2) Scalarizing Functions: SFs play a fundamental role in MOEA/D and its variants. They can significantly affect the search ability of the evolving popula- tion and the quality of the res[r]

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In the view of the present work, it is worthy to mention several studies available in the literature that employ angle information during the search. In [41] , unit vectors generated in advance are used during the evolution to decompose the population into subpopulations accord- ing to the closeness of individuals to these vectors. For this purpose, the angle between the corresponding vectors in the objective space is utilized. The selection procedure employs a nondominated sorting to trim subpopulations if the number of available slots is exceeded. Similar **decomposition** approach is used in [42] for many-objective optimization. In this approach, subpopulations are also de ﬁned by associating individuals to closest reference directions using angles between reference and solution vectors. The di ﬀerence lies in the selection where for each population member a ﬁtness value is calculated using an angle-penalized distance measure. This measure takes into consideration the individual's convergence that is penalized by the angle between its objective values and the reference vector. In [43] , constraints **based** on angles between individuals and correspond- ing weight vectors are introduced into MOEA/D. A new precedence relation is established for updating subproblems. When comparing two individuals, the one satisfying constraint is preferred. If both are infeasible in terms of angle-**based** constraint, the one having a smaller constraint violation value is favored. Otherwise, the **scalarizing** func- tion values are used for decision as usual. Another variant of MOEA/D using angle information was suggested in [44] . To improve the balance between convergence and diversity, this approach employs decomposi- tion-**based** sorting for controlling convergence and angle-**based** selec- tion for maintaining diversity. The former de ﬁnes diﬀerent solution sets **based** on **scalarizing** function values. The latter selects solutions from the set, which cannot be completely accommodated into the new population, **based** on angles so that the minimum angle is maximized. The common feature of the above approaches is that vectors deﬁning directions of search are generated in advance and used for associating population members with certain directions, depending on how small is the angle between the direction and solution vectors. Although the herein proposed approach uses angle information, the important di ﬀerence with existing approaches resides in the fact that no direction vectors are generated before and used during the search. To certain extent, this reduces a human labor. The proposed MOEA/ VAN is the ﬁrst algorithm demonstrating that the search can be e ﬀectively performed only exploiting the angle information between population members without associating them with predeﬁned direc- tions.

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This paper proposed an SAS as the selection operator for MOEA/D to address MOPs. In SAS, the balance between convergence and diversity is achieved by two components, DBS and ABS. Different from other selection schemes, e.g., global STM model, DBS only conducts sorting within the local neighboring solutions, which drastically reduce the com- putational cost of SAS. Meanwhile, ABS utilizes the angle information in the objective space to maintain a fine-grained diversity. Different from many other MOEA/D variants, SAS allows one subproblem to associate with any number of solu- tions, or even no solutions, which makes it more flexible for MOPs with different shapes of PFs. SAS is integrated into MOEA/D and the algorithm, called MOEA/D-SAS, is com- pared with four classical (NSGA-II, MSOPS-II, MOEA/D, and MOEA/D-DE) and three state-of-the-art MOEAs (MOEA/D- STM, NSGA-III, and MOEA/D-AWA) on continuous MOPs or MaOPs. The experimental results show that MOEA/D-SAS outperforms other compared **algorithms**. In addition, the com- putational efficiency of DBS and the effects of ABS are also discussed in this paper in detail.

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This work considers six MOEAs from the literature and compares their performance on the proposed test suite to assess the property and difficulty of these benchmark **functions**. These **algorithms** are classified into two groups. The first group includes four well-known dynamic MOEAs: dynamic multi-objective particle swarm optimization (DMOPSO) al- gorithm [110], dynamic non-dominated sorting genetic algorithm II (DNSGA-II) [44], dynamic competitive-cooperative coevolutionary algorithm (dCOEA) [61], and popula- tion prediction strategy (PPS) [196]. Each algorithm in this group has a mechanism of dealing with dynamism for dynamic optimization. Note that, there are two versions of DNSGA-II, and DNSGA-II with randomly created solutions whenever a change occurs is adopted here. The second group includes two classic MOEAs: strength pareto evo- lutionary algorithm II (SPEA2) [202] and MOEA **based** on **decomposition** (MOEA/D) [188], and they are high-performance **algorithms** for static **multiobjective** optimization. To handle dynamic environments, the **algorithms** of this group adopt the following strat- egy in our experiments: 10% randomly selected population members are re-evaluated for change detection, and the restart scheme is employed for change response. The parameter settings for all the tested **algorithms** are inherited from the referenced papers.

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Contrarily to the majority of classical methods, **evolutionary** **algorithms** are population- **based** optimization techniques. This feature makes them particularly suitable for solving **multiobjective** optimization problems. The main aspects to be taken into consideration when implementing MOEAs are: (i) fitness assignment, (ii) diversity preservation, and (iii) elitism. Although the discussed **algorithms** are organized on the basis of their variation operators, one can trace how each of these aspects was included into MOEAs since the first pioneering studies appeared in the mid 1980s. Regarding the fitness assignment, one can distinguish the following most popular techniques: dominance-**based**, **scalarizing**-**based**, and indicator-**based** fitness assignment. Furthermore, diversity of solutions within the current Pareto set approximation is maintained by incorporating density information into the selection process of EMO **algorithms**. Some widely used diversity preserving methods are: (i) fitness sharing, (ii) hypergrid, (iii) clustering, (iv) nearest neighbor, (v) crowding distance. Recently, the use of quality indicators as the diversity estimator has become very popular. Elitism is addressed to the issue of maintaining good solutions found during the optimization process. The elite preserving operator is usually included in MOEAs to make them better convergent to the Pareto front. One way to deal with this problem is to combine the parent and offspring populations and to apply selection procedure to select a new population. Alternatively, the external archive can be maintained where the promising solutions are stored during the search.

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We analyze the properties of the ADET for sparse ensemble learning and several state-of-the-art many- objective optimization **algorithms** are applied to solve **multiobjective** ADCH maximization problems, in- cluding the two-archive algorithm (Two Arch2) [30], which focuses on convergence and diversity sepa- rately, the **decomposition** **based** **algorithms**, such as NSGA-III [31], the **evolutionary** **algorithms** **based** on both dominance and **decomposition** (MOEA / DD) [32], the reference vector guided **evolutionary** algorithm

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Total operation cost of the proposed method has been compared with the well-known **algorithms** such as particle swarm optimization (PSO) and genetic algorithm (GA) methods. The results are reported in Table II, that show the cost effectiveness of the proposed method.

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On the other hand, despite the substantial progress achieved in the last decade, there are relatively few systematic investiga- tions aiming at providing a better understanding of what makes a particular approach distinguishable from others, and under which optimization scenario, in terms of budget and problem diiculty, it is recommended. We can cite [5] which provides a coarse-grained taxonomy of surrogate-assisted MOEAs with a particular focus on constrained MOPs. In [7], one can also ind a complementary taxonomy, and a discussion on extending existing approaches to support the production of a batch of multiple points that can be eval- uated in parallel. Besides considering a simple parallel extension of ParEGO [12], using a set of weight vectors, the most advanced MOEAs studied therein fall into the class of dominance-**based** and indicator-**based** **algorithms** such as SMS-EGO [18]. This taxonomy came with an interesting empirical analysis of the diferent consid- ered extensions with a focus on the potential gain in terms of paral- lelism. Nonetheless, a class of state-of-the-art EMO approaches was only considered at a small extent, namely, surrogate-assisted vari- ants of the so-called MOEA **based** on **Decomposition** (MOEAD) [26]. The MOEAD (**multiobjective** **evolutionary** algorithm **based** on **decomposition**) framework belongs to the class of aggregation- **based** approaches. The original MOP is transformed into a number of (single-objective) sub-problems, deined using a **scalarizing** func- tion, that are solved cooperatively. Due to its eiciency and lexi- bility in integrating existing single-objective optimizers into the **multiobjective** setting, MOEAD has become one of the most studied approaches. Nonetheless, when dealing with an expensive setting, we are not aware of any systematic analysis of what makes decom- position beneicial and how to efectively integrate surrogates in the original cooperative solving process of MOEAD. Interestingly, the most recent surrogate-assisted approaches are **based** on lever- aging concepts from **multiobjective** **decomposition** [4, 9, 16, 25, 27]. This is precisely where the contribution of this paper lies.

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Managing the metamodels or evolution control is also very important as it affects the performance of the metamodel used. As shown in Figure 3, the fixed evolution control strategy was used more than the adaptive evolution control strategy. As mentioned in Section 2, using an adaptive evolution control strategy depends on the accuracy of the metamodel used and using a fixed evolution control strategy implies that either the metamodel is accurate enough for approximation or the metamodel accuracy is not important or not checked. There are only five **algorithms** [86, 94, 95, 127, 151], where adaptive evolution control was used. For instance in [86], after using Kriging models, uncertainty of the approximated values was compared with a predefined value and if that uncertainty was acceptable, then only Kriging models were further used otherwise original function were used. A similar strategy was followed in [94], where accuracy of neural networks was measured after every generation and compared with a predefined value. In [95], uncertainties of offspring was compared with the uncertainties of parents and **based** on the comparison, the decision was made to use Kriging models or original **functions**. Singh et al. [127] and Zhu et al. [151] the approximation accuracy of metamodels for using them in subsequent evaluations. Moreover, the training time for different metamodels was not considered in many papers though training time can be substantially high in some cases, particularly, when the metamodel approximation accuracy is important. Among EAs, dominance **based** **algorithms** are more widely used than other **algorithms** e.g. **decomposition** **based** or indicator **based**. Moreover, in dominance **based** EAs, NSGA-II was used more often than other EAs.

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Many real-world problems are dynamic **multiobjective** optimization prob- lems (DMOPs), with conflicts among multiple objectives as well as objective **functions** that change over time [1]. Tracking the Pareto optimal solution set after a change is an important and challenging issue. On these issues, the researched objectives often change intricately with time. The goal of the tra- ditional **evolutionary** **algorithms** is to make the population gradually converge to get a satisfactory solution set, but this makes the population lose diversity. Especially, in the later stages of the evolution, the population will gradually lose the ability to adapt to the environmental changes, which is a challenge of the traditional **evolutionary** **algorithms** in the dynamic environment [2, 3, 24, 4, 5]. In order to track the optimal solution set in a timely manner after a change, researchers need to make some adjustments on the traditional static multiobjec- tive **algorithms** [6, 7, 8, 9], so that they can quickly respond to the environmental changes.

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In MOEA/D, there are three widely-used **scalarizing** **functions**, i.e., weighted sum, weighted Tchebycheff, and penalty boundary intersection (PBI), to aggregate M differ- ent objectives (Zhang and Li, 2007). Compared with PBI, the weighted sum and weighted Tchebycheff are easier to implement and less computationally expensive. A recent study (Ishibuchi et al., 2013) reported that the weighted sum shows better search performance than the weighted Tcheby- cheff in many-objective problems. However, the weighted sum is not effective to approximate problems with the entire concave Pareto front (Zhang and Li, 2007). The weighted Tchebycheff approach has received intensive research inter- est due to its ability to approximate both convex and concave POFs. Despite its great success for solving standard bench- mark problems like ZDT (Zitzler et al., 2000) or DTLZ (Deb et al, 2005), some recent investigations have revealed that this approach has difficulties in uniformly distributing solu- tions on boundary regions of the POF for complex problems (Qi et al., 2014; Jiang and Yang, 2015). On the other hand, the PBI approach gains a firm foothold in MOEA/D be- cause it can provide a more uniform distribution of POF than the weighted Tchebycheff for three- and higher-dimensional problems (Li et al., 2015a; Zhang and Li, 2007; Deb and Jain, 2014; Gomez and Coello Coello, 2015).

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Copyright © 2015 Zhiming Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. As is known, the Pareto set of a continuous **multiobjective** optimization problem with 𝑚 objective **functions** is a piecewise continuous (𝑚 − 1)-dimensional manifold in the decision space under some mild conditions. However, how to utilize the regularity to design **multiobjective** optimization **algorithms** has become the research focus. In this paper, **based** on this regularity, a model-**based** **multiobjective** **evolutionary** algorithm with regression analysis (MMEA-RA) is put forward to solve continuous **multiobjective** optimization problems with variable linkages. In the algorithm, the optimization problem is modelled as a promising area in the decision space by a probability distribution, and the centroid of the probability distribution is (𝑚 − 1)-dimensional piecewise continuous manifold. The least squares method is used to construct such a model. A selection strategy **based** on the nondominated sorting is used to choose the individuals to the next generation. The new algorithm is tested and compared with NSGA-II and RM-MEDA. The result shows that MMEA-RA outperforms RM-MEDA and NSGA-II on the test instances with variable linkages. At the same time, MMEA-RA has higher efficiency than the other two **algorithms**. A few shortcomings of MMEA- RA have also been identified and discussed in this paper.

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The TMEA was tested on the following test problems, each with different degrees of difficulty and different characteristics with respect to convexity and continuity in the non- inferior set, discreteness in the decision space, and degree of constraints. The set of con- strained test problems (CTPs) by Deb [11], the extended 0/1 knapsack problem presented by Zitzler [27] and two of the commonly used unconstrained test problems are used to test TMEA. A complete list and details of the test problems that were used in this study is pro- vided in Table 4.1. The parameter settings used for all the **algorithms** is provided in the Table 4.2. All objective and constraint **functions** values were normalized, and correspond- ingly the weight values were kept to a [0, 1] range. The constraints, where applicable,

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A number of **multiobjective** optimization **evolutionary** al- gorithms (MOEAs) have been developed for finding a set of solutions to approximate the PF in a single run [2]–[6]. Most MOEAs such as NSGA-II [4] mainly rely on Pareto dominance to guide their search, particularly, their selection operators. In contrast, MOEA/D (**multiobjective** **Evolutionary** Algorithm **based** on **Decomposition**) [6] makes use of tradi- tional aggregation methods to transform the task of approxi- mating the PF into a number of single objective optimization subproblems. Then a population **based** algorithm is employed to solve these subproblems in a collaborative way. Some MOEA/D variants have been proposed for dealing with various MOPs (e.g. [7], [8]). MOEA/D has also been used as a basic element in some hybrid **algorithms** (e.g. [9]–[11]).

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Multi-objective optimization problems are often subject to the presence of objectives that require expensive resampling for their computation. This is the case of many robustness metrics, frequently used as an additional objective, that accounts for the reliability of specific sections of the solution space. Typical robustness measurements use resampling, but the number of samples that constitute a precise dispersion measure has a potentially large impact on the computational cost of an algorithm. This paper proposes the integration of dominance **based** statistical testing methods as part of the selection mechanism of MOEAs with the aim of reducing the amount of fitness evaluations. The performance of the approach is tested on five classical benchmark **functions** integrating it into two well-known **algorithms** NSGA-II and SPEA2. The experimental results show a significant reduction in the number of fitness evaluations while, at the same time, maintaining the quality of the solutions.

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In [33], an enhanced version of MOEA / D [72] is developed in which the Simulated Binary Crossover (SBX) [27] has been replaced with Di ﬀ erential Evolution (DE) [53]. This gave MOEA / D-DE, [33]. The purpose of this replacement is to produce a solution while inducing two di ﬀ erent neighbourhoods, one with each child solution. One of these solutions is then allowed to replace a very small number of old solutions. In [73], resources are allocated dynamically to each sub-problem as used in the MOEA / D paradigm. In [29, 42, 28, 46, 73], the impact of multiple search operators coupled to a self-adaptive scheme has been studied. It has then been tested on instances designed for the special session on MOEA competition at the Congress of **Evolutionary** Computing of 2009 (CEC’09), [74]. In [40, 44], DE and PSO [16] have been used simultaneously within the framework of MOEA / D, [72]. This variant was then applied to five standard ZDT test problems [79] as well as the CEC’09 test instances [74]. In [39, 37, 43, 45], MOEA / D [72] and NSGA-II [13], two di ﬀ erent MOEA approaches have been used synergetically at population and generation levels. These two **algorithms** have also been used in [48] to solve hard **multiobjective** optimization problems. Fuzzy Dominance (FD) concepts have been introduced in [50] to further improve the algorithmic behavior of the MOEA / D paradigm. The e ﬀ ect of the combined use of neighbourhood sizes with a self-adaptive strategy has been investigated in [75]. For more details please refer to [38, 41, 76].

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All of the methods shown are either implemented in the R-package emoa or can easily be constructed from the **functions** contained in the emoa and relations packages. In the future, we would like to study other ranking operators as well **based** on parametric models of the data. Examples would include the Bradley-Terry model also used by [2] to estimate the abilities of machine learning **algorithms**. Furthermore, a comprehensive survey of voting literature could reveal more criteria which might ease the choice of a consensus method.

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The Non-dominated Sorting Genetic Algorithm, NSGA-II, is undoubtedly the most well-known and referenced algorithm in the multi-objective literature. It is a GA with random mating of individuals within a population. It is **based** on obtaining a new population from the original one by applying the typical genetic operators (selection, crossover and mutation); then, the individuals in the two populations are sorted ac- cording to their rank, and the best solutions are chosen to create a new population. In the case of having to select some individuals with the same rank, a density estimation **based** on measuring the crowding distance to the surrounding individuals belonging to the same rank is used to get the most promising solutions [32]. In 2014 a new version of this algorithm was introduced **based** on adaptive updating and including new refer- ence points on the fly. The resulting adaptive NSGA-III is shown to provide a denser representation of the Pareto-optimal front [39, 40].

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Watanabe and Sakakibara [ 44 ] presented two methods to trans- form a COP into an MOP: the first one considers a penalty function as an additional objective function and the second one adds noise to the original objective function or decision variables. Dong and Wang [ 10 ] converted a COP into the following BOP: (f (x) + εG(x), G(x)). The theoretical anal- ysis reveals that when ε tends to infinity, this BOP has the unique Pareto optimal vector, which exactly corresponds to the optimal solution of a COP. They claimed that this BOP could be solved by a traditional **multiobjective** optimization EA without biases. In the implementation phase, ε exponen- tially increases and Pareto ranking is employed as the selection criterion. Xu et al. [ 11 ] proposed a novel **multiobjective** model with helper objective **functions** for constrained optimization. In addition to (f (x), G(x)), an auxiliary objective function is constructed. Then a three-objective-**based** CMODE [ 7 ] is implemented. The experimental results show that the helper objective function is able to improve the performance of CMODE. Gao et al. [ 45 ] recast a COP as (G 1 (x), . . . , G m (x))

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