A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although convergence to the Pareto optimal front (PF) for such algorithms can often be superior to that of Pareto-based alternatives, the problem of effectively distributing Pareto optimal solutions in a high-dimensional space has not been solved. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying search algorithm toward specific regions of interest, or the entire Pareto front, with the desired distribution of Pareto optimal solutions. The method simplifies many-objective problems by unifying the three performance objectives of an a posteriori multi-objective optimizer - convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front - to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method, is created to illustrate the benefits of the proposed concept for many-objective problems. The algorithm - MACE-gD - is shown to be highly competitive with the existing best-in-class decomposition-based algorithm (MOEA/D) and a more elaborate EDA method (RM-MEDA). <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA/D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA/D-M2M with classic MOEA/D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA/D-M2M performs better. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Decomposition-based evolutionary algorithms have been quite successful in solving optimization problems involving two and three objectives. Recently, there have been some attempts to exploit the strengths of decomposition-based approaches to deal with many objective optimization problems. Performance of such approaches are largely dependent on three key factors: 1) means of reference point generation; 2) schemes to simultaneously deal with convergence and diversity; and 3) methods to associate solutions to reference directions. In this paper, we introduce a decomposition-based evolutionary algorithm wherein uniformly distributed reference points are generated via systematic sampling, balance between convergence and diversity is maintained using two independent distance measures, and a simple preemptive distance comparison scheme is used for association. In order to deal with constraints, an adaptive epsilon formulation is used. The performance of the algorithm is evaluated using standard benchmark problems, i.e., DTLZ1-DTLZ4 for 3, 5, 8, 10, and 15 objectives, WFG1-WFG9, the car side impact problem, the water resource management problem, and the constrained ten-objective general aviation aircraft design problem. Results of problems involving redundant objectives and disconnected Pareto fronts are also included in this paper to illustrate the capability of the algorithm. The study clearly highlights that the proposed algorithm is better or at par with recent reference direction-based approaches for many objective optimization. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Achieving balance between convergence and diversity is a key issue in evolutionary multiobjective optimization. Most existing methodologies, which have demonstrated their niche on various practical problems involving two and three objectives, face significant challenges in many-objective optimization. This paper suggests a unified paradigm, which combines dominance- and decomposition-based approaches, for many-objective optimization. Our major purpose is to exploit the merits of both dominance- and decomposition-based approaches to balance the convergence and diversity of the evolutionary process. The performance of our proposed method is validated and compared with four state-of-the-art algorithms on a number of unconstrained benchmark problems with up to 15 objectives. Empirical results fully demonstrate the superiority of our proposed method on all considered test instances. In addition, we extend this method to solve constrained problems having a large number of objectives. Compared to two other recently proposed constrained optimizers, our proposed method shows highly competitive performance on all the constrained optimization problems. <s> BIB004 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> The decomposition-based multiobjective evolutionary algorithms (MOEAs) generally make use of aggregation functions to decompose a multiobjective optimization problem into multiple single-objective optimization problems. However, due to the nature of contour lines for the adopted aggregation functions, they usually fail to preserve the diversity in high-dimensional objective space even by using diverse weight vectors. To address this problem, we propose to maintain the desired diversity of solutions in their evolutionary process explicitly by exploiting the perpendicular distance from the solution to the weight vector in the objective space, which achieves better balance between convergence and diversity in many-objective optimization. The idea is implemented to enhance two well-performing decomposition-based algorithms, i.e., MOEA, based on decomposition and ensemble fitness ranking. The two enhanced algorithms are compared to several state-of-the-art algorithms and a series of comparative experiments are conducted on a number of test problems from two well-known test suites. The experimental results show that the two proposed algorithms are generally more effective than their predecessors in balancing convergence and diversity, and they are also very competitive against other existing algorithms for solving many-objective optimization problems. <s> BIB005 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Decomposition is a well-known strategy in traditional multiobjective optimization. However, the decomposition strategy was not widely employed in evolutionary multiobjective optimization until Zhang and Li proposed multiobjective evolutionary algorithm based on decomposition (MOEA/D) in 2007. MOEA/D proposed by Zhang and Li decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them in a collaborative manner using an evolutionary algorithm (EA). Each subproblem is optimized by utilizing the information mainly from its several neighboring subproblems. Since the proposition of MOEA/D in 2007, decomposition-based MOEAs have attracted significant attention from the researchers. Investigations have been undertaken in several directions, including development of novel weight vector generation methods, use of new decomposition approaches, efficient allocation of computational resources, modifications in the reproduction operation, mating selection and replacement mechanism, hybridizing decomposition- and dominance-based approaches, etc. Furthermore, several attempts have been made at extending the decomposition-based framework to constrained multiobjective optimization, many-objective optimization, and incorporate the preference of decision makers. Additionally, there have been many attempts at application of decomposition-based MOEAs to solve complex real-world optimization problems. This paper presents a comprehensive survey of the decomposition-based MOEAs proposed in the last decade. <s> BIB006 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Decomposition based evolutionary algorithms have achieved great success in solving many-objective optimization problems. However, the design of proper decomposition vectors is not an easy task, especially in high dimensional objective space. In this paper, we study how to better use these decomposition vectors. We first show that for any given decomposition vector, new dominance relationship and crowding measurement strategy can be well defined. Based on them, we then propose a new evolutionary algorithm for many objective optimization. By this way, the utilization efficiency of decomposition vectors is enhanced and thus the task of weights design is alleviated accordingly. Experiments are conducted to compare the proposed algorithm with four state-of-the-art decomposition based evolutionary algorithms on a set of well-known many-objective test problems with 5 to 10 objectives. The simulation results show that the proposed algorithm can achieve comparable results with fewer decomposition vectors. <s> BIB007 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Recently, a number of high performance many-objective evolutionary algorithms with systematically generated weight vectors have been proposed in the literature. Those algorithms often show surprisingly good performance on widely used DTLZ and WFG test problems. The performance of those algorithms has continued to be improved. The aim of this paper is to show our concern that such a performance improvement race may lead to the overspecialization of developed algorithms for the frequently used many-objective test problems. In this paper, we first explain the DTLZ and WFG test problems. Next, we explain many-objective evolutionary algorithms characterized by the use of systematically generated weight vectors. Then we discuss the relation between the features of the test problems and the search mechanisms of weight vector-based algorithms such as multiobjective evolutionary algorithm based on decomposition (MOEA/D), nondominated sorting genetic algorithm III (NSGA-III), MOEA/dominance and decomposition (MOEA/DD), and $\theta $ -dominance based evolutionary algorithm ( $\theta$ -DEA). Through computational experiments, we demonstrate that a slight change in the problem formulations of DTLZ and WFG deteriorates the performance of those algorithms. After explaining the reason for the performance deterioration, we discuss the necessity of more general test problems and more flexible algorithms. <s> BIB008 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> In this paper, we propose an evolutionary multi-and many-objective optimization algorithm combining I_SDE + and region decomposition. It decomposes the objective space into a number of sub-regions by a set of direction vectors and independently calculates the indicator I_SDE + by using the corresponding direction vector in each subregion. Thus, the convergence direction of each sub-region is relatively adjusted. In this way, the proposed algorithm can adapt to various of Pareto Front shapes. The inferior individuals are eliminated according to the value of I_SDE + of each individual one by one. In the experiments, we compare the proposed algorithm with four evolutionary multi-and many-objective optimization algorithms on WFG series with different number of objectives. The result shows that the proposed algorithm promotes diversity and convergence. <s> BIB009 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Decomposition-based evolutionary algorithms have shown great potential in many-objective optimization. However, the lack of theoretical studies on decomposition methods has hindered their further development and application. In this paper, we first theoretically prove that weight sum, Tchebycheff, and penalty boundary intersection decomposition methods are essentially interconnected. Inspired by this, we further show that highly customized dominance relationship can be derived from decomposition for any given decomposition vector. A new evolutionary algorithm is then proposed by applying the customized dominance relationship with adaptive strategy to each subpopulation of multiobjective to multiobjective framework. Experiments are conducted to compare the proposed algorithm with five state-of-the-art decomposition-based evolutionary algorithms on a set of well-known scaled many-objective test problems with 5 to 15 objectives. Simulation results have shown that the proposed algorithm can make better use of the decomposition vectors to achieve better performance. Further investigations on unscaled many-objective test problems verify the robust and generality of the proposed algorithm. <s> BIB010 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) STUDIES BASED ON DECOMPOSITION <s> Decomposition-based many-objective evolutionary algorithms generally decompose the objective space into multiple subregions with the help of a set of reference vectors. The resulting subregions are fixed since the reference vectors are usually predefined. When the optimization problem has a complicated Pareto front (PF), this decomposition may decrease the algorithm performance. To deal with this problem, this paper proposes a dynamical decomposition strategy. Instead of using predefined reference vectors, solution themselves are used as reference vectors. Thus, they are adapted to the shape of PF automatically. Besides, the subregions are produced one by one through successively bipartitioning the objective space. The resulting subregions are not fixed but dynamically determined by the population solutions as well as the subregions produced previously. Based on this strategy, a solution ranking method, named dynamical-decomposition-based ranking method (DDR), is proposed which can be employed in the mating selection and environmental selection in commonly used algorithm frameworks. Compared with those in the other decomposition-based algorithms, DDR has the following properties: 1) no predefined reference vectors are required; 2) less parameters are involved; and 3) the ranking results can not only be utilized directly to select solutions but also serve as a secondary criterion in traditional Pareto-based algorithms. In this paper, DDR is equipped in two algorithm frameworks for handling many-objective optimization problems. Comparisons with five state-of-the-art algorithms on 31 widely used test problems are carried out to test the performance of the proposed approach. The experimental results have shown the effectiveness of the proposed approach in keeping a good tradeoff between convergence and diversity. <s> BIB011
For MaOPs, how to select a set of weight vectors to produce the expected distribution of PS is of great significance and challenge. Giagkiozis et al.  , BIB001 defined a mathematical programming model with respect to the optimization of weight vectors. Different from a nonlinear one used for the TCH decomposition, the objective function in this model is simply a linear transformation of the weight vector, which means that the transformed vector is also a part of the convex set because of the convexity of the weight vector, and thus the model remains convex. On this basis, a generalized decomposition method was proposed, which improves the TCH approach and selects weight vectors by using the defined programming model. It produces the Pareto optimal vectors along the PF according to user preferences, and provides a framework in which DM can guide the basic EAs to a specific region or the entire region of the PF. Therefore, the generalized decomposition method can be more easily applied to a wide range of EAs. The significant advantage of this method is that it simplifies MaOPs by unifying three performance criteria (convergence, diversity, and coverage BIB006 ) of multiobjective optimization algorithms into only convergence. Facing the third challenge just mentioned, Yuan et al. BIB005 presented an MOEA/D variant with a distance-based updating (MOEA/D-DU) strategy, which fundamentally aims at clearly maintaining the solutions diversity during an evolution. Specifically, the K weight vectors closest to the new solution S new mapping vector are determined according to the vertical distance between S new mapping vector and weight vectors in the objective space, and then their corresponding solutions are replaced with S new by comparing aggregation function values. This differs from BIB003 that directly uses the vertical distance as a measure of the solution comparison. The solution update process of the proposed algorithm is stable, i.e., S new may replace only one old solution in the current population, resulting in a better PF coverage. In addition, the value of parameter K was determined experimentally. A larger K value puts more emphasis on the aggregation function for encouraging convergence, while a smaller K value highlights the vertical distance for promoting population diversity. MOEA/D-M2M, first proposed by Liu et al. BIB002 , is also suitable for solving MaOPs. In fact, many practical problems have m objectives, while the true PF is less than m-dimension due to the redundancy of some objectives. For the degenerated MaOPs with redundant objectives and very low-dimensional PS, the uniform and fixed weight vectors generated in BIB002 waste computational resources. Many researchers have improved the original MOEA/D-M2M with different mechanisms. Liu et al.  introduced an adaptive mechanism into the MOEA/D-M2M framework, which adaptively adjusts weight vectors of each subregion with the Max-Min method according to the distribution information of the solutions found so far, thus enabling a region to be decomposed adaptively. Chen et al. BIB007 designed a new dominance relation called D-dominance, which combines the advantages of decomposition and domination. Then the D-dominance was employed into the MOEA/D-M2M as the selection criterion for each subpopulation. Besides, the resulting algorithm further utilizes the weight vector and reduces its design difficulty. Lin et al. BIB009 proposed an algorithm combining MOEA/D-M2M and I SDE + (the indicator based on the shift-based density estimation) that contains both information on individual distribution and convergence. The algorithm calculates the indicator I SDE + independently in each subregion, which can effectively maintain population diversity and reduce the computational cost. Li et al. BIB004 combined the decomposition with the Pareto dominance for distinguishing and selecting candidate solutions, and termed the resulting algorithm as MOEA/D with domination archive (MOEA/DD). Each weight vector in this algorithm specifies not only a subproblem for fitness evaluation but also a subregion for estimating the local density of a population. In the original MOEA/D, the nondominated solutions are selected by comparing the aggregation function values between solutions, while in MOEA/DD they are selected according to both Pareto domination and crowding distance sorting. In addition, the update mechanism of the parent population is stable, i.e., it considers only one offspring to update the population each time, and performs multiple rounds of update procedure if more than one offspring solution has been generated. To make parents closer to each other and thus improve the effectiveness of crossover operators, a subregion-based mating restriction scheme was introduced into MOEA/DD. By applying D-dominance to the MOEA/ D-M2M framework, Chen et al. BIB010 proposed an adaptive D-dominance relationship-based evolutionary algorithm called DrEA. It employs a customized D-dominance with adjustable parameters to each subpopulation independently, which enables parallelism of D-dominance in each subpopulation. Therefore, the balance between population convergence and diversity can be achieved better compared with MOEAD/D. Ishibuchi et al. BIB008 demonstrated that the shape and the size of the PF have a large effect on the performance of the weight vector-based algorithms that perform well only when the distribution of weight vectors is consistent with the PF shape. When the PF shape is irregular, the design of weight vectors to ensure the diversity of solutions is challenging. He et al. BIB011 proposed a dynamical decomposition strategy for many-objective optimization. Different from existing decomposition methods, this method decomposes the objective space into subregions dynamically without employing a set of predefined reference vectors. Instead, the solutions themselves are considered as reference vectors. Thus, the performance of the proposed algorithm is less dependent on the PF shapes and remains robust especially in solving MaOPs with irregular PFs.