A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) METHODS BASED ON ADAPTIVE ADJUSTMENT <s> MOEA/D is a recently proposed methodology of Multiobjective Evolution Algorithms that decomposes multiobjective problems into a number of scalar subproblems and optimizes them simultaneously. However, classical MOEA/D uses same weight vectors for different shapes of Pareto front. We propose a novel method called Pareto-adaptive weight vectors (paλ) to automatically adjust the weight vectors by the geometrical characteristics of Pareto front. Evaluation on different multiobjective problems confirms that the new algorithm obtains higher hypervolume, better convergence and more evenly distributed solutions than classical MOEA/D and NSGA-II. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) METHODS BASED ON ADAPTIVE ADJUSTMENT <s> Recently, MOEA/D multi-objective evolutionary algorithm based on decomposition has achieved great success in the field of evolutionary multi-objective optimization and has attracted a lot of attention. It decomposes a multi-objective optimization problem MOP into a set of scalar subproblems using uniformly distributed aggregation weight vectors and provides an excellent general algorithmic framework of evolutionary multi-objective optimization. Generally, the uniformity of weight vectors in MOEA/D can ensure the diversity of the Pareto optimal solutions, however, it cannot work as well when the target MOP has a complex Pareto front PF; i.e., discontinuous PF or PF with sharp peak or low tail. To remedy this, we propose an improved MOEA/D with adaptive weight vector adjustment MOEA/D-AWA. According to the analysis of the geometric relationship between the weight vectors and the optimal solutions under the Chebyshev decomposition scheme, a new weight vector initialization method and an adaptive weight vector adjustment strategy are introduced in MOEA/D-AWA. The weights are adjusted periodically so that the weights of subproblems can be redistributed adaptively to obtain better uniformity of solutions. Meanwhile, computing efforts devoted to subproblems with duplicate optimal solution can be saved. Moreover, an external elite population is introduced to help adding new subproblems into real sparse regions rather than pseudo sparse regions of the complex PF, that is, discontinuous regions of the PF. MOEA/D-AWA has been compared with four state of the art MOEAs, namely the original MOEA/D, Adaptive-MOEA/D, -MOEA/D, and NSGA-II on 10 widely used test problems, two newly constructed complex problems, and two many-objective problems. Experimental results indicate that MOEA/D-AWA outperforms the benchmark algorithms in terms of the IGD metric, particularly when the PF of the MOP is complex. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) METHODS BASED ON ADAPTIVE ADJUSTMENT <s> MOEA/D decomposes a multi-objective optimization problem (MOP) into a set of scalar sub-problems with evenly spread weight vectors. Recent studies have shown that the fixed weight vectors used in MOEA/D might not be able to cover the whole Pareto front (PF) very well. Due to this, we developed an adaptive weight adjustment method in our previous work by removing subproblems from the crowded parts of the PF and adding new ones into the sparse parts. Although it performs well, we found that the sparse measurement of a subproblem which is determined by the m-nearest (m is the dimensional of the object space) neighbors of its solution can be more appropriately defined. In this work, the neighborhood relationship between subproblems is defined by using Delaunay triangulation (DT) of the points in the population. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) METHODS BASED ON ADAPTIVE ADJUSTMENT <s> Multiobjective evolutionary algorithms based on decomposition (MOEA/D) decomposes a multiobjective optimization problem (MOP) into a number of scalar optimization subproblems, and solves each subproblem in a collaborative manner. For a MOP, different subproblems often have different difficulty to be approximated, especially when the MOP is extremely complex or has a discontinuous optimal front. This paper proposes a weight vector adaptation strategy for this issue, which changes the weight vectors and optimizes their computational resource allocation to fit the MOP. The experimental results on a variety of MOP test instances show that the proposed algorithm is competitive in comparison with three state-of-the-art decomposition multiobjective evolutionary algorithms. <s> BIB004
In real-world problems, the search complexity of each subproblem is usually different, and thus, weight vectors need to be dynamically adjusted according to the computational resources required by each subproblem. Harada et al.  improved MOEA/D by adding a method of adaptively assigning weight vectors to subproblems. The method identifies the subproblems that are hard to search by using the EP and weight vectors, and then divides each of these subproblems into several subproblems to assign multiple weight vectors, implying that more computational resources are allocated in the direction of the subproblem with search difficulty. Therefore, the search speed of each subproblem can be adjusted reasonably to find a wider and more uniform PS distribution. Li et al. BIB004 proposed an improved MOEA/D with a weight vector adaptation (WAD) strategy, which first identifies weight vectors that waste computational resources. Specifically, the resulting solution mapping vectors of their corresponding subproblems may fall into local optimum or point to discontinuous PF regions. Then the strategy adjusts these weight vectors and optimizes their computational resource allocations to accommodate specific MOPs. For MOPs with concave PFs, Jiang et al. BIB001 proposed a novel method, termed Pareto-adaptive weight vectors (paλ), to automatically adjust the weight vectors according to the geometrical characteristics of the PF. This method uses the mixed uniform design to generate an arbitrary number of initial weight vectors with gradients denoted by λ lines, which produce N intersection points along the PF. Then weight vectors are adjusted so that the N intersection points have the maximum hypervolume to realize their uniform distribution, where the hypervolume is used to evaluate the distribution of these points and to drive the paλ method. When MOPs have an irregular PF, the fixed weight vectors cannot guide the algorithm to obtain the solutions distributed uniformly and widely. Particularly, in the discontinuous parts of a PF, several subproblems have the same optimal solution. While in the peak or low-tail region of a PF, many solution mapping vectors are distributed in a narrow region of the objective space. Qi et al. BIB002 proposed an improved MOEA/D algorithm with AWA. The AWA strategy identifies real sparse regions instead of discontinuous parts of the complex PF, and then introduces the elite population that helps add new subproblems in real sparse regions and remove redundant subproblems from the crowded parts of the PF. The sparseness of a subproblem in the AWA method is determined by m (the number of objectives) nearest neighbors of its solution, which results in the AWA being inaccurate when the solution of a subproblem has several close neighbors within the objective space in similar directions. To overcome this deficiency in BIB002 , Qi et al. BIB003 defined a new neighborhood relation between subproblems by using the Delaunay triangulation net. Then the sparseness measurement of a subproblem is determined by not only Euclidean distances between solutions but also their distribution.