to the field of multi-objective optimization by Thierens, Bosman and etc in 2001. EDAs use the important speciality that the Pareto sets'distributing of certain continuous multi-objective optimization problems present definite regulation, and reach good solutions on the high dimensional multi-objective optimization problem.
EDAs don't use crossover or mutation opteration like conventional evolvement algorithms but extract globally statistical information from the selected solutions and build probability distribution models of promising solutions based on this extracted information. New solutions are sampled from the models thus built. Qingfu Zhang etc. have developed a regularity model-based multi-objective estimation of distribution (RM-MEDA) in 2007 which captures and utilizes the regularity of the Pareto set in the decision space. Systematic experiments show that, overall RM-MEDA outperforms GDE2, PCX-NSGA-Ⅱand MIDEA, on a set of test instances with variable linkages.
However, RM-MEDA has its drawbacks and shortcomings. Clustering and building probability distribution models of algorithm based on model are more complex and cost much time. Adopt probability distribution models to produce new individuals at early of algorithm on some test problems usually make the search direction far away from objective direction. For example, the regularity model-based multi-objective estimation of distribution (RM-MEDA) is not convergence on ZDT4, ZDT6, DTLZ1and DTLZ3, which make algorithm not stability on some degree.
This paper first presents the research background, significance and actuality in and out of problem, discussing multi-objective optimization problem and analyzing advantage and shortcoming of conventional evolvement algorithms and estimation of distribution algorithms. Study of model based algorithm is emphasized and two kinds of multi-objective optimization problem are brought in and expatiated. The manifold of PF and PS for the firs kind of multi-objective problem are all (m-1)-D, while the The manifold of PF for the second kind of multi-objective problem is (m-1)-D and the dimension of its PS is higher than (m-1)-D.
Then we improve and design model based algorithm for the two kinds of algorithm including:1) strategy of manifold initialization population, construct a upotian Pareto front(UPF) and choose N(size of population) individuals that are closestt to UPF from current samples as initialized samples. This strategy make the individuals of population trying their best to approach Pareto front uniformly; 2) revise parameter of convergence criterion in the first kind of optimization problem based on anterior work that producing individuals in the way of GA+EDA, which enhance the proportion of using EDA to produce individuals in the precondition of keeping even improving quality of solutions; 3) make sure of latent dimension and design convergence criterion for the second kind optimization problem, when the latent dimension of K clusters in population are not far away from each other we decide the distribution of population present certain regulation, and this time we can use EDA to creat the next generation.
In addition, we construct three new test functions ZDT2.3, DTLZ2.3 and DTLZ2.4 with more complex PS(multimodal curve or multimodal curved surface) and supply IGD measurement based on convergence measurement and diversity measurement. The IGD measurement evaluates convergence and distributing uniformity at the same time which renew performance evaluate of three-objective problem.
We compare improvement algorithm m-RM-MEDA with NSGA-Ⅱ、NSGA-Ⅱwith manifold initialization(m-NSGA-Ⅱ)、RM-MEDA and IORM-MEDA. After analyzing the result we found that the advantage of model-based algorithm will be weaken when the complexity of problems' regulation is low while the advantage of m-NSGA-Ⅱis prominence. However when the distribution regulation of PS of problems become more complex, the model-based algorithm appear advantage gradually, at the same time the convergence of NSGA-Ⅱand m-NSGA-Ⅱare down and the their diversity are off form. As the PS of problems become more complex, use EDA to creat the next generation at early make against search in all solutions space for algorithm. But it is aslo bad if we just use GA to creat child generation. So it is a good choice to merge GA and EDA to produce child. The experimental results aslo show that m-RM-MEDA is efficient on convergence and distributing uniformity.
The fifth chapter use m-RM-MEDA to resolve low earth orbit (LEO) constellation optimization problem and compared with m-NSGA-Ⅱ、RM-MEDA and IORM-MEDA. The experiment results show that the convergence percentage of our improved algorithm is better than other algorithms, and can better meet the regional durative coverage demand, providing a useful reference for decision maker dealing with constellation optimization.
引文
[1]Horn J, Nafpliotis N. Multi-objective optimization using the niched Pareto genetic algorithm. University of Illinois at Urbana Champaign, Urbana, Illinois, USA:Technical Report, IlliGAL Report 93005,1993.
[2]Srinivas N, Deb K. Multi-objective optimization using non-dominated sorting in genetic algorithms. Evolutionary Computation,1994,2 (3):221-248.
[3]Deb K., Agrawal S., Pratap A. Meyarivan T. A fast and elitist multi-objective genetic algorithm:NSGAII. Danpur Genetic Algorithms Laboratory (KanGAL), Indian Institute of Technology, Kanpur:Technical Report 200001,2000.
[4]Zitzler E, Thiele L. Multi-objective evolutionary algorithms:A comparative case study and the strength Pareto approach. IEEE Transactions on Evolutionary Computation,1999, 3(4):257-271.
[5]Zitzler E, Laumanns M, Thiele L. SPEA2:Improving the Strength Pareto Evolutionary Algorithm. Technical Report 103, Computer Engineering and Networks Laboratory(TIK), Swiss Federal Institute of Technology(ETH) Zurich, Gloriastrasse 35, CH-8092 Zurich, Switzerland,2001.
[6]Knowles J D, Corne D W. Approximating the Non-dominated Front Using the Pareto Archived Evolution Strategy. Evolutionary Computation,2000.8(2):149-172.
[7]张雅声,张育林.一种有效的重托圆规刀卫星星座设计与分析.装备指挥技术学院学报,2006,17(4)：42～47.
[8]王小平,曹立明.遗传算法——理论,应用与软件实现.第一版.西安：西安交通大学出版社,2002,1.
[9]J. D. Schaffer, "Multiple objective optimization with vector evaluated genetic algorithms," in Proc.1st Int. Conf. Genetic Algorithms, Pitts-burgh, PA,1985,93-100.
[10]Fonseca C M, Fleming P J. Genetic algorithms for multiobjective optimization: formulation, discussion and generation. In:Proceedings of the 5th International Conference on Genetic Algorithms, San Mateo, California,1993.416-423.
[11]Srinivas N, Deb K. Multi-objective optimization using non-dominated sorting in genetic algorithms. Evolutionary Computation,1994,2 (3):221-248.
[12]Deb K, Pratap A,A garwal S etal. A fast and elitist multi-objective genetic algorithm: NSGA-Ⅱ. IEEE Transactions on Evolutionary Computation,2002.6(2):182-197.
[13]D. Thierens and P. A. N. Bosman, "Multi-objective mixture-based iterated density estimation evolutionary algorithms," in Proceedings of the Genetic and Evolutionary Computation Conference. San Francisco, California:Morgan Kaufmann,2001,663-670.
[14]Larra-naga, P. and Lozano, J. A., Estimation of Distribution Algorithms.A New Tool for Evolutionary Computation, Kluwer Publishers,2002.
[15]Costa, M. and Minisci, E., MOPED:A Multi-objective Parzen-based Estimation of Distribution Algorithm for Continuous Problems, In Fonseca C. M. et al., editors, Proceedings of the Second International Conference on Evolutionary Multi-Criterion Optimization,2003,282-294.
[16]Pelikan, M., Goldberg, D. E. and Lobo, F., A Survey of Optimization by Building and Using Probabilistic Models, Technical Report 99018, University of Illinois,1999.
[17]J. M. P. S'anchez, V. Robles, P. Larra~naga, V. Herves, F. Rosales, and M. S. P'erez, "GA-EDA:Hybrid evolutionary algorithm using genetic and estimation of distribution algorithms," in The 17th International Conference on Industrial & Engineering Applications of Artificial Intelligence & Expert Systems,2004,361-371.
[18]FRAYSSINHES E. Alcatel espace, investigating new satellite constellation geometries with genetic algorithmsp. AIAA.1996.
[19]Aimin Zhou, Qingfu Zhang, Yaochu Jin, Edward Tsang.A Model-Based Evolutionary Algorithm for Bi-objective Optimization.cec2005.
[20]Aimin Zhou, Yaochu Jin, Qingfu Zhang, Bernhard Sendhoff, Edward Tsang. Combining Model-based and Genetics-based Offspring Generation for Multi-objective Optimization Using a Convergence Criterion.cec2006.
[21]Aimin Zhou, Qingfu Zhang, Yaochu Jin, Bernhard Sendhoff, and Edward Tsang. Modelling the population distribution in multi-objective optimization by generative topographic mapping. In Parallel Problem Solving From Nature (PPSN IX), volume 4193 of Lecture Notes in Computer Science, Reykjavik, Iceland, September 2006,443-452. Springer-Verlag.
[22]Q. Zhang, A. Zhou, and Y. Jin, "RM-MEDA:A regularity model based multiobjective estimation of distribution algorithm," IEEE Transactions on Evolutionary Computation, vol.12, no.1,2008,41-63.
[23]Aimin Zhou, Qingfu Zhang and Yaochu Jin, "Approximating the Set of Pareto Optimal Solutions in Both the Decision and Objective Spaces by an Estimation of Distribution Algorithm,"
[24]BALLARD A H. Rosette constellations of earth satellite. IEEE Transaction on Aerospace and Electronic Sysems,1980,16(5):656-673.
[25]HANSON J M, EVANS M J. Designing good partial coverage satellite constellations. Journal of Astronautical Sciences,1992,20(2):215-239.
[26]Eric Frayssinhes, Alcatel Espace. Investigating New Satellite Constellation Geometries with Genetic Algorithmsp. AIAA 96-3636, AAS/AIAA Astrodynamics Specialist Conference, San Diego, CA,1996.
[27]Mason W J, Victor IA C. Optimal earth orbiting satellite constellation via a pareto genetic algorithm. In:AAS 00-139, AAS/AIAA Astrodynamics Specialist Conference and Exhibit. San Diego:American Institute of Aeronautics and Astronautics Inc,1998.
[28]李文华,袁建平,赵育善.低地轨道小卫星星座优化问题研究.飞行力学,2001,19(2)： 41-44.
[29]阎志伟,田箐,李汗铃.基于改进的NSGA-Ⅱ算法的区域覆盖卫星星座优化,空间技术学报,2004,24(1)：43～50.
[30]郦苏丹,朱江,李广侠.采用遗传算法的中轨区域通信星座优化设计.系统仿真学报2005,17(6)：1366～1369.
[31]吴延勇,吴诗其.区域覆盖共地面轨迹星座的优化设计.电子与信息学报.2006，8.
[32]吴延勇,吴诗其.基于遗传算法的区域覆盖共地面轨迹卫星星座的优化设计.系统仿真学报,2007,19(11)：2583～2586.
[33]郑蔚.基于模型的多目标进化算法在星座优化设计中的应用研究.中国地质大学(武汉,2007.5.
[34]李晓萌.基于Pareto算法的卫星星座优化设计中的应用研究.中国地质大学(武汉),2007.5.
[35]张璟诚.基于指标的进化算法在卫星星座优化设计中的应用.中国地质大学(武汉),2008.5.
[36]王剑文.基于规则模型的分布估计多目标算法及应用.中国地质大学(武汉),2009.5.
[37]崔逊学.多目标进化算法及其应用.国防工业出版社,2006.6.
[38]Y.Jin, editor. Knowledge Incorporation in Evolutionary Computation. Springer, Berlin, 2005.
[39]P.A.N. Bosman and D.Thierens. Numerical optimization with real-valued estimation of distribution algorithms. In M. Pelikan, K. Sastry, and E. Cantu-Paz, editors, Scalable Optimization via Probabilistic Modeling. Springer,2006.
[40]P. Larra naga and J. A. Lozano, editors. Estimation of Distribution Algorithms:A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Norwell, MA,2001.
[41]Y.Jin and B. Sendho. Connectedness, regularity and thesuccess of local search in evolutionary multi-objective optimization. In Proceedings of the Congress on Evolutionary Computation (CEC 2003), Canberra, Australia,2003,1910-1917. IEEE Press.
[42]K. Miettinen. Nonlinear Multiobjective Optimization, volume12of Kluwer's International Series in Operations Research & Management Science.Kluwer Academic Publishers, 1999.
[43]O. Sch"utze, S. Mostaghim, M. Dellnitz, and J. Teich. Covering Pareto sets bymultilevel evolutionary subdivision techniques. In Second International Conference on Evolutionary Multi-Criterion Optimization (EMO 2003), Faro, Portugal,2003,118-132. Springer, LNCS 2632.
[44]Trevor, Hastie and Werner Stuetzle. Principal curves. Journal of the American Statistical Association,84(406):502-516, June 1989.
[45]何晓群.现代统计分析方法与应用.北京：中国人民大学出版社,1998.1.
[46]Nandakishore Kambhatla and Todd K. Leen. Dimension Reduction by Local Principal Component Analysis. Neural Computation,9(7):1493-1516.
[47]Simon Huband, Phil Hingston, Luigi Barone, etc. A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit. IEEE Transactions on Evolution computation,2006.5(10):477-506.
[48]S. Huband, L. Barone, L. While, and P. Hingston, "A scalable multi-objective test problem toolkit," in Lecture Notes in Computer Science.Berlin, Germany:Springer-Verlag, Mar. 2005, vol.3410, Proc. Evolu-tionary Multi-Criterion Optimization:3rd Int. Conf., 280-294.
[49]程凤周.卫星星座仿真设计.西安：西北工业大学.2001.
[50]章仁为.卫星轨道姿态动力学与控制.北京：北京航空航天大学出版社.1997,4.
[51]杨嘉墀,范剑锋.航天器轨道动力学与控制(上).北京：中国宇航出版社.1995.
[52]李大耀.卫星对地面覆盖区域的通用求解方法.运载火箭与返回技术,1991,1：19～21.
[53]郦苏丹,朱江,李广侠.采用遗传算法的低轨区域通信星座优化设计.通信学报,2005,26(8)：122～128.
[2]Srinivas N, Deb K. Multi-objective optimization using non-dominated sorting in genetic algorithms. Evolutionary Computation,1994,2 (3):221-248.
[3]Deb K., Agrawal S., Pratap A. Meyarivan T. A fast and elitist multi-objective genetic algorithm:NSGAII. Danpur Genetic Algorithms Laboratory (KanGAL), Indian Institute of Technology, Kanpur:Technical Report 200001,2000.
[4]Zitzler E, Thiele L. Multi-objective evolutionary algorithms:A comparative case study and the strength Pareto approach. IEEE Transactions on Evolutionary Computation,1999, 3(4):257-271.
[5]Zitzler E, Laumanns M, Thiele L. SPEA2:Improving the Strength Pareto Evolutionary Algorithm. Technical Report 103, Computer Engineering and Networks Laboratory(TIK), Swiss Federal Institute of Technology(ETH) Zurich, Gloriastrasse 35, CH-8092 Zurich, Switzerland,2001.
[6]Knowles J D, Corne D W. Approximating the Non-dominated Front Using the Pareto Archived Evolution Strategy. Evolutionary Computation,2000.8(2):149-172.
[7]张雅声,张育林.一种有效的重托圆规刀卫星星座设计与分析.装备指挥技术学院学报,2006,17(4)：42～47.
[8]王小平,曹立明.遗传算法——理论,应用与软件实现.第一版.西安：西安交通大学出版社,2002,1.
[9]J. D. Schaffer, "Multiple objective optimization with vector evaluated genetic algorithms," in Proc.1st Int. Conf. Genetic Algorithms, Pitts-burgh, PA,1985,93-100.
[10]Fonseca C M, Fleming P J. Genetic algorithms for multiobjective optimization: formulation, discussion and generation. In:Proceedings of the 5th International Conference on Genetic Algorithms, San Mateo, California,1993.416-423.
[11]Srinivas N, Deb K. Multi-objective optimization using non-dominated sorting in genetic algorithms. Evolutionary Computation,1994,2 (3):221-248.
[12]Deb K, Pratap A,A garwal S etal. A fast and elitist multi-objective genetic algorithm: NSGA-Ⅱ. IEEE Transactions on Evolutionary Computation,2002.6(2):182-197.
[13]D. Thierens and P. A. N. Bosman, "Multi-objective mixture-based iterated density estimation evolutionary algorithms," in Proceedings of the Genetic and Evolutionary Computation Conference. San Francisco, California:Morgan Kaufmann,2001,663-670.
[14]Larra-naga, P. and Lozano, J. A., Estimation of Distribution Algorithms.A New Tool for Evolutionary Computation, Kluwer Publishers,2002.
[15]Costa, M. and Minisci, E., MOPED:A Multi-objective Parzen-based Estimation of Distribution Algorithm for Continuous Problems, In Fonseca C. M. et al., editors, Proceedings of the Second International Conference on Evolutionary Multi-Criterion Optimization,2003,282-294.
[16]Pelikan, M., Goldberg, D. E. and Lobo, F., A Survey of Optimization by Building and Using Probabilistic Models, Technical Report 99018, University of Illinois,1999.
[17]J. M. P. S'anchez, V. Robles, P. Larra~naga, V. Herves, F. Rosales, and M. S. P'erez, "GA-EDA:Hybrid evolutionary algorithm using genetic and estimation of distribution algorithms," in The 17th International Conference on Industrial & Engineering Applications of Artificial Intelligence & Expert Systems,2004,361-371.
[18]FRAYSSINHES E. Alcatel espace, investigating new satellite constellation geometries with genetic algorithmsp. AIAA.1996.
[19]Aimin Zhou, Qingfu Zhang, Yaochu Jin, Edward Tsang.A Model-Based Evolutionary Algorithm for Bi-objective Optimization.cec2005.
[20]Aimin Zhou, Yaochu Jin, Qingfu Zhang, Bernhard Sendhoff, Edward Tsang. Combining Model-based and Genetics-based Offspring Generation for Multi-objective Optimization Using a Convergence Criterion.cec2006.
[21]Aimin Zhou, Qingfu Zhang, Yaochu Jin, Bernhard Sendhoff, and Edward Tsang. Modelling the population distribution in multi-objective optimization by generative topographic mapping. In Parallel Problem Solving From Nature (PPSN IX), volume 4193 of Lecture Notes in Computer Science, Reykjavik, Iceland, September 2006,443-452. Springer-Verlag.
[22]Q. Zhang, A. Zhou, and Y. Jin, "RM-MEDA:A regularity model based multiobjective estimation of distribution algorithm," IEEE Transactions on Evolutionary Computation, vol.12, no.1,2008,41-63.
[23]Aimin Zhou, Qingfu Zhang and Yaochu Jin, "Approximating the Set of Pareto Optimal Solutions in Both the Decision and Objective Spaces by an Estimation of Distribution Algorithm,"
[24]BALLARD A H. Rosette constellations of earth satellite. IEEE Transaction on Aerospace and Electronic Sysems,1980,16(5):656-673.
[25]HANSON J M, EVANS M J. Designing good partial coverage satellite constellations. Journal of Astronautical Sciences,1992,20(2):215-239.
[26]Eric Frayssinhes, Alcatel Espace. Investigating New Satellite Constellation Geometries with Genetic Algorithmsp. AIAA 96-3636, AAS/AIAA Astrodynamics Specialist Conference, San Diego, CA,1996.
[27]Mason W J, Victor IA C. Optimal earth orbiting satellite constellation via a pareto genetic algorithm. In:AAS 00-139, AAS/AIAA Astrodynamics Specialist Conference and Exhibit. San Diego:American Institute of Aeronautics and Astronautics Inc,1998.
[28]李文华,袁建平,赵育善.低地轨道小卫星星座优化问题研究.飞行力学,2001,19(2)： 41-44.
[29]阎志伟,田箐,李汗铃.基于改进的NSGA-Ⅱ算法的区域覆盖卫星星座优化,空间技术学报,2004,24(1)：43～50.
[30]郦苏丹,朱江,李广侠.采用遗传算法的中轨区域通信星座优化设计.系统仿真学报2005,17(6)：1366～1369.
[31]吴延勇,吴诗其.区域覆盖共地面轨迹星座的优化设计.电子与信息学报.2006，8.
[32]吴延勇,吴诗其.基于遗传算法的区域覆盖共地面轨迹卫星星座的优化设计.系统仿真学报,2007,19(11)：2583～2586.
[33]郑蔚.基于模型的多目标进化算法在星座优化设计中的应用研究.中国地质大学(武汉,2007.5.
[34]李晓萌.基于Pareto算法的卫星星座优化设计中的应用研究.中国地质大学(武汉),2007.5.
[35]张璟诚.基于指标的进化算法在卫星星座优化设计中的应用.中国地质大学(武汉),2008.5.
[36]王剑文.基于规则模型的分布估计多目标算法及应用.中国地质大学(武汉),2009.5.
[37]崔逊学.多目标进化算法及其应用.国防工业出版社,2006.6.
[38]Y.Jin, editor. Knowledge Incorporation in Evolutionary Computation. Springer, Berlin, 2005.
[39]P.A.N. Bosman and D.Thierens. Numerical optimization with real-valued estimation of distribution algorithms. In M. Pelikan, K. Sastry, and E. Cantu-Paz, editors, Scalable Optimization via Probabilistic Modeling. Springer,2006.
[40]P. Larra naga and J. A. Lozano, editors. Estimation of Distribution Algorithms:A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Norwell, MA,2001.
[41]Y.Jin and B. Sendho. Connectedness, regularity and thesuccess of local search in evolutionary multi-objective optimization. In Proceedings of the Congress on Evolutionary Computation (CEC 2003), Canberra, Australia,2003,1910-1917. IEEE Press.
[42]K. Miettinen. Nonlinear Multiobjective Optimization, volume12of Kluwer's International Series in Operations Research & Management Science.Kluwer Academic Publishers, 1999.
[43]O. Sch"utze, S. Mostaghim, M. Dellnitz, and J. Teich. Covering Pareto sets bymultilevel evolutionary subdivision techniques. In Second International Conference on Evolutionary Multi-Criterion Optimization (EMO 2003), Faro, Portugal,2003,118-132. Springer, LNCS 2632.
[44]Trevor, Hastie and Werner Stuetzle. Principal curves. Journal of the American Statistical Association,84(406):502-516, June 1989.
[45]何晓群.现代统计分析方法与应用.北京：中国人民大学出版社,1998.1.
[46]Nandakishore Kambhatla and Todd K. Leen. Dimension Reduction by Local Principal Component Analysis. Neural Computation,9(7):1493-1516.
[47]Simon Huband, Phil Hingston, Luigi Barone, etc. A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit. IEEE Transactions on Evolution computation,2006.5(10):477-506.
[48]S. Huband, L. Barone, L. While, and P. Hingston, "A scalable multi-objective test problem toolkit," in Lecture Notes in Computer Science.Berlin, Germany:Springer-Verlag, Mar. 2005, vol.3410, Proc. Evolu-tionary Multi-Criterion Optimization:3rd Int. Conf., 280-294.
[49]程凤周.卫星星座仿真设计.西安：西北工业大学.2001.
[50]章仁为.卫星轨道姿态动力学与控制.北京：北京航空航天大学出版社.1997,4.
[51]杨嘉墀,范剑锋.航天器轨道动力学与控制(上).北京：中国宇航出版社.1995.
[52]李大耀.卫星对地面覆盖区域的通用求解方法.运载火箭与返回技术,1991,1：19～21.
[53]郦苏丹,朱江,李广侠.采用遗传算法的低轨区域通信星座优化设计.通信学报,2005,26(8)：122～128.