摘要
近年来,对NP难的组合优化问题寻求高效的解决方法已成为优化领域的一个极具挑战性的研究课题。除了传统的运筹学方法,现代启发式方法正在得到越来越多的研究人员的关注和重视,已经广泛地应用于基础研究和实际工程领域。现有的大多数启发式方法,如进化算法、人工生命、模拟退火算法和禁忌算法等,都是从生物进化、统计物理和人工智能等领域发展而来。
极值动力学优化算法(Extremal Optimization,EO)是近年来出现的一种新颖的、通用的、基于局部搜索的启发式方法,该方法是从统计物理学发展而来。众所周知,模拟退火算法(Simulated An-nealing,SA)是模拟系统处于平衡态的一种优化方法,与SA不同的是,EO算法的理论基础建筑在Bak-Sneppen生物进化模型之上,该模型模拟处于远离平衡态的系统,具备自组织临界性(Self-Organized Criti-cality,SOC)。SOC是指不管系统处于何种初始状态,不需要调整任何参数,整个系统就可以演化到一个自组织临界状态,在该状态下,系统呈现出幂律分布(Power-law)。遗传算法通过对交配池中的所有可能解实施选择、杂交和变异等遗传操作来达到寻优的目的,而EO算法总是不断地变异近似解的最差组成部分(即所谓的极值动力学机制)来达到寻优的目的。正是这种内在的极值动力学机制,使得EO具备很强的爬山能力,尤其在求解带有相变点(Phase transitions)的组合优化问题时EO更是展现出强大的优势。EO算法的特点是收敛速度快,局部搜索能力强,只有变异算子,无可调参数(对于基本EO算法)或只有一个可调参数τ(对于τ-EO算法)。目前EO算法已经被成功地应用于求解一些NP难的组合优化问题,如二分图,旅行商问题,图着色,旋转玻璃和动态组合优化问题。但是,国外对于EO算法在数值优化和多目标优化问题方面的研究并不多,国内学者对EO算法的研究更少之甚少。
本文主要研究求解无约束或带约束数值优化问题的EO算法,并将求解单目标优化问题的EO算法扩展到多目标优化领域。本文的主要工作包括:
(1)本文从分析EO算法的机理入手,提出了一种求解约束连续优化问题的新算法――带自适应Le′vy变异的基于种群的EO算法(PEO),通过求解6个经典的约束连续优化问题,实验结果证实了PEO能与3种流行的优化算法相匹敌,不失为一种求解数值约束优化问题的有效方法。
(2)为了弥补标准粒子群算法容易陷入局部极值点的不足,本文提出了一种新颖的混合粒子群-极值动力学优化算法(PSO-EO),该算法有效地结合了PSO的全局搜索能力和EO的局部搜索能力,使得标准PSO算法可以跳出局部极值点,从而弥补了标准PSO算法的不足。迄今为止,还没有文献提出将EO和PSO结合起来的优化算法。通过求解6个经典的复杂单峰/多峰函数,PSO-EO算法被证实了具有避免早熟收敛的特点,是一种求解复杂数值优化问题的有效算法。
(3)由于EO算法只有变异操作,因此,变异算子对EO算法的性能好坏起到了重要作用。本文将高斯变异和柯西变异有效地结合起来,提出了一种新颖的适合于求解数值优化问题的变异算子――混合高斯-柯西变异,该算子将“粗调”和“微调”很好地结合起来,并且省去了决定何时在不同变异之间进行切换的麻烦。
(4)本文将基于Pareto支配概念的适应度评价方法引入到EO,提出了一种新颖的多目标极值动力学优化方法(Multiobjective Extremal Op-timization,MOEO),使EO算法成功地扩展到多目标优化领域。接着,用MOEO算法解决了多目标连续优化问题(包括无约束问题和带约束问题),实验结果表明MOEO非常适合于求解多目标连续优化问题,能够与3种经典的多目标进化算法(即NSGA-II,SPEA2和PAES)相匹敌。最后,提出了一种适合于求解多目标0/1背包问题的MOEO算法。实验结果表明MOEO算法具有快速的收敛能力和良好的多样化性能,具有与3种经典的多目标进化算法(即NSGA,SPEA和NPGA)相竞争的优势。
(5)本文利用MOEO算法解决了4个经典的机械组件设计问题。实验结果表明:MOEO算法找到的非劣解集在收敛性和多样性方面有着良好的性能,能够与3种经典的多目标进化算法(NSGA-II,SPEA2和PAES)相匹敌。因此,MOEO算法是一个能解决实际工程优化问题的行之有效的方法。
(6)本文将MOEO算法应用于求解5个经典的股票投资组合优化问题。实验结果表明:MOEO找到的近似Pareto前沿具有良好的收敛性能和多样化性能,能够与算法NSGA-II和SPEA2相匹敌,比PAES更优。因此,MOEO算法是一个能解决实际管理决策优化问题的有效方法。
In recent years, the studies on NP-hard combinatorial optimization problems have beena challenge subject in optimization community. In addition to traditional operations research,the modern heuristics have been attractive in fundamental research and real applications. Theapproaches to evolutionary algorithm (EA), artificial life, simulated annealing (SA) and Tabusearch et al. are developed from the natures of biological evolution, statistical physics andartificial intelligence et al..
Recently, a novel general-purpose local search optimization approach, so-called“Ex-tremal Optimization (EO)”has been proposed based on the fundamentals of statisticalphysics. In contrast to SA which is inspired by equilibrium statistical physics, EO is basedon Bak-Sneppen (BS) model of biological evolution which simulates far-from equilibriumdynamics in statistical physics. The BS model is one of the models that show the natureof self-organized criticality (SOC). The SOC means that regardless of the initial state, thesystem always tunes itself to a critical point having a power-law behavior without any tun-ing control parameter. In contrast to genetic algorithms (GAs) which operate on an entire“gene-pool”of huge number of possible solutions, EO successively eliminates those worstcomponents in the sub-optimal solutions. The extremal dynamics mechanism in EO pro-vide significant hill-climbing ability, which enables EO to perform well particularly at thephase transitions. EO has many advanced features,such as fast convergence speed, strongability of local search, only mutation operator, no adjustable parameter for basic EO or onlyone adjustable parameterτforτ-EO. EO has been successfully applied to some NP-hardcombinatorial optimization problems such as graph bi-partitioning, traveling salesman prob-lem(TSP), graph coloring, spin glasses and dynamic combinatorial problems. However, sofar there have been only limited papers studying on the mechanism of EO applied to numer-ical optimization problems or multiobjective optimization problems(MOPs).
This paper makes an investigation on the mechanism of EO for solving the uncon-strained or constrained numerical optimization problems, and further extends EO to dealwith the MOPs. The main research work includes:
(1) Based on the analysis on the mechanism of EO, this paper presents a new algorithmnamed population-based EO with adaptive Le′vy mutation (PEO). Compared with threestate-of-the-art stochastic search methods with six benchmark functions, it has beenshown that our approach is a good choice to deal with the numerical constrained opti-mization problems.
(2) To overcome the limitation of standard PSO, this paper proposes a novel hybrid algo-rithm, called hybrid PSO-EO algorithm, through introducing EO to PSO. The hybridapproach elegantly combines the exploration ability of PSO with the exploitation abil-ity of EO and thus the hybrid algorithm can help standard PSO to jump out of the localoptima. The proposed approach is validated using six complex unimodal/multimodalbenchmark functions. The simulation results demonstrate that the proposed approachis capable of avoiding premature convergence. Thus, the hybrid PSO-EO algorithmcan be considered a good alternative to solve complex numerical function optimiza-tion problems.
(3) Since there is merely mutation operation in EO, the mutation operator plays a keyrole in EO search. In this paper, we present a new mutation method based on mixingGaussian mutation and Cauchy mutation, called hybrid Gaussian-Cauchy mutation.The new mutation operator combines the advantages of coarse-grained search andfine-grained search. Unlike some switching algorithms which have to decide whento switch between different mutations during search, the hybrid GC mutation does notneed to make such decisions.
(4) In order to extend EO to solve the MOPs, in this work, we propose a novel multiob-jective optimization algorithm, called Multiobjective Extremal Optimization (MOEO),through introducing the fitness assignment based on Pareto optimality to EO. In thispaper, MOEO has been successfully applied to solving unconstrained or constrainedMOPs. The simulation results indicate that the proposed approach is highly competi-tive with three state-of-the-art multiobjective evolutionary algorithms (MOEAs), i.e.,NSGA-II, SPEA2 and PAES. This paper also successfully extends MOEO to solvethe multiobjective 0/1 knapsack problems. The experimental results demonstrate thatthe proposed approach is highly competitive with three state-of-the-art MOEAs, i.e.,NSGA,SPEA and NPGA.
(5) The proposed MOEO algorithm is applied to handling four mechanical component de-sign problems reported in the specialized literature. The experimental results demon- strate that MOEO can find the nondominated solutions with good performance in bothaspects of convergence and diversity. The results also show that MOEO is highlycompetitive with NSGA-II, SPEA2 and PAES. Thus MOEO can be considered a goodalternative to solve engineering MOPs.
(6) The MOEO algorithm is applied to solving five benchmark stock portfolio optimiza-tion problems. It is demonstrated that MOEO is highly competitive with NSGA-II andSPEA2, and superior to PAES. As a consequence, MOEO is an effective method fordealing with MOPs in the fields of managements and decision-making.
引文
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