进 化 多 目 标 优 化 算 法 及 其 应 用 研 究 陈 碧 黎 指 导 教 师 曾 文 华 教 授 厦 门 大 学
厦门大学博硕士论文摘要库
学校编码：10384 分类号 TP18 密级
学 号：24320120153757 UDC
博 士 学 位 论 文
进化多目标优化算法及其应用研究
Research on Evolutionary Multi-objective Optimization
Algorithms and their Applications
陈碧黎
指导教师姓名：
曾 文 华 教 授
专 业 名 称：
计算机科学与技术
论文提交日期：
2 0 1 5 年 4 月
论文答辩日期：
2 0 1 5 年 5 月
学位授予日期：
2 0 1 5 年 6 月
指 导 教 师：
答辩委员会主席：
2015 年 4 月
厦门大学博硕士论文摘要库
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厦门大学博硕士论文摘要库
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厦门大学博硕士论文摘要库
摘 要 I
摘 要
现实世界中的许多优化问题通常可以转化为多目标优化问题。所谓的多目标 优化问题是指那些需要同时优化多个相互作用且相互冲突的目标函数的优化问 题。目前，采用进化算法求解多目标优化问题已经成为了多目标优化领域中的研 究热点。许多研究学者相继提出了一些进化多目标优化算法用于求解各种类型的 多目标优化问题，一些优秀算法已经成功地应用在实际工程项目中。本文在全面 介绍进化多目标优化算法的理论知识以及研究现状的基础上，主要针对不同特征 的多目标优化问题设计出效果显著的求解策略。本论文的主要研究工作如下： （1）针对具有简单几何形状Pareto最优解集的多目标优化问题，分析多目 标差分进化算法在处理该类多目标优化问题的不足之处，如过快收敛以及容易陷 入局部最优，提出基于模拟退火机制的多目标差分进化算法（Multi-ObjectiveDifferential Evolution with Simulated Annealing，简称MODESA）。该算法引入模
拟退火机制以及基于生命值概念的优先策略来保留一些潜力个体使其优先进入 下一代进化中，从而提高算法所求最优解集的收敛性和分布性。此外，算法还采 用一种基于邻近距离的动态修剪算法作为分布度维持策略。本文将所提算法 MODESA 用于求解五个双目标函数以及两个三目标函数的多目标优化问题，并 与其他三个算法进行对比分析，实验结果表明：改进后的多目标差分进化算法 MODESA能适当地提高算法的收敛性和分布性。 （2）针对具有复杂几何形状Pareto最优解集的多目标优化问题，提出一种 全新的基于非支配排序和局部搜索的进化多目标优化算法（Non-dominated
Sorting and Local Search based algorithm，简称NSLS）。该算法是基于迭代的：在
每一次迭代中，给定一个进化种群P，采用基于差分算子的双方向局部搜索策略 产生一个更好的进化种群 ，接着在合并种群 上执行非支配排序获取下一 代进化种群。这种通过以局部搜索为主的搜索策略使算法可以很好地逼近Pareto 最优前沿。此外，受采样理论中的最优候选点算法启发，提出最远候选点算法作 为分布度维持策略，使得算法所获最优解集在目标空间上能够沿着Pareto最优前 沿均匀分布。本文将算法NSLS同其他四个代表性经典算法进行了实验对比分析， 结果表明：在求解具有复杂几何形状的 Pareto 最优解集的多目标优化问题上，
厦门大学博硕士论文摘要库
摘 要 II NSLS算法具有更好的收敛性和分布性。 （3）针对具有较多局部Pareto最优前沿以及Pareto最优前沿属于非连续或 分布不均匀的多目标优化问题，本文提出劣值表概念和基于劣值表的搜索策略， 并将其集成到第四章所提出的算法NSLS中，构成算法NSLS-BTM（NSLS with
Bad Table Mechanism）。该算法将进化过程中产生的劣解通过劣值表这一存储结
构进行记忆保存。当无法获得更优秀的个体时，算法利用这些劣解开辟新的搜索 领域，最终使得算法所求最优解集在目标空间上能够更逼近Pareto最优前沿。实 验结果与分析表明：该方法可以有效地开辟新的搜索区域，从而在整体上提高算 法所求最优解集的收敛性和分布性。 （4）针对高维多目标优化问题的求解难点，在多目标进化算法中分别引入 基于优胜关系的替换策略以及基于目标值比例和的修剪策略，提出一种适用于求 解高维多目标优化问题的进化多目标优化算法（称为 m-NSLS）。首先，采用基 于优胜关系的替换策略可以更好地指导算法进行替换判定，提高替换准确性，从 而提高收敛性能。其次，基于非支配排序的进化多目标优化算法在求解高维多目 标优化问题时，最后一层边界集在进化后期的非支配解数量急剧增加，本文通过 预先采用基于目标值比例和的修剪策略，可以有效减少非支配解集的个体数目， 从而提高分布度维持策略的效率。实验结果与分析表明：所提出的算法与当前优 秀的高维多目标进化算法相比，具有良好的竞争能力。 （5）针对证券投资组合优化问题，采用一种修正的基于非支配排序和局部搜 索的进化多目标优化算法（称为 e-NSLS）来处理该类问题。通过求解不同参数 设置下的五组Benchmark测试数据，实验结果表明：e-NSLS能够比其他三个算 法获得更好的最优解集，供决策者根据相关偏好信息进行选择。 总之，本论文主要针对不同特征的多目标优化问题，提出一系列有效的解决 方法。这些工作在一定程度上提高了进化多目标优化算法的性能，也促进了进化 多目标优化算法的深入研究，并且对进化多目标优化算法的实际应用也有着重要 的指导意义。 关键字：进化多目标优化算法；模拟退火；非支配排序；局部搜索；证券投资组 合优化问题
厦门大学博硕士论文摘要库
Abstract
III
Abstract
Many optimization problems in the real word can be transformed into multi-objective optimization problems (MOPs). The MOPs have many interactive and conflicting objective functions. At present, using evolutionary algorithms to solve MOPs has become a researching hotspot in the field of multi-objective optimization. A lot of research scholars have proposed many evolutionary multi-objective optimization (EMO) algorithms to solve various kinds of MOPs. Some of them have been successfully applied in the actual projects. Based on a comprehensive survey of the state-of-the-art of EMO algorithms, this dissertation emphasizes designing remarkable methods to solve the MOPs. The major researching work and contributions can be summarized as follows:
（1）To solve the MOPs with simple Pareto-optimal sets, the disadvantages of the multi-objective differential evolution evolutionary algorithms are analyzed. For example, it has a too fast convergence speed and it is easy to trap into local optimum. In this dissertation, a multi-objective differential evolution evolutionary algorithm using simulated annealing (named MODESA) and priority strategy with the concept of prior life value is proposed to overcome the disadvantages of multi-objective differential evolution evolutionary algorithm. The simulated annealing and priority strategy can be used to keep some potential individuals and permit them preferentially entering to the next process of evolution for the purpose of improving the convergence and diversity of the algorithm. In the proposed simulated annealing approach, a new acceptance probability computation function based on domination is presented and these potential individuals are assigned a prior life value to have a priority to be selected to the next generation. In addition, the algorithm applies an efficient diversity maintenance approach to get a good distributed Pareto-optimal front. The feasibility of the proposed algorithm is investigated on a set of five bi-objective and two tri-objective optimization problems and the results are compared with three other algorithms. The experimental results show the effectiveness of the
Abstract
IV
proposed algorithm in terms of convergence and diversity.
（2）A new multiobjective optimization framework based on non-dominated sorting and local search (named NSLS) is introduced to solve the MOPs with complicate Pareto-optimal sets. The proposed algorithm is based on iterations. At each iteration, given a population P, a simple local search method is used to get a better population
, and then the non-dominated sorting is applied on the combined population to get a better population for the next iteration. The local search-based searching mechanism is good for the algorithm converging to the Pareto-optimal front. In addition, the farthest-candidate approach is combined with the fast non-dominated sorting to get the new population to increase the diversity. The experimental results reveal that NSLS is able to find a better spread of solutions and a better convergence to the true Pareto-optimal front compared to four other good algorithms.
（3）A new concept of bad table and the corresponding searching mechanism based on the bad table (named NSLS-BTM) are proposed to solve the MOPs with complicate Pareto-optimal sets and Pareto-optimal fronts. In the proposed algorithm, these bad solutions generated during the process of evolution are selectively stored in the bad table. When the algorithm could not find a better solution, the bad table based mechanism is used for opening up a new searching area to improve the convergent ability of the algorithm. The experimental results demonstrate that the proposed mechanism can enhance the performance of the algorithm in terms of the convergence and diversity.
（4）A replacement mechanism based on the favor relation and a prunning strategy based on the sum of objective function proportionsare presented in the EMO algoriths to solve the many-MOPs (named m-NSLS). First, the probability of the situation that two solutions cannot be compared in the replacement process is increased, therefore, applying the replacement mechanism can help the algorithm to determine whether to remove the solution or not for the purpose of increasing the accuracy of the replacement. Second, when the EMO algorithms based on non-dominated sorting is used for solving the many-MOPs, the number of non-dominated solutions in the last front is rapidly increased. The proposed repair mechanism based on the sum of the
Abstract
V
objective function proportions can effectively remove some non-dominated solutions in advance to improve the performance of the diversity maintenance mechanism. Empirical results reveal that the proposed algorithm is better than the state-of-the-art of EMO algorithms.
（5）An enhanced non-dominated sorting and local search based evolutionary multi-objective optimization algorithm is proposed to solve the multi-objective portfolio optimization problems (named e-NSLS). The method is compared with the other three good algorithms on five benchmark data sets. The experimental results with different cardinality constraints show that the proposed algorithm can provide better results than the other algorithms for the decision makers according to their preference.
In summary, this dissertation proposes some effective methods to overcome the difficulties of various characteristic MOPs. The work can improve the performance of EMO algorithms to some extent, which is not only conducive to promoting the intensive research of the EMO algorithms, but also has vital significance for the practical projects of EMO algorithms.
Key words: Evolutionary Multi-objective Optimization Algorithms; Simulated Annealing; Non-dominated Sorting; Local Search; Portfolio Optimization Problems
目 录 VI
目 录
第一章
绪 论 ... 1
1.1 研究背景与意义 ... 1 1.2 多目标优化的基本概念 ... 2 1.2.1 多目标优化问题的定义 ... 2 1.2.2 Pareto 相关概念 ... 3 1.3 进化多目标优化算法的发展与研究现状 ... 4 1.3.1 第一代进化多目标优化算法 ... 5 1.3.2 第二代进化多目标优化算法 ... 5 1.3.3 当前进化多目标优化算法的研究进展 ... 6 1.4 论文的主要研究内容 ... 8 1.5论文的主要创新点 ... 9 1.6 论文的组织结构 ... 9第二章
进化多目标优化算法的理论基础 ... 12
2.1进化多目标优化算法的基本原理和算法框架 ... 12 2.2进化多目标优化算法的主要设计目标 ... 13 2.3进化多目标优化算法的研究难点 ... 14 2.4进化多目标优化算法的标准测试函数 ... 14 2.5进化多目标优化算法的性能评价方法 ... 19 2.5.1 逆向世代距离 ... 20 2.5.2 世代距离 ... 20 2.5.3 分布度评价指标 ... 21 2.5.4 威氏符号秩次检验 ... 21 2.6 进化多目标优化算法的应用简介 ... 22 2.7 本章小结 ... 23第三章
一种基于模拟退火机制的多目标差分进化算法 ... 24
3.1 引言... 24厦门大学博硕士论文摘要库
目 录 VII 3.2 差分进化算法简介 ... 26 3.3 算法描述 ... 27 3.3.1 种群初始化 ... 27 3.3.2 选择程序中的模拟退火过程 ... 28 3.3.3 分布度维持策略 ... 30 3.3.4 算法主体描述 ... 31 3.3.5 MODESA与MODEA比较 ... 33 3.3.6 算法复杂度分析 ... 33 3.4 实验... 34 3.4.1 测试问题与算法性能评价指标 ... 34 3.4.2 实验参数设置 ... 34 3.4.3 实验结果与分析 ... 35 3.5 本章小结 ... 39
第四章
一种基于非支配排序和局部搜索的进化多目标优化算法 . 41
4.1 引言... 41 4.2 算法描述 ... 43 4.2.1 局部搜索策略 ... 43 4.2.2 基于采样理论的分布度维持策略 ... 46 4.2.3 算法主体描述 ... 48 4.2.4 算法时间复杂度分析 ... 49 4.3 实验... 50 4.3.1 测试问题与算法性能评价指标 ... 50 4.3.2 实验参数设置 ... 50 4.3.3 实验结果与分析 ... 51 4.4 本章小结 ... 65第五章
一种基于劣值表搜索策略的进化多目标优化算法 ... 67
5.1 引言... 67 5.2 算法描述 ... 67 5.2.1 劣值表定义 ... 68厦门大学博硕士论文摘要库
目 录 VIII 5.2.2 劣值表搜索策略 ... 68 5.2.3 基于劣值表搜索策略的进化多目标优化算法 ... 70 5.2.4 算法时间复杂度分析 ... 72 5.3 实验... 72 5.3.1 测试问题与算法性能评价指标 ... 72 5.3.2 实验参数设置 ... 73 5.3.3 实验结果与分析 ... 73 5.4 本章小结 ... 77
第六章
一种基于模糊支配和修剪策略的高维进化多目标优化算
法... .79
6.1 引言... 79 6.2 算法描述 ... 81 6.2.1 基于优胜关系的替换策略 ... 81 6.2.2 基于目标值比例和的修剪策略 ... 83 6.2.3 算法描述与时间复杂度分析 ... 84 6.3 实验... 85 6.3.1 测试问题与算法性能评价指标 ... 85 6.3.2 实验参数设置 ... 85 6.3.3 实验结果与分析 ... 86 6.4 本章小结 ... 87第七章
进化多目标优化算法在证券投资组合中的应用 ... 89
7.1 引言... 89 7.2 常见的证券投资组合优化问题求解算法 ... 90 7.2.1 精确算法 ... 90 7.2.2 启发式算法 ... 90 7.3 Markowitz的均值-方差组合模型 ... 92 7.4 算法描述 ... 93 7.4.1 定义与编码 ... 93厦门大学博硕士论文摘要库
目 录 IX 7.4.2 约束处理策略 ... 93 7.4.3 算法主体过程描述 ... 93 7.5实验... 94 7.5.1 测试问题与算法性能评价指标 ... 94 7.5.2 实验参数设置 ... 95 7.5.3 实验结果与分析 ... 96 7.6 本章小结 ... 105
第八章
总结与展望 ... 106
8.1 总结... 106 8.2 展望... 107参考文献 ... 109
附录 作者读博期间已发表(录用)和撰写的论文 ... 124
致谢
... 126
厦门大学博硕士论文摘要库
Contents
X
Contents
Chapter 1 Introduction
... 1
1.1 Background ... 1
1.2 Basic concepts of multi-objective optimization ... 2
1.2.1 Definition of MOPs ... 2
1.2.2 Concepts of Pareto ... 3
1.3 State of EMO Algorithms ... 4
1.3.1 First Generation of EMO Algorithms ... 5
1.3.2 Second Generation of EMO Algorithms ... 5
1.3.3 Research Hotspot of Current EMO Algorithms ... 6
1.4 Major Work and Contributions ... 9
1.5 Outline of Thesis ... 9
1.5 Innovation of Thesis ... 9
Chapter 2 Basic Theory of EMO Algorithms
... 12
2.1 Basic Theory and Framework of EMO Algorithms ... 12
2.2 Design Goals of EMO Algorithms ... 13
2.3 Research Difficulties of EMO Algorithms ... 14
2.4 Test Problems of EMO Algorithms ... 14
2.5 Performance Measures of EMO Algorithms ... 19
2.5.1 Performance of Inverted generational distance ... 20
2.5.2 Performance of Generational distance ... 20
2.5.3 Performance of Spread ... 21
2.5.4 Wilcoxon signed ranks test ... 21
2.6 Applications of EMO Algorithms ... 22
2.7 Summary ... 23
Chapter 3 A New Multi-objective Differential Evolution Algorithm
based on Simulated Annealing
... 24
Contents
XI
3.1 Introduction ... 24
3.2 An Overview of Differential Evolution Algorithms ... 26
3.3 Description of Proposed Algorithm ... 27
3.3.1 Population Initialization ... 27
3.3.2 Simulated Annealing Process during Selection Procedure ... 28
3.3.3 Diversity Maintenance Mechanism ... 30
3.3.4 Description of Proposed Algorithm ... 31
3.3.5 Comparison Between MODESA and MODEA ... 33
3.3.6 Time Complexity Analysis ... 33
3.4 Experiments ... 34
3.4.1 Test Problems and Performance Measures ... 34
3.4.2 Parameters Settings ... 34
3.4.3 Experimental Results and Analysis ... 35
3.5 Summary ... 39
Chapter 4 A Non-dominated Sorting and Local Search based EMO
Algorithm
... 41
4.1 Introduction ... 41
4.2 Description of Proposed Algorithm ... 43
4.2.1 Local Search Mechanism ... 43
4.2.2 Sampling Theory based Diversity Maintenance Mechanism ... 46
4.2.3 Description of Proposed Algorithm ... 48
4.2.4 Time Complexity Analysis of Proposed Algorithm ... 49
4.3 Experiment ... 50
4.3.1 Test Problems and Performance Measures ... 50
4.3.2 Parameters Settings ... 50
4.3.3 Experimental Results and Analysis ... 51
4.4 Summary ... 65
Chapter 5 Bad Table based Mechanism EMO Algorithm
... 67
5.1 Introduction ... 67
厦门大学博硕士论文摘要库
Contents
XII
5.2 Description of Algorithms ... 67
5.2.1 Definition of Bad Table ... 68
5.2.2 Bad Table based Searching Mechanism ... 68
5.2.3 Bad Table Searching Mechanism based EMO Algorithm ... 70
5.2.4 Time Complexity Analysisi of Proposed Algorithm ... 70
5.3 Experiments ... 72
5.3.1 Test Problems and Performance Measure ... 72
5.3.2 Parameters Settings ... 72
5.3.3 Experimental Results and Analysis ... 73
5.4 Summary ... 77
Chapter 6 A Dominate Relation and Trim Idea based EMO
Algorithm for Many MOPs
... 79
6.1 Introduction ... 79
6.2 Description of Algorithm ... 81
6.2.1 A Replacement Mechanism based on Favor Relation ... 81
6.2.2 A Prunning Mechanism based on Sum of Values of Objective Function Proportions ... 83
6.2.3 Algorithm and Time Complexity Analysis ... 84
6.3 Experiments ... 84
6.3.1 Test Problems and Mearsures ... 85
6.3.2 Parameters Settings ... 85
6.3.3 Experimental and Analysis ... 86
6.4 Summary ... 87
Chapter 7 Application of EMO Algorithms in Portfolio Optimization
problems
... 89
7.1 Introduction ... 89
7.2 Usual EMO Algorithms for Portfolio Optimization Problems ... 90
7.2.1 Exact Algorithms ... 90
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