A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) STUDIES BASED ON UNIFORM DESIGN <s> Decomposition is a basic strategy in traditional multiobjective 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 scalability and sensitivity of MOEA/D have also been experimentally investigated in this paper. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) STUDIES BASED ON UNIFORM DESIGN <s> This paper presents Differential Evolution with Self-adaptation and Local Search for Constrained Multiobjective Optimization algorithm (DECMOSA-SQP), which uses the self-adaptation mechanism from DEMOwSA algorithm presented at CEC 2007 and a SQP local search. The constrained handling mechanism is also incorporated in the new algorithm. Assessment of the algorithm using CEC 2009 special session and competition on constrained multiobjective optimization test functions is presented. The functions are composed of unconstrained and constrained problems. Their results are assessed using the IGD metric. Based on this metric, algorithm strengths and weaknesses are discussed. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) STUDIES BASED ON UNIFORM DESIGN <s> To extend multiobjective evolutionary algorithm based on decomposition (MOEA/D) in higher dimensional objective spaces, this paper proposes a new version of MOEA/D with uniform design, named the uniform design multiobjective evolutionary algorithm based on decomposition (UMOEA/D), and compares the proposed algorithm with MOEA/D and NSGA-II on some scalable test problems with three to five objectives. UMOEA/D adopts the uniform design method to set the aggregation coefficient vectors of the subproblems. Compared with MOEA/D, distribution of the coefficient vectors is more uniform over the design space, and the population size neither increases nonlinearly with the number of objectives nor considers a formulaic setting. The experimental results indicate that UMOEA/D outperforms MOEA/D and NSGA-II on almost all these many-objective test instances, especially on problems with higher dimensional objectives and complicated Pareto set shapes. Experimental results also show that UMOEA/D runs faster than NSGA-II for the problems used in this paper. In additional, the results obtained are very competitive when comparing UMOEA/D with some other algorithm on the multiobjective knapsack problems. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) STUDIES BASED ON UNIFORM DESIGN <s> Many-objective problems (MAPs) have put forward a number of challenges to classical Pareto-dominance based multi-objective evolutionary algorithms (MOEAs) for the past few years. Recently, researchers have suggested that MOEA/D (multi-objective evolutionary algorithm based on decomposition) can work for MAPs. However, there exist two difficulties in applying MOEA/D to solve MAPs directly. One is that the number of constructed weight vectors is not arbitrary and the weight vectors are mainly distributed on the boundary of weight space for MAPs. The other is that the relationship between the optimal solution of subproblem and its weight vector is nonlinear for the Tchebycheff decomposition approach used by MOEA/D. To deal with these two difficulties, we propose an improved MOEA/D with uniform decomposition measurement and the modified Tchebycheff decomposition approach (MOEA/D-UDM) in this paper. Firstly, a novel weight vectors initialization method based on the uniform decomposition measurement is introduced to obtain uniform weight vectors in any amount, which is one of great merits to use our proposed algorithm. The modified Tchebycheff decomposition approach, instead of the Tchebycheff decomposition approach, is used in MOEA/D-UDM to alleviate the inconsistency between the weight vector of subproblem and the direction of its optimal solution in the Tchebycheff decomposition approach. The proposed MOEA/D-UDM is compared with two state-of-the-art MOEAs, namely MOEA/D and UMOEA/D on a number of MAPs. Experimental results suggest that the proposed MOEA/D-UDM outperforms or performs similarly to the other compared algorithms in terms of hypervolume and inverted generational distance metrics on different types of problems. The effects of uniform weight vector initializing method and the modified Tchebycheff decomposition are also studied separately. <s> BIB004 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) STUDIES BASED ON UNIFORM DESIGN <s> Any numbers of weight vectors with good diversity can be generated by a uniform design method.A sub-population strategy is used to enhance the local search ability of the algorithm.An update strategy based on decomposition is proposed to maintain the diversity. For many-objective optimization problems, how to get a set of solutions with good convergence and diversity is a difficult and challenging work. In this paper, a new decomposition based evolutionary algorithm with uniform designs is proposed to achieve the goal. The proposed algorithm adopts the uniform design method to set the weight vectors which are uniformly distributed over the design space, and the size of the weight vectors neither increases nonlinearly with the number of objectives nor considers a formulaic setting. A crossover operator based on the uniform design method is constructed to enhance the search capacity of the proposed algorithm. Moreover, in order to improve the convergence performance of the algorithm, a sub-population strategy is used to optimize each sub-problem. Comparing with some efficient state-of-the-art algorithms, e.g., NSGAII-CE, MOEA/D and HypE, on six benchmark functions, the proposed algorithm is able to find a set of solutions with better diversity and convergence. <s> BIB005 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) STUDIES BASED ON UNIFORM DESIGN <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> BIB006 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) STUDIES BASED ON UNIFORM DESIGN <s> Many-objective optimization problems (MaOPs), in which the number of objectives is greater than three, are undoubtedly more challenging compared with the bi- and tri-objective optimization problems. Currently, the decomposition-based evolutionary algorithms have shown promising performance in dealing with MaOPs. Nevertheless, these algorithms need to design the weight vectors, which has significant effects on the performance of the algorithms. In particular, when the Pareto front of problems is incomplete, these algorithms cannot obtain a set of uniformly distribution solutions by using the conventional weight design methods. In the literature, it is well-known that the self-organizing map (SOM) can preserve the topological properties of the input data by using the neighborhood function, and its display is more uniform than the probability density of the input data. This phenomenon is advantageous to generate a set of uniformly distributed weight vectors based on the distribution of the individuals. Therefore, we will propose a novel weight design method based on SOM, which can be integrated with most of the decomposition-based algorithms for solving MaOPs. In this paper, we choose the existing state-of-the-art decomposition-based algorithms as examples for such integration. This integrated algorithms are then compared with some state-of-the-art algorithms on eleven redundancy problems and eight nonredundancy problems, respectively. The experimental results show the effectiveness of the proposed approach. <s> BIB007
The challenges resulting from the weight vector generation and population size design in the high-dimensional space can be effectively overcome by the uniform design method. The motivation of uniform design here is to extend MOEA/D to many-objective optimization. The method of simplex-lattice design used in the original MOEA/D cannot generate weight vectors with an arbitrary number, which restricts the extension of MOEA/D to solve MaOPs. To deal with this drawback, Tan et al. BIB003 proposed a transformation method and termed the resulting algorithm as UMOEA/D. A parameter M was designed to measure the nonuniformity, with a smaller M meaning that the weight vector distribution is more uniform. GLP was adopted to determine the weight vector set with a minimum M . The algorithm not only finds a set of weight vectors that are distributed evenly over the objective space, but also prevents the population size from becoming very large as the number of objective functions increases. However, when the size of the evolving population is small, UMOEA/D faces the risk of missing some boundary parts of the PF, especially for MaOPs, which can be effectively overcome by the hybrid method proposed in BIB004 . Simplex-lattice design in BIB001 assigns too many weight vectors at the boundary of the weight space. On the contrary, the transformation method BIB003 allocates no weight vector on the boundary of weight space and can generate any number of weight vectors. Ma et al. BIB004 proposed an improved MOEA/D with uniform decomposition measurement (UDM), which is called MOEA/D-UDM and combines two methods above to construct uniform weight vectors. Particularly, it first generates alternative weight vectors, most of which are located in the interior of the weight space with a few ones distributed on the boundary of the weight space. Then uniform weight vectors are selected based on the UDM mechanism from alternative weight vectors. Compared with the UMOEA/D BIB003 , the proposed MOEA/D-UDM converges faster and the uniformity of obtained solutions is better. Dai and Wang BIB005 proposed a new decomposition-based EA with uniform design (UDEA/D) to make the solution set have better convergence and diversity. The algorithm first generates uniform weight vectors with a uniform design method. Then, these weight vectors are used to divide the population into multiple subpopulations, each of which is associated with a subproblem, thus accelerating the algorithm convergence. In addition, a crossover operator based on uniform design is constructed to improve the searching ability of the algorithm. The authors also proposed the control of dominance area of solutions (CDAS)  to sort solutions of each subpopulation, guiding the search process to converge toward the PS. The simulation results indicated that the proposed algorithm has a good ability of exploration and exploitation as in BIB002 , and outperforms the three famous algorithms, MOEA/D, NSGAII-CE and HypE (hyper evolving) on six benchmark functions with 5 to 25 objectives. The design of weight vectors is a key issue for solving an MaOP with an incomplete PF, because the same POSs with different weight vectors were obtained in this case BIB006 , thus greatly reducing population diversity. Gu and Cheung BIB007 developed a novel weight design method based on the selforganizing map (SOM), which periodically trains an SOM network with N neurons by using the objective vectors of the recent individuals, where N is the population size. The weights of neurons are employed as the weight vectors. This weight design method was then integrated into M2M and MOEA/D, respectively. Experiments on the MaOPs with an incomplete PF showed that these algorithms could generate weight vectors distributed evenly based on individuals' distribution, thus leading to a set of solutions distributed uniformly.