基于全局排序的高维多目标优化研究

目标数超过4的高维多目标优化是目前进化多目标优化领域求解难度最大的问题之一,现有的多目标进化算法求解该类问题时,存在收敛性和解集分布性上的缺陷,难以满足实际工程优化需求.提出一种基于全局排序的高维多目标进化算法GR-MODE,首先,采用一种新的全局排序策略增强选择压力,无需用户偏好及目标主次信息,且避免宽松Pareto支配在排序结果合理性与可信性上的损失;其次,采用Harmonic平均拥挤距离对个体进行全局密度估计,提高现有局部密度估计方法的精确性;最后,针对高维多目标复杂空间搜索需求,设计新的精英选择策略及适应度值评价函数.将该算法与国内外现有的5种高性能多目标进化算法在标准测试函数集DTLZ{1,2, 4,5}上进行对比实验,结果表明,该算法具有明显的性能优势,大幅提升了4~30维高维多目标优化的收敛性和分布性.     

An effective hybrid particle swarm optimization algorithm for multi-objective flexible job-shop scheduling problem

Flexible job-shop scheduling problem (FJSP) is an extension of the classical job-shop scheduling problem. Although the traditional optimization algorithms could obtain preferable results in solving the mono-objective FJSP. However, they are very difficult to solve multi-objective FJSP very well. In this paper, a particle swarm optimization (PSO) algorithm and a tabu search (TS) algorithm are combined to solve the multi-objective FJSP with several conflicting and incommensurable objectives. PSO which integrates local search and global search scheme possesses high search efficiency. And, TS is a meta-heuristic which is designed for finding a near optimal solution of combinatorial optimization problems. Through reasonably hybridizing the two optimization algorithms, an effective hybrid approach for the multi-objective FJSP has been proposed. The computational results have proved that the proposed hybrid algorithm is an efficient and effective approach to solve the multi-objective FJSP, especially for the problems on a large scale.     

Multi-objective optimization with artificial weed colonies

Invasive Weed Optimization (IWO) was recently proposed as a simple but powerful metaheuristic algorithm for real parameter optimization. IWO draws inspiration from the ecological process of weeds colonization and distribution and is capable of solving general multi-dimensional, linear and nonlinear optimization problems with appreciable efficiency. This article extends the basic IWO for tackling multi-objective optimization problems that aim at achieving two or more objectives (very often conflicting) simultaneously. The concept of fuzzy dominance has been used to sort the promising candidate solutions at each iteration. The new algorithm has been shown to be statistically significantly better than some state of the art existing evolutionary multi-objective algorithms, namely NSGAIILS, DECMOSA-SQP, MOEP, Clustering MOEA, GDE3, and MOEADGM on a 12-function test-suite (including both unconstrained and constrained problems) from the IEEE CEC (Congress on Evolutionary Computation) 2009 competition and special session on multi-objective optimization algorithms. The following performance metrics were considered: IGD, Spacing, and Minimum Spacing. Our experimental results suggest that IWO holds immense promise to appear as an efficient metaheuristic for multi-objective optimization.     

一种基于偏好的多目标调和遗传算法

最近涌现了各种进化方法来解决多目标优化问题,多数方法使用Pareto优胜关系作为选择策略而没有采用偏好信息.这些算法不能有效处理目标数目许多时的优化问题.通过在不同准则之间引入偏好来解决该问题,提出一种多目标调和遗传算法MOCGA(multi-objective concordance genetic algorithm).当同时待优化的目标数目增加时,根据决策者提供的信息使用弱优胜关系进行个体优劣的比较.这种算法被证明为能收敛至全局最优.对于目标数目为很多的优化问题,测试实验结果表明了这种新算法的有效性.     

A multi-objective Markov Chain Monte Carlo cellular automata model: Simulating multi-density urban expansion in NYC

Cellular automata (CA) models have increasingly been used to simulate land use/cover changes (LUCC). Metaheuristic optimization algorithms such as particle swarm optimization (PSO) and genetic algorithm (GA) have been recently introduced into CA frameworks to generate more accurate simulations. Although Markov Chain Monte Carlo (MCMC) is simpler than PSO and GA, it is rarely used to calibrate CA models. In this article, we introduce a novel multi-chain multi-objective MCMC (mc-MO-MCMC) CA model to simulate LUCC. Unlike the classical MCMC, the proposed mc-MO-MCMC is a multiple chains method that imports crossover operation from classical evolutionary optimization algorithms. In each new chain, after the initial one, the crossover operator generates the initial solution. The selection of solutions to be crossed over are made according to their fitness score. In this paper, we chose the example of New York City (USA) to apply our model to simulate three conflicting objectives of changes from non-urban to low-, medium- or high-density urban between 2001 and 2016 using USA National Land Cover Database (NLCD). Elevation, slope, Euclidean distance to highways and local roads, population volume and average household income are used as LUCC causative factors. Furthermore, to demonstrate the efficiency of our proposed model, we compare it with the multi-objective genetic algorithm (MO-GA) and standard single-chain multi-objective MCMC (sc-MO-MCMC). Our results demonstrate that mc-MO-MCMC produces accurate simulations of land use dynamics featured by faster convergence to the Pareto frontier comparing to MO-GA and sc-MO-MCMC. The proposed multi-objective cellular automata model should efficiently help to simulate a trade-off among multiple and, possibly, conflicting land use change dynamics at once.     

Constrained multi-objective optimization algorithms: Review and comparison with application in reinforced concrete structures

Engineering design problems are often multi-objective in nature, which means trade-offs are required between conflicting objectives. In this study, we examine the multi-objective algorithms for the optimal design of reinforced concrete structures. We begin with a review of multi-objective optimization approaches in general and then present a more focused review on multi-objective optimization of reinforced concrete structures. We note that the existing literature uses metaheuristic algorithms as the most common approaches to solve the multi-objective optimization problems. Other efficient approaches, such as derivative-free optimization and gradient-based methods, are often ignored in structural engineering discipline. This paper presents a multi-objective model for the optimal design of reinforced concrete beams where the optimal solution is interested in trade-off between cost and deflection. We then examine the efficiency of six established multi-objective optimization algorithms, including one method based on purely random point selection, on the design problem. Ranking and consistency of the result reveals a derivative-free optimization algorithm as the most efficient one.     

Machining scheme selection of digital manufacturing based on genetic algorithm and AHP

The selection of machining scheme for a part is an important and strategic problem. It involves multiple and conflicting objectives such as cost, time, quality, service level, resource utilization, etc. The selection is always affected by subjective factors such as the knowledge and experiences of decision maker in conventional machining. This paper proposes a method based on genetic algorithms (GA) to find out the set of Pareto-optimal solutions for multi-objective digital machining scheme selection. To deal with multi-objective and enable the engineer to make decision on different demands, an analytic hierarchy process (AHP) is implemented in the proposed procedure to determine the weight value of evaluation indexes. Three conflicting objectives: cost, quality and operation time are simultaneously optimized. An application sample is developed and its results are analyzed. The optimization results show that the hybrid algorithm is reliable and robust.     

Multi-objective evolutionary optimization of polynomial neural networks for modelling and prediction of explosive cutting process

In this paper, evolutionary algorithms (EAs) are deployed for multi-objective Pareto optimal design of group method of data handling (GMDH)-type neural networks which have been used for modelling an explosive cutting process using some input–output experimental data. In this way, multi-objective EAs (non-dominated sorting genetic algorithm, NSGA-II) with a new diversity-preserving mechanism are used for Pareto optimization of such GMDH-type neural networks. The important conflicting objectives of GMDH-type neural networks that are considered in this work are, namely, training error (TE), prediction error (PE), and number of neurons (N) of such neural networks. Different pairs of theses objective functions are selected for 2-objective optimization processes. Therefore, optimal Pareto fronts of such models are obtained in each case which exhibit the trade-off between the corresponding pair of conflicting objectives and, thus, provide different non-dominated optimal choices of GMDH-type neural networks models for explosive cutting process. Moreover, all the three objectives are considered in a 3-objective optimization process, which consequently leads to some more non-dominated choices of GMDH-type models representing the trade-offs among the training error, prediction error, and number of neurons (complexity of network), simultaneously. The overlay graphs of these Pareto fronts also reveal that the 3-objective results include those of the 2-objective results and, thus, provide more optimal choices for the multi-objective design of GMDH-type neural networks in terms of minimum training error, minimum prediction error, and minimum complexity.     

高维多目标进化算法研究综述

传统的多目标进化算法能够有效地解决2个或3个目标的优化问题,但当优化目标超过4维即具有高维目标时,其优化效果将大大下降,因此高维多目标进化算法的研究得到了较多的关注.鉴于此,对高维多目标进化算法的研究进展进行系统地分类综述,分析了高维目标对优化算法造成的困难以及改进的可视化技术;总结了各类算法的特点与缺陷,并给出进一步可能的研究方向.     

An objective reduction algorithm using representative Pareto solution search for many-objective optimization problems

In recent years, many-objective optimization problems (i.e. more than three objectives) have attracted the interests of many researchers. The main difficulties of many-objective optimization problems lie in high computational cost, stagnation in search process, etc. It is almost impossible to design an algorithm effective for all problems. However, for some problems, especially for problems with redundant objectives, it is possible to design effective algorithms by removing the redundant objectives and keeping the non-redundant objectives so that the original problem becomes the one with much fewer objectives. To do so, first, a multi-objective evolutionary algorithm-based decomposition is adopted to generate a smaller number of representative non-dominated solutions widely distributed on the Pareto front. Then the conflicting objective pairs are identified through these non-dominated solutions, and the redundant objectives are determined by these pairs and then removed. Based on these, a fast non-redundant objectives generation algorithm is proposed in this paper. Finally, the experiments are conducted on a set of benchmark test problems and the results indicate the effectiveness and efficiency of the proposed algorithm.     

Self-adaptive differential evolution algorithm with α-constrained-domination principle for constrained multi-objective optimization

Real-world problems are inherently constrained optimization problems often with multiple conflicting objectives. To solve such constrained multi-objective problems effectively, in this paper, we put forward a new approach which integrates self-adaptive differential evolution algorithm with α-constrained-domination principle, named SADE-αCD. In SADE-αCD, the trial vector generation strategies and the DE parameters are gradually self-adjusted adaptively based on the knowledge learnt from the previous searches in generating improved solutions. Furthermore, by incorporating domination principle into α-constrained method, α-constrained-domination principle is proposed to handle constraints in multi-objective problems. The advantageous performance of SADE-αCD is validated by comparisons with non-dominated sorting genetic algorithm-II, a representative of state-of-the-art in multi-objective evolutionary algorithms, and constrained multi-objective differential evolution, over fourteen test problems and four well-known constrained multi-objective engineering design problems. The performance indicators show that SADE-αCD is an effective approach to solving constrained multi-objective problems, which is basically enabled by the integration of self-adaptive strategies and α-constrained-domination principle.     

免疫克隆多目标优化算法求解约束优化问题

针对现有的约束处理技术的一些不足之处,提出一种用于求解约束优化问题的算法——免疫克隆多目标优化算法(immune clonal multi-objective optimization algorithm,简称ICMOA).算法的主要特点是通过将约束条件转化为一个目标,从而将问题转化为两个目标的多目标优化问题.引入多目标优化中的Pareto-支配的概念,每一个个体根据其被支配的程度进行克隆、变异及选择等操作.克隆操作实现了全局择优,有利于得到高质量的解;变异操作提高算法的局部搜索能力,有利于所得解的多样性;选择操作有利于算法向着最优搜索,而且加快了收敛速度.基于抗体群的随机状态转移过程,证明该算法具有全局收敛性.通过对13个标准测试问题的测试,并与已有算法进行比较。结果表明,该算法在收敛速度和求解精度上均具有一定的优势.     

ADAPTIVE MULTI-OBJECTIVE OPTIMIZATION BASED ON NONDOMINATED SOLUTIONS

An adaptive hybrid model (AHM) based on nondominated solutions is presented in this study for multi-objective optimization problems (MOPs). In this model, three search phases are devised according to the number of nondominated solutions in the current population: 1) emphasizing the dominated solutions when the population contains very few nondominated solutions; 2) maintaining the balance between nondominated and dominated solutions when nondominated ones become more; 3) when the population consists of adequate nondominated solutions, dominated ones could be ignored and the isolated nondominated ones are allocated more computational budget by their crowding distance values for heuristic search. To exploit local information efficiently, a local incremental search algorithm, LISA, is proposed and merged into the model. This model maintains the adaptive mechanism between the optimization process by the online discovered nondominated solutions. The proposed model is validated using five ZDT and five DTLZ problems. Compared with three other state-of-the-art multi-objective algorithms, namely NSGA-II, SPEA2, and PESA-II, AHM achieves comparable results in terms of convergence and diversity metrics. Finally, the sensitivity of introduced parameters and scalability to the number of objectives are investigated.     

Particle swarm optimization with preference order ranking for multi-objective optimization

A new optimality criterion based on preference order (PO) scheme is used to identify the best compromise in multi-objective particle swarm optimization (MOPSO). This scheme is more efficient than Pareto ranking scheme, especially when the number of objectives is very large. Meanwhile, a novel updating formula for the particle’s velocity is introduced to improve the search ability of the algorithm. The proposed algorithm has been compared with NSGA-II and other two MOPSO algorithms. The experimental results indicate that the proposed approach is effective on the highly complex multi-objective optimization problems.     

网络社区发现的多目标分解粒子群优化算法

通过分析社会网络中社区发现问题的优化目标,构造了社区发现的多目标优化模型,提出一种网络社区发现的多目标分解粒子群优化算法。该算法采用切比雪夫法将多目标优化问题分解为多个单目标优化子问题,使用粒子群优化(PSO)算法对社区结构进行挖掘,并引入了一种新颖的基于局部搜索的变异策略以提高算法的搜索效率和收敛速度,该算法克服了单目标优化算法存在的解单一以及难以发现社区层次结构的缺陷。人工网络及真实网络上的实验结果表明,该算法能够快速准确地挖掘网络社区并揭示社区的层次结构。     

基于参考点的高维多目标粒子群算法

高维多目标优化问题一般指目标个数为4个 或以上时的多目标优化问题.由于种群中非支配解数量随着目标数量的增加而急剧增多,导致进化算法的进化压力严重降低,求解效率低.针对该问题,提出一种基于粒子群的高维多目标问题求解方法,在目标空间中引入一系列的参考点,根据参考点筛选出能兼顾多样性和收敛性的非支配解作为粒子的全局最优,以增大选择压力.同时,提出了基于参考点的外部档案维护策略,以保持最后所得解集的多样性.在标准测试函数DTLZ2上的仿真结果表明,所提方法在求解高维多目标问题时能够得到收敛性和分布性都较好的解集.     

Objective reduction based on nonlinear correlation information entropy

It is hard to obtain the entire solution set of a many-objective optimization problem (MaOP) by multi-objective evolutionary algorithms (MOEAs) because of the difficulties brought by the large number of objectives. However, the redundancy of objectives exists in some problems with correlated objectives (linearly or nonlinearly). Objective reduction can be used to decrease the difficulties of some MaOPs. In this paper, we propose a novel objective reduction approach based on nonlinear correlation information entropy (NCIE). It uses the NCIE matrix to measure the linear and nonlinear correlation between objectives and a simple method to select the most conflicting objectives during the execution of MOEAs. We embed our approach into both Pareto-based and indicator-based MOEAs to analyze the impact of our reduction method on the performance of these algorithms. The results show that our approach significantly improves the performance of Pareto-based MOEAs on both reducible and irreducible MaOPs, but does not much help the performance of indicator-based MOEAs.     

多目标拆卸线平衡问题的Pareto细菌觅食算法*

拆卸线平衡问题的优化涉及多个目标,为克服传统方法在求解多目标拆卸线平衡问题时不能很好处理各子目标间冲突及易于早熟等不足,提出了一种多目标细菌觅食优化算法。算法采用Pareto非劣排序技术对种群进行分级,并结合拥挤距离机制评价同级个体的优劣。为提高算法收敛性能,在趋向性操作结束后引入精英保留策略保留优秀个体,并采用全局信息共享策略引导菌群不断向均匀分布的Pareto最优前沿趋近。通过不同规模算例的对比验证表明了算法的有效性与优越性。     

改进的r支配高维多目标粒子群优化算法

高维多目标优化问题是广泛存在于实际应用中的复杂优化问题,目前的研究方法大都限于进化算法.本文利用粒子群优化算法求解高维多目标优化问题,提出了一种基于r支配的多目标粒子群优化算法.采用r支配关系进行粒子的比较与选择,并结合粒子群优化算法收敛速度快的优势,使得算法在目标个数增加时仍保持较强的搜索能力;为了弥补由此造成的群体多样性的丢失,优化非r支配阈值的取值策略;此外,引入决策空间的拥挤距离测度,并给出新的外部存储器更新方法,从而进一步防止算法陷入局部最优.对多个基准测试函数的仿真结果表明所得解集在收敛性、多样性以及围绕参考点的分布性上均优于其他两种算法.     

A surrogate-assisted evolution strategy for constrained multi-objective optimization

In many real-world optimization problems, several conflicting objectives must be achieved and optimized simultaneously and the solutions are often required to satisfy certain restrictions or constraints. Moreover, in some applications, the numerical values of the objectives and constraints are obtained from computationally expensive simulations. Many multi-objective optimization algorithms for continuous optimization have been proposed in the literature and some have been incorporated or used in conjunction with expert and intelligent systems. However, relatively few of these multi-objective algorithms handle constraints, and even fewer, use surrogates to approximate the objective or constraint functions when these functions are computationally expensive. This paper proposes a surrogate-assisted evolution strategy (ES) that can be used for constrained multi-objective optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. Such an algorithm can be incorporated into an intelligent system that finds approximate Pareto optimal solutions to simulation-based constrained multi-objective optimization problems in various applications including engineering design optimization, production management and manufacturing. The main idea in the proposed algorithm is to generate a large number of trial offspring in each generation and use the surrogates to predict the objective and constraint function values of these trial offspring. Then the algorithm performs an approximate non-dominated sort of the trial offspring based on the predicted objective and constraint function values, and then it selects the most promising offspring (those with the smallest predicted ranks from the non-dominated sort) to become the actual offspring for the current generation that will be evaluated using the expensive objective and constraint functions. The proposed method is implemented using cubic radial basis function (RBF) surrogate models to assist the ES. The resulting RBF-assisted ES is compared with the original ES and to NSGA-II on 20 test problems involving 2–15 decision variables, 2–5 objectives and up to 13 inequality constraints. These problems include well-known benchmark problems and application problems in manufacturing and robotics. The numerical results showed that the RBF-assisted ES generally outperformed the original ES and NSGA-II on the problems used when the computational budget is relatively limited. These results suggest that the proposed surrogate-assisted ES is promising for computationally expensive constrained multi-objective optimization.