Inverted Generational Distance (IGD) has been widely considered as a reliable performance indicator to concurrently quantify the convergence and diversity of multi-and many-objective evolutionary algorithms. In this paper, an IGD indicator-based evolutionary algorithm for solving many-objective optimization problems (MaOPs) has been proposed. Specifically, the IGD indicator is employed in each generation to select the solutions with favorable convergence and diversity. In addition, a computationally efficient dominance comparison method is designed to assign the rank values of solutions along with three newly proposed proximity distance assignments. Based on these two designs, the solutions are selected from a global view by linear assignment mechanism to concern the convergence and diversity simultaneously. In order to facilitate the accuracy of the sampled reference points for the calculation of IGD indicator, we also propose an efficient decomposition-based nadir point estimation method for constructing the Utopian Pareto front which is regarded as the best approximate Pareto front for real-world MaOPs at the early stage of the evolution. To evaluate the performance, a series of experiments is performed on the proposed algorithm against a group of selected state-of-the-art many-objective optimization algorithms over optimization problems with 8-, 15-, and 20-objective. Experimental results measured by the chosen performance metrics indicate that the proposed algorithm is very competitive in addressing MaOPs.
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Decomposition is a basic strategy in traditional mul-tiobjective optimization. However, it has not yet been widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is optimized by only using information from its several neighboring subproblems, which makes MOEA/D have lower computational complexity at each generation than MOGLS and nondominated sorting genetic algorithm II (NSGA-II). Experimental results have demonstrated that MOEA/D with simple decomposition methods outperforms or performs similarly to MOGLS and NSGA-II on multiobjective 0-1 knapsack problems and continuous multiobjective optimization problems. It has been shown that MOEA/D using objective normalization can deal with disparately-scaled objectives, and MOEA/D with an advanced decomposition method can generate a set of very evenly distributed solutions for 3-objective test instances. The ability of MOEA/D with small population, the scal-ability and sensitivity of MOEA/D have also been experimentally investigated in this paper.
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多目标进化算法(MOEAs),尤其是基于分解的MOEAs,近年来引起了广泛的关注。最近的研究表明,分解方法和统治方法的良好设计组合可以改善性能,即收敛性和多样性。经济部。本文提出了一种结合分解方法和支配方法的新方法。更确切地说,使用一组权重向量来分解给定的多目标优化问题(MaOP),并且提出基于惩罚的边界交叉函数和优势的混合方法来比较由权重向量定义的子群中的局部解。基于混合方法的MOEA在两个着名的测试套件(即DTLZ和WFG)中选择的问题上实施和测试。实验结果表明,我们的算法在处理MaOP时非常有竞争力。随后,我们的算法被扩展到求解约束MaOP,并且我们的算法的约束版本在收敛性和多样性方面也表现出良好的性能。这些揭示了在局部使用优势​​并将其与分解方法相结合可以有效地提高MOEA的性能。
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现有研究表明,在解决一些多目标优化问题时,基于分解的传统多目标进化算法(MOEAs)可能会失去种群多样性。在本文中,提出了一种基于分解的简单MOEA,它具有局部迭代更新(LIU)。 LIU策略有两个特征,预计将推动人口接近Pareto Front并具有良好的分布。一个是当前社区中最糟糕的解决方案被新生的后代换掉,防止人口被少数人的副本占用。另一个是它的迭代过程有助于为子问题分配更好的解决方案,这有利于充分利用相邻子问题的相似解决方案,并探索搜索空间中的局部区域。此外,所提出的算法的时间复杂度与MOEA / D的时间复杂度相同,并且低于其他已知MOEA的时间复杂度,因为它在每次更新时仅考虑当前邻域内的个体。该算法与两个着名的测试套件DTLZ和WFG中选择的问题的几个最佳MOEAs进行了比较。实验结果表明,在DTLZ4上只有少数运行算法的实例失去了它们的种群多样性。更重要的是,该算法在运行时间和解决方案质量的大多数测试实例中获胜,表明它在解决MaOP方面非常有效。
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Domination-based sorting and decomposition are two basic strategies used in multiobjective evolutionary optimization. This paper proposes a hybrid multiobjective evolutionary algorithm integrating these two different strategies for combinatorial optimization problems with two or three objectives. The proposed algorithm works with an internal (working) population and an external archive. It uses a decomposition-based strategy for evolving its working population and uses a domination-based sorting for maintaining the external archive. Information extracted from the external archive is used to decide which search regions should be searched at each generation. In such a way, the domination-based sorting and the decomposition strategy can complement each other. In our experimental studies, the proposed algorithm is compared with a domination-based approach, a decomposition-based one, and one of its enhanced variants on two well-known multiobjective combinatorial optimization problems. Experimental results show that our proposed algorithm outperforms other approaches. The effects of the external archive in the proposed algorithm are also investigated and discussed.
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Under mild conditions, it can be induced from the Karush-Kuhn-Tucker condition that the Pareto set, in the decision space, of a continuous multiobjective optimization problem is a piecewise continuous (1)-D manifold, where is the number of objectives. Based on this regularity property, we propose a regularity model-based multiobjective estimation of distribution algorithm (RM-MEDA) for continuous multiobjective optimization problems with variable linkages. At each generation, the proposed algorithm models a promising area in the decision space by a probability distribution whose centroid is a (1)-D piecewise continuous manifold. The local principal component analysis algorithm is used for building such a model. New trial solutions are sampled from the model thus built. A nondominated sorting-based selection is used for choosing solutions for the next generation. Systematic experiments have shown that, overall, RM-MEDA outperforms three other state-of-the-art algorithms, namely, GDE3, PCX-NSGA-II, and MIDEA, on a set of test instances with variable linkages. We have demonstrated that, compared with GDE3, RM-MEDA is not sensitive to algorithmic parameters, and has good scalability to the number of decision variables in the case of nonlinear variable linkages. A few shortcomings of RM-MEDA have also been identified and discussed in this paper.
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In some expensive multiobjective optimization problems, several function evaluations can be carried out at one time. Therefore, it is very desirable to develop methods which can generate several test points simultaneously. This paper proposes such a method, called MOEA/D-EGO, for dealing with expensive multiobjective optimization. MOEA/D-EGO decomposes a MOP in question into a number of single objective optimization subproblems. A predictive model is built for each subproblem based on the points already evaluated. Effort has been made to save the overhead for modeling and to improve the prediction quality. At each generation, MOEA/D is used for maximizing the expected improvement metric values of all the subproblems and then several test points are selected for evaluation. Experimental results on a number of test instances have shown that MOEA/D-EGO is very promising.
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A multiobjective optimization problem involves several conflicting objectives and has a set of Pareto optimal solutions. By evolving a population of solutions, multiobjective evolutionary algorithms (MOEAs) are able to approximate the Pareto optimal set in a single run. MOEAs have attracted a lot of research effort during the last 20 years, and they are still one of the hottest research areas in the field of evolutionary computation. This paper surveys the development of MOEAs primarily during the last eight years. It covers algorithmic frameworks such as decomposition-based MOEAs (MOEA/Ds), memetic MOEAs, coevolutionary MOEAs, selection and offspring reproduction operators, MOEAs with specific search methods, MOEAs for multimodal problems, constraint handling and MOEAs, computationally expensive multiobjective optimization problems (MOPs), dynamic MOPs, noisy MOPs, combinatorial and discrete MOPs, benchmark problems, performance indicators, and applications. In addition, some future research issues are also presented.
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研究表明,在维持大学解决多目标优化问题(MaOPs)的同时,难以获得邻近性。真正的Pareto Front(PF)的复杂性对于基于参考矢量的算法提出了严峻的挑战,因为它们对真实PF的特性的适应性不具有先验性。本文提出了一种具有两个交互过程的多目标优化算法:级联聚类和参考点增量学习(CLIA)。在基于级联聚类的人口选择过程中,利用增量学习过程提供的参考向量,非主导和支配的个体以不同的方式进行聚类和分类,并通过循环法选择更好的接近和多样性。在基于参考点增量学习的参考矢量自适应过程中,利用来自聚类过程的反馈,通过增量学习逐渐获得参考点的适当分布,并相应地重新定位参考矢量。 CLIA的优势不仅在于其有效和高效的性能,还在于处理真正PF的各种特性的多功能性,只使用两个过程之间的相互作用而不会产生额外的评估。许多基准问题的实验研究表明CLIA具有竞争力与现有技术相比,高效且通用。
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Multi-objective optimization problems arise frequently in applications, but can often only be solved approximately by heuristic approaches. Evolutionary algorithms have been widely used to tackle multi-objective problems. These algorithms use different measures to ensure diversity in the objective space but are not guided by a formal notion of approximation. We present a framework for evolutionary multi-objective optimization that allows to work with a formal notion of approximation. This approximation-guided evolutionary algorithm (AGE) has a worst-case runtime linear in the number of objectives and works with an archive that is an approximation of the non-dominated objective vectors seen during the run of the algorithm. Our experimental results show that AGE finds competitive or better solutions not only regarding the achieved approximation, but also regarding the total hypervolume. For all considered test problems, even for many (i.e., more than ten) dimensions, AGE discovers a good approximation of the Pareto front. This is not the case for established algorithms such as NSGA-II, SPEA2, and SMS-EMOA. In this paper we compare AGE with two additional algorithms that use very fast hypervolume-approximations to guide their search. This significantly speeds up the runtime of the hypervolume-based algorithms, which now allows a comparison of the underlying selection schemes.
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Over the last decade, a variety of evolutionary algorithms (EAs) have been proposed for solving multi-objective optimization problems. Especially more recent multi-objective evolutionary algorithms (MOEAs) have been shown to be efficient and superior to earlier approaches. In the development of new MOEAs, the strive is to obtain increasingly better performing MOEAs. An important question however is whether we can expect such improvements to converge onto a specific efficient MOEA that behaves best on a large variety of problems. The best MOEAs to date behave similarly or are individually preferable with respect to different performance indicators. In this paper, we argue that the development of new MOEAs cannot converge onto a single new most efficient MOEA because the performance of MOEAs shows characteristics of multi-objective problems. While we will point out the most important aspects for designing competent MOEAs in this paper, we will also indicate the inherent multi-objective trade-off in multi-objective optimization between proximity and diversity preservation. We will discuss the impact of this trade-off on the concepts and design of exploration and exploitation operators. We also present a general framework for competent MOEAs and show how current state-of-the-art MOEAs can be obtained by making choices within this framework. Furthermore, we show an example of how we can separate non-domination selection pressure from diversity preservation selection pressure and discuss the impact of changing the ratio between these components.
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分解已成为进化多目标优化(EMO)的一种日益流行的技术。通常设计基于分解的EMO算法来近似整个帕累托最优前沿(PF)。然而,在实践中,决策者(DM)可能仅对她/他的兴趣区域(ROI)感兴趣,即PF的一部分。外面的解决方案可能对决策程序无用甚至嘈杂。此外,在解决许多客观问题时,无法找到首选解决方案。本文开发了一个基于分解的EMO算法的交互框架,将DM引导到他/他选择的首选解决方案。它由三个模块组成,即咨询,偏好和优化。具体地,在每几代之后,要求DM在咨询会话中对几个候选解决方案进行评分。此后,从DM的行为逐渐学习用于对DM的偏好信息建模的近似值函数。在偏好请求会话中,在咨询模块中学习的偏好信息被转换成可以在基于分解的EMO算法中使用的形式,即,偏向于ROI的一组参考点。优化模块原则上可以是任何基于分解的EMO算法,利用偏置参考点来指导其搜索过程。对三到十个目标的基准问题进行了广泛的实验,充分证明了我们提出的寻找DM首选解决方案的方法的有效性。
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The hypervolume measure (or S metric) is a frequently applied quality measure for comparing the results of evolutionary multiobjective optimisation algorithms (EMOA). The new idea is to aim explicitly for the maximisation of the dominated hypervolume within the optimisation process. A steady-state EMOA is proposed that features a selection operator based on the hypervolume measure combined with the concept of non-dominated sorting. The algorithm's population evolves to a well-distributed set of solutions, thereby focussing on interesting regions of the Pareto front. The performance of the devised S metric selection EMOA (SMS-EMOA) is compared to state-of-the-art methods on two-and three-objective benchmark suites as well as on aeronautical real-world applications.
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Multi-objective optimization problems having more than three objectives are referred to as many-objective optimization problems. Many-objective optimization brings with it a number of challenges that must be addressed, which highlights the need for new and better algorithms that can efficiently handle the growing number of objectives. This article reviews the different challenges associated with many-objective optimization and the work that has been done in the recent-past to alleviate these difficulties. It also highlights how the existing methods and body of knowledge have been used to address the different real world many-objective problems. Finally, it brings focus to some future research opportunities that exist with many-objective optimization. We report in this article what is commonly used, be it algorithms or test problems, so that the reader knows what are the benchmarks and also what other options are available. We deem this to be especially useful for new researchers and for researchers from other fields who wish to do some work in the area of many-objective optimization.
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We propose a surrogate-assisted reference vector guided evolutionary algorithm for com-putationally expensive optimization problems with more than three objectives. The proposed algorithm is based on a recently developed evolutionary algorithm for many-objective optimization that relies on a set of adaptive reference vectors for selection. The proposed surrogate-assisted evolutionary algorithm uses Kriging to approximate each objective function to reduce the computational cost. In managing the Kriging models, the algorithm focuses on the balance of diversity and convergence by making use of the uncertainty information in the approximated objective values given by the Kriging models, the distribution of the reference vectors as well as the location of the individuals. In addition, we design a strategy for choosing data for training the Kriging model to limit the computation time without impairing the approximation accuracy. Empirical results on comparing the new algorithm with the state-of-the-art surrogate-assisted evolutionary algorithms on a number of benchmark problems demonstrate the competitiveness of the proposed algorithm.
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Recently the inverted generational distance (IGD) measure has been frequently used for performance evaluation of evolutionary multi-objective optimization (EMO) algorithms on many-objective problems. When the IGD measure is used to evaluate an obtained solution set of a many-objective problem, we have to specify a set of reference points as an approximation of the Pareto front. The IGD measure is calculated as the average distance from each reference point to the nearest solution in the solution set, which can be viewed as an approximate distance from the Pareto front to the solution set in the objective space. Thus the IGD-based performance evaluation totally depends on the specification of reference points. In this paper, we illustrate difficulties in specifying reference points. First we discuss the number of reference points required to approximate the entire Pareto front of a many-objective problem. Next we show some simple examples where the uniform sampling of reference points on the known Pareto front leads to counter-intuitive results. Then we discuss how to specify reference points when the Pareto front is unknown. In this case, a set of reference points is usually constructed from obtained solutions by EMO algorithms to be evaluated. We show that the selection of EMO algorithms used to construct reference points has a large effect on the evaluated performance of each algorithm.
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在本文中,我们提出了一种并行多目标进化算法,称为基于并行准则的分区MOEA(PCPMOEA),并应用于多目标背包问题(MOKP)。建议的搜索策略基于潜在有效解决方案的周期性划分,其被分配到多个多目标进化算法(MOEAs)。每个MOEA都致力于一个唯一的目标,即将基于标准和基于优势的方法结合起来。建议的算法解决了两个主要的子目标:最小化当前非支配解决方案与理想点之间的距离,并确保潜在有效解决方案的扩展。实验结果包括在内,我们使用众所周知的多目标元启发式方法,与最先进的结果相比较,对上述子目标评估建议算法的性能。
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The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most powerful evolutionary algorithms for real-valued single-objective optimization. In this paper , we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMA-ES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.
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The Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele 1999) is a relatively recent technique for finding or approximating the Pareto-optimal set for multiobjective optimization problems. In different studies (Zitzler and Thiele 1999; Zitzler, Deb, and Thiele 2000) SPEA has shown very good performance in comparison to other multiobjective evolutionary algorithms, and therefore it has been a point of reference in various recent investigations, e.g., (Corne, Knowles, and Oates 2000). Furthermore, it has been used in different applications, e.g., (La-hanas, Milickovic, Baltas, and Zamboglou 2001). In this paper, an improved version , namely SPEA2, is proposed, which incorporates in contrast to its predecessor a fine-grained fitness assignment strategy, a density estimation technique, and an enhanced archive truncation method. The comparison of SPEA2 with SPEA and two other modern elitist methods, PESA and NSGA-II, on different test problems yields promising results.
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结构化进化算法已经研究了一段时间。但是,它们在多目标优化领域的研究还很少。尽管它们具有良好的效果,但复杂动力学和结构的使用使它们的理解和采用率降低。在这里,我们提出了一般的子群体框架,它能够无限制地集成优化算法,并有助于结构化算法的设计。所提出的框架能够在其形式化下概括大多数结构化进化算法,例如细胞算法,岛模型,空间捕食者 - 猎物和基于限制交配的算法。此外,我们提出了基于一般子群体框架的两种算法,证明了通过简单加入一些单目标差分进化算法,即使在测试时单独评估组合算法时,结果也会大大改善。最重要的是,子群体算法与其相关的泛化算法之间的比较表明,人口内部不同策略之间的竞争可能对算法产生有害后果,并揭示使用子群体框架的强大益处。 SAN的代码,建议的多目标算法,在最难的基准测试中具有当前最佳结果,可在以下网址获得:http://github.com/zweifel/zweifel
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