高维多目标减少算法的比较与研究
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摘要
在实际优化过程中,许多优化问题都需要同时考虑多个目标,并且这些目标往往是相互冲突的,因此,多目标优化受到更多的关注。进化算法是模拟生物自然进化的全局智能搜索算法,广泛地应用于求解高度复杂的非线性问题。研究者们针对不同的应用问题,提出了不同的多目标进化算法,比如:NSGA-II、ε-MOEA等,它们都能很好地处理多目标优化问题。
但是,许多文献只是考虑两个或者三个目标的低维问题,而在实际中,往往包括的目标数非常大(4维或者更多)。当目标数量超过3维时,分析Pareto面比较困难。甚至有研究表明,基于Pareto优化的MOEA在高维情况下不易找到好的Pareto面,原因之一就是非支配解的比率随目标维数增加迅速增长。这意味着,许多算法在选择过程中都是随机的。目前处理高维问题有两种方法:松驰Pareto支配关系的方法和目标减少算法。本文针对第二种方法作了一些比较研究,主要工作包括以下三个方面。
第一、分析比较了目前已经提出的三种目标减少算法。
本文首先对三种同类算法作一个简介,然后分析了这些算法的性能,并与本文将提出的新算法进行对比,从另一面验证本文算法的可行性。
第二、提出了基于最小二乘法的目标减少算法,并通过实验证明它的可行性。
本文将从决策者角度出发提出一个新的目标减少算法,该算法采用最小二乘法将目标空间中每个目标函数拟合为多条直线段,然后两两比较各直线段的斜率,确定最冗余目标对,并将冗余目标从目标集中删除。在算法设计的每一步,本文将详细对它介绍与分析,得出其时间复杂度。另外,通过大量的比较实验证明,本文的算法是一种有效的算法。
第三、提出了两种目标减少算法的评价方法。即:
(1)在目标减少前后,用支配关系改变的比率来衡量它的优劣; (2)将目标函数拟合为多条直线段,用空间分布相似程度来评价它的好坏。
考虑到目标减少算法目前暂时缺少专门的评价方法这个问题,本文提出了上述两种评价方法,评价的数值结果与图的直观反映结合验证评价方法的可行性;并且,本文已将评价方法(1)用于评价各目标减少算法。最后,通过与已有评价方法进行比较,实验结果表明本文提出的两种方法能准确地评价目标减少算法。
另外,本文还改进了单目标遗传算法求解一些实际问题:如旅行商问题,数值优化问题等。
In the real world, many optimization problems conside lots of objectives simultaneously, where the objectives are conflict, and so multi-objective optimization are concerned. Evolutionary algorithm is a global intelligent search algorithm simulating natural evolution. For solving highly complex and nonlinear problems, it has been widely used. Researchers put forward their own multi-objective evolutionary algorithm for different applications, for example: NSGA-II,ε-MOEA, and so on, they can deal with a wide range of optimization problems better.
Nonetheless, most of the publications consider problems with two or three objectives, in spite of the fact that many real-world problems involve a large number of objectives (4 or more). Besides the difficulty to analyze the Pareto front when there are more than three objectives, recent studies have shown that MOEAs based on Pareto optimality have difficulties to find a good Pareto front approximation in higher dimensions. One of the reasons for this is that the proportion of non-dominated solutions (i.e., equally good solutions) in a population increases rapidly with the number of objectives. This implies that in the presence of many objectives the selection of new solutions is carried out almost at random since a large number of the solutions are equally good in the Pareto sense. Currently, there are mainly two approaches to deal with problems involving many objectives, namely: i) to propose relaxed forms of Pareto optimality ,and ii) to reduce the number of objectives of the problem to ease the decision making or the search processes . This paper focuses on the second method, and its main tasks are the following three aspects:
First, analyze and compare three algorithms similar to our approach.
At the beginning, this paper introduces those algorithms simpely, and then analyzes their performance. Comparing with our new one, this proves that our algorithm is feasible from another point of view.
Second, we propose Objective Reduction based on the Least Square Method,the experiment result with other similar algorithm shows our algorithm is competitive.
Our algorithm fits every objective’s valus into multi-strip lines, then determines the most redundant objective couples between each two-slope vector, and considers the most redundant objective, then deletes it from the redundant objective set. At every step of the new algorithm, this paper will introduce and analyze detailedly, and educe the time complexity. Othermore, lots of experiments show that our new algorithm is effective.
Last, propose two methods for estimating redundant objective reducing algorithms.
1) Use the changed dominant relation radio pre and post objective reducing to measure them;
2) Fit every function to multi lines and use space similarity radio to survey them.
Based on less of metrics on these algorithms, there are two new ways. The numerical value and the figure hang together,which show the two methods are useful. And also, metric (1) was used for estimating all the objective reduction algorithms.Last, compare with existed method, experiment shows that the new method can be used for evaluating non-redundant objective reducing algorithm.
In addition, this paper improved single objective genetic algorithm. For example trivial sales problem, and function optimal problems, and so on.
但是,许多文献只是考虑两个或者三个目标的低维问题,而在实际中,往往包括的目标数非常大(4维或者更多)。当目标数量超过3维时,分析Pareto面比较困难。甚至有研究表明,基于Pareto优化的MOEA在高维情况下不易找到好的Pareto面,原因之一就是非支配解的比率随目标维数增加迅速增长。这意味着,许多算法在选择过程中都是随机的。目前处理高维问题有两种方法:松驰Pareto支配关系的方法和目标减少算法。本文针对第二种方法作了一些比较研究,主要工作包括以下三个方面。
第一、分析比较了目前已经提出的三种目标减少算法。
本文首先对三种同类算法作一个简介,然后分析了这些算法的性能,并与本文将提出的新算法进行对比,从另一面验证本文算法的可行性。
第二、提出了基于最小二乘法的目标减少算法,并通过实验证明它的可行性。
本文将从决策者角度出发提出一个新的目标减少算法,该算法采用最小二乘法将目标空间中每个目标函数拟合为多条直线段,然后两两比较各直线段的斜率,确定最冗余目标对,并将冗余目标从目标集中删除。在算法设计的每一步,本文将详细对它介绍与分析,得出其时间复杂度。另外,通过大量的比较实验证明,本文的算法是一种有效的算法。
第三、提出了两种目标减少算法的评价方法。即:
(1)在目标减少前后,用支配关系改变的比率来衡量它的优劣; (2)将目标函数拟合为多条直线段,用空间分布相似程度来评价它的好坏。
考虑到目标减少算法目前暂时缺少专门的评价方法这个问题,本文提出了上述两种评价方法,评价的数值结果与图的直观反映结合验证评价方法的可行性;并且,本文已将评价方法(1)用于评价各目标减少算法。最后,通过与已有评价方法进行比较,实验结果表明本文提出的两种方法能准确地评价目标减少算法。
另外,本文还改进了单目标遗传算法求解一些实际问题:如旅行商问题,数值优化问题等。
In the real world, many optimization problems conside lots of objectives simultaneously, where the objectives are conflict, and so multi-objective optimization are concerned. Evolutionary algorithm is a global intelligent search algorithm simulating natural evolution. For solving highly complex and nonlinear problems, it has been widely used. Researchers put forward their own multi-objective evolutionary algorithm for different applications, for example: NSGA-II,ε-MOEA, and so on, they can deal with a wide range of optimization problems better.
Nonetheless, most of the publications consider problems with two or three objectives, in spite of the fact that many real-world problems involve a large number of objectives (4 or more). Besides the difficulty to analyze the Pareto front when there are more than three objectives, recent studies have shown that MOEAs based on Pareto optimality have difficulties to find a good Pareto front approximation in higher dimensions. One of the reasons for this is that the proportion of non-dominated solutions (i.e., equally good solutions) in a population increases rapidly with the number of objectives. This implies that in the presence of many objectives the selection of new solutions is carried out almost at random since a large number of the solutions are equally good in the Pareto sense. Currently, there are mainly two approaches to deal with problems involving many objectives, namely: i) to propose relaxed forms of Pareto optimality ,and ii) to reduce the number of objectives of the problem to ease the decision making or the search processes . This paper focuses on the second method, and its main tasks are the following three aspects:
First, analyze and compare three algorithms similar to our approach.
At the beginning, this paper introduces those algorithms simpely, and then analyzes their performance. Comparing with our new one, this proves that our algorithm is feasible from another point of view.
Second, we propose Objective Reduction based on the Least Square Method,the experiment result with other similar algorithm shows our algorithm is competitive.
Our algorithm fits every objective’s valus into multi-strip lines, then determines the most redundant objective couples between each two-slope vector, and considers the most redundant objective, then deletes it from the redundant objective set. At every step of the new algorithm, this paper will introduce and analyze detailedly, and educe the time complexity. Othermore, lots of experiments show that our new algorithm is effective.
Last, propose two methods for estimating redundant objective reducing algorithms.
1) Use the changed dominant relation radio pre and post objective reducing to measure them;
2) Fit every function to multi lines and use space similarity radio to survey them.
Based on less of metrics on these algorithms, there are two new ways. The numerical value and the figure hang together,which show the two methods are useful. And also, metric (1) was used for estimating all the objective reduction algorithms.Last, compare with existed method, experiment shows that the new method can be used for evaluating non-redundant objective reducing algorithm.
In addition, this paper improved single objective genetic algorithm. For example trivial sales problem, and function optimal problems, and so on.
引文
[1]郑金华.多目标进化算法及其应用[M].北京:科学出版社,2007:2
[2] Carlos A. Coello Coello. 20 Years of Evolutionary Multi-Objective Optimization: What Has Been Done and What Remains to be Done [J], in Gary Y. Yen and David B. Fogel (editors), Computational Intelligence: Principles and Practice, Chapter 4, pp. 73--88, IEEE Computational Intelligence Society, 2006
[3] Rosenberg R. S. 1967. Simulation of genetic populations with biochemical properties [D]. PhD thesis, University of Michigan, Ann Harbor, Michigan
[4] Schaffer J. D. 1984. Some experiments in machine learning using vector evaluated genetic algorithms [D]. PhD thesis, Vanderbilt University
[5] Fonseca Carlos M. and Peter J. Fleming. 1993. Genetic Algorithm for Multiobjective Optimization: Formulation, Discussion and Generalization [J]. In Stephanie Forrest, editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 416-423, San Mateo, California. University of Illinois at Urbana-Champaign, Morgan Kauffman Publishers
[6] Horn J., Nafpliotis N., & Goldberg D. E. (1994). A Niched Pareto genetic Algorithm for Multiobjective Optimization [C]. Proceeding of the first IEEE Conference on Evolutionary Computation, 82-87
[7] N. Srinivas and Kalyanmoy Deb. Multiobjective optimization using nondominated sorting in genetic algorithms [R]. Technical report, Department of Mechanical Engineering, Indian Institute of Technology, Kanput, India, 1993
[8] Zitzler, E. and L. Thiele (1999). Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach[J]. IEEE Transactions on Evolutionary Computation, 3(4): 257-271, November 1999
[9] E. Zitzler, M. Laumanns, and L. Thiele. SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. EUROGEN 2001 - Evolutionary Methods for Design, Optimisation and Control with Applications to Industrial Problems, September 2001.
[10] Kalyanmoy Deb, Amrit Pratap, Sameer Agrawal and T. Meyrivan. A Fast and Elitist Multi-objective Genetic Algorithm: NSGA-II [J]. IEEE Transactions on Evolutionary Computation, 6(2): 182-197, April 2002
[11] Knowles, J., D. and Corne, D., W. Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy Evolutionary Computation [J], 2000, 149-172
[12] Carlos A. Coello Coello, et al. Evolutionary Algorithms for Solving Multi-Objective Problems [M]. Kluwer Acedemic/Plenum Publishers, 2002
[13] D. E. Goldberg. Genetic algorithms for search, optimization, and machine learning [M]. Reading, MA: Addison-Wesley, 1989
[14] Kita H. et al. Mult-Objective optimazition by Means of the Thermodynamical Genetic Algorithm [J]. In Voigt H.-M., et al., editors, Parallel problem Solving from Nature-PPSN IV, pages 504-512. Springer-Verlag. Lecture Notes on Computer Science No.1141. Berlin Germany
[15] Marvin N. An evolutionary approach to constructing prognostic models [J]. Artificial Intellegence in Medicine, 15(2): 155-165
[16] Corne, D.W., Jerram, N.R., Knowles, J.D., and Oates, M.J. (2001) PESA-II: Region-based Selection in Evolutionary Multiobjective Optimization[C]. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), pp. 283-290
[17] Arturo Hernádez Aguirre, Salvador Botello Rionda, Carlos A. Coello Coello, Giovanni Lizáraga Lizáraga, and Efrén Mezura Montes, Handling Constraints using Multiobjective Optimization Concepts [J], International Journal for Numerical Methods in Engineering, Volume 59, No. 15, pp. 1989-2017, April 2004
[18] David A. Van Veldhuizen and G. B. Lamont. Multiobjective Evolutionary Algorithm Test Suites [J]. In Carroll J., et al., editors, Proceedings of the 1999 ACM Symposium on Applied Computing, pages: 351-357
[19] Kalyanmoy Deb. Multi-Objective Genetic Algorithms: Problem difficulties and Construction of Test Problems [J]. Evolutionary Computation, 7(3): 205-230
[20]崔逊学.一种求解高维优化问题的多目标遗传算法及其收敛性分析[J].计算机研究与发展, 2003, 40(7): 901-906
[21] G. Rudolph. Evolutionary Search under Partially ordered Fitness Sets [C]. In Procedings of the Internaional NAISO Congress on Information Science Innovations (ISI2001), pages818-822. ICSC Academic Press: Millet/Sliedrecht
[22] Hanne T. On the Convergence of Multiobjective Evolutionary Algorithms [J]. European Journal of Operational Reseach, 117(3): 553-564
[23] Joshua Knowles and David Corne. Properties of an Adaptive Archiving Algorithm for Storing Nondominated Vectors [J], IEEE Transactions on Evolutionary Computation, Vol. 7, No. 2, pp. 100-116, April 2003
[24] Qingfu Zhang, H.Muhlenbein. On the convergence of a class of estimation of distribution algorithms [J]. IEEE Transactions on Evolutionary Computation, 2004, 8(2): 127- 136
[25]郑金华,史忠植.基于聚类的快速多目标遗传算法[J].计算机研究与发展, 2004,41(7), 1081-1087
[26] Jinhua Zheng, Charles Ling, Zhongzhi Shi, Juan Xue, Xuyong Li: A Multi-objective Genetic Algorithm Based on Quick Sort [C]. Canadian Conference on AI 2004: 175-186
[27] Jinhua Zheng, Zhongzhi Shi, Charles X.Ling, Yong Xie. A New Method to Construct the Non-Dominated Set in Multi-Objective Genetic Algorithms [C]. IIP 2004, 2004.10, Beijing
[28] Mikkel T. Jensen. Reducing the Run-Time Complexity of Multiobjective EAs: The NSGA-II and Other Algorithms [J], IEEE Transactions on Evolutionary Computation, Vol. 7, No. 5, pp. 503--515, October 2003
[29] E.K.Burke, S.Gustafson, G.Kenda. Diversity in genetic programming: an analysis of measures and correlation with fitness [J]. IEEE Transactions on Evolutionary Computation, 2004, 8(1): 47- 62
[30]崔逊学,李淼,方廷健.基于免疫原理的多目标进化算法群体多样性研究[J].模式识别与人工智能, 2001,14(3):291-295
[31] K.C. Tan, T.H. Lee and E.F. Khor. Evolutionary Algorithms for Multi-Objective Optimization: Performance Assessments and Comparisons [C], Artificial Intelligence Review, Vol. 17, No. 4, pp.253--290, June 2002
[32]陈国良,王煦法,庄镇泉,王东生.遗传算法及其应用[M].北京:人民邮电出版社,1996
[33]刘勇,康立山,陈毓屏.非数值并行算法——遗传算法[M].北京:科学出版社,1998
[34]李敏强,寇纪淞,林丹,李书全.遗传算法的基本理论应用[M].北京:科学出版社,2002
[35]王小平,曹立明.遗传算法——理论、应用与软件实现[M].西安:西安交通大学出版社,2002
[36]王凌.车间调度及其遗传算法[M].北京:清华大学出版社,2003
[37]张文修,梁怡.遗传算法的数学基础[M].西安:西安交通大学出版社,2000
[38]玄光男,程润伟(于歆杰,周根贵译).遗传算法与工程优化[M].北京:清华大学出版社,2004
[39]崔逊学.多目标进化算法及其应用[J].国防工业出版社, 2006.6
[40] Carlos A. Coello Coello, The EMOO repository: a resource for doing research in evolutionary multiobjective optimization [J], IEEE Computational Intelligence Magazine, Vol. 1, No. 1, pp. 37-45, February 2006
[41] Carlos A. Coello Coello. 20 Years of Evolutionary Multi-Objective Optimization: What Has Been Done and What Remains to be Done [C], in Gary Y. Yen and David B. Fogel (editors), Computational Intelligence: Principles and Practice, Chapter 4, pp. 73--88, IEEE Computational Intelligence Society, 2006
[42] C. A. C. Coello, D. A. Vanveldhuizen, and G. Lamont. Evolutionary Algorithms For Solving MultiObjective Problems [M]. Boston, MA: Kluwer Academic Publishers, 2002
[43] P. J. Agrell. On redundancy in MCDM [J]. Europen Journal of operational Research, 1997, 98(3), 571-586
[44] D. Brockhoff, E. Zitzler. Are All objectives Necessary? On Dimensionality Reduction in Evolutionary MultiObjective Optimization [J]. In parallel problem solving from Nature, Springer. 2006, 533-542
[45] Jian J.Dai, Linh lieu, and David Rocke. Dimension Reduction for Classification with RGene Expression Microarray Data [J]. In Statistical Applications in Genetics and Molecular Biology, Vol 5, 2006 No.1, Article 6
[46] Luis Marti, Jesus Garci and Jose Molina. Modal building Algorithms for mnltiobjective EDAs: Directions for Improvement [C]. In congress on Evolutionary Computation (CEC 2008) IEEE Press. 2848-2853
[47] D. K. Saxena and K. Deb. Dimensionality Reduction of Objectives and constraints in MultiObjective Optimization Problems [C], A System Design Perspective. In congress on Evolutionary Computation (CEC 2008) IEEE Press. 3203-3210
[48] K. Deb and D. K. Saxena. Searching for Pareto optimal Solutions through Dimensionality Reduction for Certain Large Dimensional MultiObjective Optimization Problem [C]. In congtess on Evolutionary Computation (CEC 2006) IEEE Press. 3352-3360
[49] D. Brockhoff, and E. Zitzler. Dimensionality Reduction in MutiObjective Optimization, the Minimum Objective Subject problem [J]. In Operations Research Proceedings. 2007, 423-429
[50] F.di piero. Many Objectives Evolutionary Algorithms and Applications to Water Resources Engineering [D]. PhD Thesis School of Engineering, Computer Science and Mathematics, 2006
[51] P.J.Fleming, R.C.Purshouse, and R.J.Lygoe. Many objective optimizations: an engineering design perspective [C]. In Proc EMO 2005, 2005(3410): 14-32
[52] D. K. Saxena and K. Deb. Non Line Dimensionality Reduction Procedures for Certain large Dimensional MultiObjective optimization problem [C]. In Evolutionary Multi criterion Optimization (EMO 2007) 272-787
[53] D. Brockhoff, and E. Zitzler. Dimensionality Reduction in MutiObjective Optimization, the Minimum Objective Subject problem [J]. In Operations Research Proceedings. 2007, 423-429
[54] Brockhoff, D., Zitzler, E.: Dimensionality Reduction in Multiobjective Optimization with (Partial) Dominance Structure Preservation: Generalized Minimum Objective Subset Problems [R]. TIK Report 247, ETH Zurich, Zurich, Switzerland (2006)
[55] D.Brockhoff, and E.Zitzler. Are All objectives Necessary? On Dimensionality Reduction in Evolutionary MultiObjective Optimization [J]. In solving from Nature, Springer. 2006, 533-542
[56] A.L.Jaines, C.A.C.Coello and D.Chakraborty. Objective Reduction Using a feature Selection Technique [J]. GECCO 08. 673--680, ACM Press, Atlanta, USA, July 2008
[57] K.Deb and D.K.Saxena. On Finding Pareto-Optimal Solutions Through Dimensionality Reduction for Certain Large- Dimensional Multi-Objective Optimization Problems [R]. KanGAL 2005011
[58] D. Brockhoff and E. Zitzler. Objective Reduction in Evolutionary Multiobjective Optimization: Theory and Applications [J]. Evolutionary Computation, 2009
[59] D. Brockhoff and E. Zitzler. Improving Hypervolume-based Multiobjective Evolutionary Algorithms by Using Objective Reduction Methods [C]. In Congress on Evolutionary Computation (CEC 2007), pages 2086-2093. IEEE Press, 2007
[60] Laumanns, M., Thiele, L., Deb, K., and Zitzler, E. (2002). Combining convergence and diversity in evolutionary multi-objective optimization [J]. Evolutionary Computation, 10(3): 263-282
[61] Lin S,Kernighan B W. An effective heuristic algorithm for the traveling salesman problem. Operational Research,1971,19:486-515
[62] Gen M. Crossover in intensive search and traveling salesman problem. Institute of technology,1994,568-679
[63]唐立新.旅行商问题(TSP)的改进遗传算法[J].东北大学报.1999,1:40-43
[64] David Greenhalgh, Stephen Marshall. Convergence criteria for genetic algorithms[J]. SIAM J. Comput. 2000, 30(1): 269-282
[65] Radu Belea. Genetic algorithm convergence analysis using a unified representation of genes and the hamming distance [J]. International Journal of Knowledge-based and Intelligent Engineering Systems, 2006,10(1):29-40
[66]孟佳娜,王立宏.基于多种群的强者进化遗传算法[J].计算机工程与应用,2004(14):41-42.
[67]邓长春,朱儒明,李咏霞,许波.一种求解TSP问题的多种群并行遗传算法[J].计算机仿真,2008,25(9):187-190
[68]巩敦卫,孙晓燕.变搜索区域多种群遗传算法[J].控制理论与应用,2006,23(2):256-260
[69]李军华,黎明,袁丽华.一种改进的双种群遗传算法[J],小型微型计算机系统,2008,11:2099-2102
[70]郑金华等.基于空间交配的遗传算法.模式识别与人工智能[J], 2003, 16(4): 482-485
[2] Carlos A. Coello Coello. 20 Years of Evolutionary Multi-Objective Optimization: What Has Been Done and What Remains to be Done [J], in Gary Y. Yen and David B. Fogel (editors), Computational Intelligence: Principles and Practice, Chapter 4, pp. 73--88, IEEE Computational Intelligence Society, 2006
[3] Rosenberg R. S. 1967. Simulation of genetic populations with biochemical properties [D]. PhD thesis, University of Michigan, Ann Harbor, Michigan
[4] Schaffer J. D. 1984. Some experiments in machine learning using vector evaluated genetic algorithms [D]. PhD thesis, Vanderbilt University
[5] Fonseca Carlos M. and Peter J. Fleming. 1993. Genetic Algorithm for Multiobjective Optimization: Formulation, Discussion and Generalization [J]. In Stephanie Forrest, editor, Proceedings of the Fifth International Conference on Genetic Algorithms, pages 416-423, San Mateo, California. University of Illinois at Urbana-Champaign, Morgan Kauffman Publishers
[6] Horn J., Nafpliotis N., & Goldberg D. E. (1994). A Niched Pareto genetic Algorithm for Multiobjective Optimization [C]. Proceeding of the first IEEE Conference on Evolutionary Computation, 82-87
[7] N. Srinivas and Kalyanmoy Deb. Multiobjective optimization using nondominated sorting in genetic algorithms [R]. Technical report, Department of Mechanical Engineering, Indian Institute of Technology, Kanput, India, 1993
[8] Zitzler, E. and L. Thiele (1999). Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach[J]. IEEE Transactions on Evolutionary Computation, 3(4): 257-271, November 1999
[9] E. Zitzler, M. Laumanns, and L. Thiele. SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. EUROGEN 2001 - Evolutionary Methods for Design, Optimisation and Control with Applications to Industrial Problems, September 2001.
[10] Kalyanmoy Deb, Amrit Pratap, Sameer Agrawal and T. Meyrivan. A Fast and Elitist Multi-objective Genetic Algorithm: NSGA-II [J]. IEEE Transactions on Evolutionary Computation, 6(2): 182-197, April 2002
[11] Knowles, J., D. and Corne, D., W. Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy Evolutionary Computation [J], 2000, 149-172
[12] Carlos A. Coello Coello, et al. Evolutionary Algorithms for Solving Multi-Objective Problems [M]. Kluwer Acedemic/Plenum Publishers, 2002
[13] D. E. Goldberg. Genetic algorithms for search, optimization, and machine learning [M]. Reading, MA: Addison-Wesley, 1989
[14] Kita H. et al. Mult-Objective optimazition by Means of the Thermodynamical Genetic Algorithm [J]. In Voigt H.-M., et al., editors, Parallel problem Solving from Nature-PPSN IV, pages 504-512. Springer-Verlag. Lecture Notes on Computer Science No.1141. Berlin Germany
[15] Marvin N. An evolutionary approach to constructing prognostic models [J]. Artificial Intellegence in Medicine, 15(2): 155-165
[16] Corne, D.W., Jerram, N.R., Knowles, J.D., and Oates, M.J. (2001) PESA-II: Region-based Selection in Evolutionary Multiobjective Optimization[C]. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), pp. 283-290
[17] Arturo Hernádez Aguirre, Salvador Botello Rionda, Carlos A. Coello Coello, Giovanni Lizáraga Lizáraga, and Efrén Mezura Montes, Handling Constraints using Multiobjective Optimization Concepts [J], International Journal for Numerical Methods in Engineering, Volume 59, No. 15, pp. 1989-2017, April 2004
[18] David A. Van Veldhuizen and G. B. Lamont. Multiobjective Evolutionary Algorithm Test Suites [J]. In Carroll J., et al., editors, Proceedings of the 1999 ACM Symposium on Applied Computing, pages: 351-357
[19] Kalyanmoy Deb. Multi-Objective Genetic Algorithms: Problem difficulties and Construction of Test Problems [J]. Evolutionary Computation, 7(3): 205-230
[20]崔逊学.一种求解高维优化问题的多目标遗传算法及其收敛性分析[J].计算机研究与发展, 2003, 40(7): 901-906
[21] G. Rudolph. Evolutionary Search under Partially ordered Fitness Sets [C]. In Procedings of the Internaional NAISO Congress on Information Science Innovations (ISI2001), pages818-822. ICSC Academic Press: Millet/Sliedrecht
[22] Hanne T. On the Convergence of Multiobjective Evolutionary Algorithms [J]. European Journal of Operational Reseach, 117(3): 553-564
[23] Joshua Knowles and David Corne. Properties of an Adaptive Archiving Algorithm for Storing Nondominated Vectors [J], IEEE Transactions on Evolutionary Computation, Vol. 7, No. 2, pp. 100-116, April 2003
[24] Qingfu Zhang, H.Muhlenbein. On the convergence of a class of estimation of distribution algorithms [J]. IEEE Transactions on Evolutionary Computation, 2004, 8(2): 127- 136
[25]郑金华,史忠植.基于聚类的快速多目标遗传算法[J].计算机研究与发展, 2004,41(7), 1081-1087
[26] Jinhua Zheng, Charles Ling, Zhongzhi Shi, Juan Xue, Xuyong Li: A Multi-objective Genetic Algorithm Based on Quick Sort [C]. Canadian Conference on AI 2004: 175-186
[27] Jinhua Zheng, Zhongzhi Shi, Charles X.Ling, Yong Xie. A New Method to Construct the Non-Dominated Set in Multi-Objective Genetic Algorithms [C]. IIP 2004, 2004.10, Beijing
[28] Mikkel T. Jensen. Reducing the Run-Time Complexity of Multiobjective EAs: The NSGA-II and Other Algorithms [J], IEEE Transactions on Evolutionary Computation, Vol. 7, No. 5, pp. 503--515, October 2003
[29] E.K.Burke, S.Gustafson, G.Kenda. Diversity in genetic programming: an analysis of measures and correlation with fitness [J]. IEEE Transactions on Evolutionary Computation, 2004, 8(1): 47- 62
[30]崔逊学,李淼,方廷健.基于免疫原理的多目标进化算法群体多样性研究[J].模式识别与人工智能, 2001,14(3):291-295
[31] K.C. Tan, T.H. Lee and E.F. Khor. Evolutionary Algorithms for Multi-Objective Optimization: Performance Assessments and Comparisons [C], Artificial Intelligence Review, Vol. 17, No. 4, pp.253--290, June 2002
[32]陈国良,王煦法,庄镇泉,王东生.遗传算法及其应用[M].北京:人民邮电出版社,1996
[33]刘勇,康立山,陈毓屏.非数值并行算法——遗传算法[M].北京:科学出版社,1998
[34]李敏强,寇纪淞,林丹,李书全.遗传算法的基本理论应用[M].北京:科学出版社,2002
[35]王小平,曹立明.遗传算法——理论、应用与软件实现[M].西安:西安交通大学出版社,2002
[36]王凌.车间调度及其遗传算法[M].北京:清华大学出版社,2003
[37]张文修,梁怡.遗传算法的数学基础[M].西安:西安交通大学出版社,2000
[38]玄光男,程润伟(于歆杰,周根贵译).遗传算法与工程优化[M].北京:清华大学出版社,2004
[39]崔逊学.多目标进化算法及其应用[J].国防工业出版社, 2006.6
[40] Carlos A. Coello Coello, The EMOO repository: a resource for doing research in evolutionary multiobjective optimization [J], IEEE Computational Intelligence Magazine, Vol. 1, No. 1, pp. 37-45, February 2006
[41] Carlos A. Coello Coello. 20 Years of Evolutionary Multi-Objective Optimization: What Has Been Done and What Remains to be Done [C], in Gary Y. Yen and David B. Fogel (editors), Computational Intelligence: Principles and Practice, Chapter 4, pp. 73--88, IEEE Computational Intelligence Society, 2006
[42] C. A. C. Coello, D. A. Vanveldhuizen, and G. Lamont. Evolutionary Algorithms For Solving MultiObjective Problems [M]. Boston, MA: Kluwer Academic Publishers, 2002
[43] P. J. Agrell. On redundancy in MCDM [J]. Europen Journal of operational Research, 1997, 98(3), 571-586
[44] D. Brockhoff, E. Zitzler. Are All objectives Necessary? On Dimensionality Reduction in Evolutionary MultiObjective Optimization [J]. In parallel problem solving from Nature, Springer. 2006, 533-542
[45] Jian J.Dai, Linh lieu, and David Rocke. Dimension Reduction for Classification with RGene Expression Microarray Data [J]. In Statistical Applications in Genetics and Molecular Biology, Vol 5, 2006 No.1, Article 6
[46] Luis Marti, Jesus Garci and Jose Molina. Modal building Algorithms for mnltiobjective EDAs: Directions for Improvement [C]. In congress on Evolutionary Computation (CEC 2008) IEEE Press. 2848-2853
[47] D. K. Saxena and K. Deb. Dimensionality Reduction of Objectives and constraints in MultiObjective Optimization Problems [C], A System Design Perspective. In congress on Evolutionary Computation (CEC 2008) IEEE Press. 3203-3210
[48] K. Deb and D. K. Saxena. Searching for Pareto optimal Solutions through Dimensionality Reduction for Certain Large Dimensional MultiObjective Optimization Problem [C]. In congtess on Evolutionary Computation (CEC 2006) IEEE Press. 3352-3360
[49] D. Brockhoff, and E. Zitzler. Dimensionality Reduction in MutiObjective Optimization, the Minimum Objective Subject problem [J]. In Operations Research Proceedings. 2007, 423-429
[50] F.di piero. Many Objectives Evolutionary Algorithms and Applications to Water Resources Engineering [D]. PhD Thesis School of Engineering, Computer Science and Mathematics, 2006
[51] P.J.Fleming, R.C.Purshouse, and R.J.Lygoe. Many objective optimizations: an engineering design perspective [C]. In Proc EMO 2005, 2005(3410): 14-32
[52] D. K. Saxena and K. Deb. Non Line Dimensionality Reduction Procedures for Certain large Dimensional MultiObjective optimization problem [C]. In Evolutionary Multi criterion Optimization (EMO 2007) 272-787
[53] D. Brockhoff, and E. Zitzler. Dimensionality Reduction in MutiObjective Optimization, the Minimum Objective Subject problem [J]. In Operations Research Proceedings. 2007, 423-429
[54] Brockhoff, D., Zitzler, E.: Dimensionality Reduction in Multiobjective Optimization with (Partial) Dominance Structure Preservation: Generalized Minimum Objective Subset Problems [R]. TIK Report 247, ETH Zurich, Zurich, Switzerland (2006)
[55] D.Brockhoff, and E.Zitzler. Are All objectives Necessary? On Dimensionality Reduction in Evolutionary MultiObjective Optimization [J]. In solving from Nature, Springer. 2006, 533-542
[56] A.L.Jaines, C.A.C.Coello and D.Chakraborty. Objective Reduction Using a feature Selection Technique [J]. GECCO 08. 673--680, ACM Press, Atlanta, USA, July 2008
[57] K.Deb and D.K.Saxena. On Finding Pareto-Optimal Solutions Through Dimensionality Reduction for Certain Large- Dimensional Multi-Objective Optimization Problems [R]. KanGAL 2005011
[58] D. Brockhoff and E. Zitzler. Objective Reduction in Evolutionary Multiobjective Optimization: Theory and Applications [J]. Evolutionary Computation, 2009
[59] D. Brockhoff and E. Zitzler. Improving Hypervolume-based Multiobjective Evolutionary Algorithms by Using Objective Reduction Methods [C]. In Congress on Evolutionary Computation (CEC 2007), pages 2086-2093. IEEE Press, 2007
[60] Laumanns, M., Thiele, L., Deb, K., and Zitzler, E. (2002). Combining convergence and diversity in evolutionary multi-objective optimization [J]. Evolutionary Computation, 10(3): 263-282
[61] Lin S,Kernighan B W. An effective heuristic algorithm for the traveling salesman problem. Operational Research,1971,19:486-515
[62] Gen M. Crossover in intensive search and traveling salesman problem. Institute of technology,1994,568-679
[63]唐立新.旅行商问题(TSP)的改进遗传算法[J].东北大学报.1999,1:40-43
[64] David Greenhalgh, Stephen Marshall. Convergence criteria for genetic algorithms[J]. SIAM J. Comput. 2000, 30(1): 269-282
[65] Radu Belea. Genetic algorithm convergence analysis using a unified representation of genes and the hamming distance [J]. International Journal of Knowledge-based and Intelligent Engineering Systems, 2006,10(1):29-40
[66]孟佳娜,王立宏.基于多种群的强者进化遗传算法[J].计算机工程与应用,2004(14):41-42.
[67]邓长春,朱儒明,李咏霞,许波.一种求解TSP问题的多种群并行遗传算法[J].计算机仿真,2008,25(9):187-190
[68]巩敦卫,孙晓燕.变搜索区域多种群遗传算法[J].控制理论与应用,2006,23(2):256-260
[69]李军华,黎明,袁丽华.一种改进的双种群遗传算法[J],小型微型计算机系统,2008,11:2099-2102
[70]郑金华等.基于空间交配的遗传算法.模式识别与人工智能[J], 2003, 16(4): 482-485