A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) OTHER RESEARCHES ON MANY-OBJECTIVE OPTIMIZATION <s> Decomposition-Based Multi-Objective Evolutionary Algorithms (DBMOEA), such as Multiple Single Objective Pareto Sampling (MSOPS) and Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D), have been successfully applied in finding Pareto-optimal fronts in Multiobjective Optimization Problems (MOPs), two or three-objective in general. DBMOEA decomposes one MOP into multiple Single-objective Optimization Problems (SOPs) where the convergence of approximated front is facilitated by finding the optimal solution of each SOP and its diversity is preserved by a group of well distributed SOPs. However, when solving problems with many objectives, one single solution can be the optimal solution of multiple SOPs which inadvertently leads to a severe loss of population diversity. In this paper, we propose a new diversity improvement method incorporated into a modified DBMOEA to directly handle this challenge. The design includes two steps. First, a few number of weight vectors guide the whole population towards a small number of solutions nearby the true Pareto front. Afterwards, initialize a subpopulation around each solution and diversify them toward well distribution. As a case study, a new algorithm based on this design is compared with three state-of-the-art DBMOEAs, MOEA/D, MSOPS, and MO-NSGA-II. Experimental results show that the proposed methods exhibit better performance in both convergence and diversity than the chosen competitors for solving many-objective optimization problems. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) OTHER RESEARCHES ON MANY-OBJECTIVE OPTIMIZATION <s> In evolutionary multi-objective optimization, each solution in the population generally has two roles. The first one is to approximate a part of the Pareto front, and the second one is to be a variable information resource to generate offspring. In many-objective optimization involving four or more conflicting objectives, solutions in the population have to be sparsely distributed in the objective space and the variable space to approximate a high-dimensional Pareto front, and each solution faces the difficulty to play the second role since variables are drastically individualized in the population. To overcome this problem, we focus on MOEA/D algorithm framework and propose a method to introduce supplemental weight vectors and solutions which maintain variable information resource to enhance the solution search for each part of the Pareto front. Experimental results using many-objective knapsack problems show that the supplemental weight vectors and solutions improves the search performance of MOEA/D by improving the diversity of the obtained solutions. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) OTHER RESEARCHES ON MANY-OBJECTIVE OPTIMIZATION <s> Decomposition-based multiobjective evolutionary algorithm has shown its advantage in addressing many-objective optimization problem (MaOP). To further improve its convergence on MaOPs and its diversity for MaOPs with irregular Pareto fronts (PFs, e.g., degenerate and disconnected ones), we proposed a decomposition-based many-objective evolutionary algorithm with two types of adjustments for the direction vectors (MaOEA/D-2ADV). At the very beginning, search is only conducted along the boundary direction vectors to achieve fast convergence, followed by the increase of the number of the direction vectors for approximating a more complete PF. After that, a Pareto-dominance-based mechanism is used to detect the effectiveness of each direction vector and the positions of ineffective direction vectors are adjusted to better fit the shape of irregular PFs. The extensive experimental studies have been conducted to validate the efficiency of MaOEA/D-2ADV on many-objective optimization benchmark problems. The effects of each component in MaOEA/D-2ADV are also investigated in detail. <s> BIB003
To address the third challenge, He and Yen BIB001 proposed a new algorithm, termed MOEA/D with diversity improvement (MOEA/D-DI). The algorithm consists mainly of two steps for different purposes. In the first step, a small number of Pareto solution vectors distributed uniformly on the PF boundary are generated, focusing on the algorithm convergence. The second step concerns population diversity. After finding the target solutions, a population is initialized around each solution, and the individuals in the population are well spread and distributed by using the proposed diversity method. In every iteration, two parents are selected from the whole population, and offsprings are generated after reproduction, and the next generation of individuals is then selected from the parent and its offspring. Similarly, Cai et al. BIB003 utilized two types of direction vectors, one aiming to approximate a more complete PF by expanding the number of direction vectors after the fast convergence along the boundary direction vectors, and the other trying to adjust the position of ineffective direction vectors for MaOPs with irregular PFs. In the high-dimensional space, weight vectors in the weight space become sparse, resulting in insufficient search resources for each part of a PF. Sato et al. BIB002 introduced supplemental weight vectors and solutions into the original MOEA/D. The algorithm assigns supplemental weight vectors around each original weight vector and generates the corresponding supplemental solutions to enhance the detailed search in the objective space. Differing from the original MOEA/D, parents are selected from the supplemental solutions that have more similar objective values and variable information even in a higher dimensional objective space.