A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) REPLACEMENT MECHANISM <s> This paper studies the replacement schemes in MOEA/D and proposes a new replacement named global replacement. It can improve the performance of MOEA/D. Moreover, trade-offs between convergence and diversity can be easily controlled in this replacement strategy. It also shows that different problems need different trade-offs between convergence and diversity. We test the MOEA/D with this global replacement on three sets of benchmark problems to demonstrate its effectiveness. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) REPLACEMENT MECHANISM <s> Multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem into a set of scalar optimization subproblems and optimizes them in a collaborative manner. Subproblems and solutions are two sets of agents that naturally exist in MOEA/D. The selection of promising solutions for subproblems can be regarded as a matching between subproblems and solutions. Stable matching, proposed in economics, can effectively resolve conflicts of interests among selfish agents in the market. In this paper, we advocate the use of a simple and effective stable matching (STM) model to coordinate the selection process in MOEA/D. In this model, subproblem agents can express their preferences over the solution agents, and vice versa. The stable outcome produced by the STM model matches each subproblem with one single solution, and it tradeoffs convergence and diversity of the evolutionary search. Comprehensive experiments have shown the effectiveness and competitiveness of our MOEA/D algorithm with the STM model. We have also demonstrated that user-preference information can be readily used in our proposed algorithm to find a region that decision makers are interested in. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) REPLACEMENT MECHANISM <s> A decomposition approach decomposes a multiobjective optimization problem into a number of scalar objective optimization subproblems. It plays a key role in decomposition-based multiobjective evolutionary algorithms. However, many widely used decomposition approaches, originally proposed for mathematical programming algorithms, may not be very suitable for evolutionary algorithms. To help decomposition-based multiobjective evolutionary algorithms balance the population diversity and convergence in an appropriate manner, this letter proposes to impose some constraints on the subproblems. Experiments have been conducted to demonstrate that our proposed constrained decomposition approach works well on most test instances. We further propose a strategy for adaptively adjusting constraints by using information collected from the search. Experimental results show that it can significantly improve the algorithm performance. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) REPLACEMENT MECHANISM <s> Multiobjective evolutionary algorithms based on decomposition (MOEA/D) decompose a multiobjective optimization problem into a set of simple optimization subproblems and solve them in a collaborative manner. A replacement scheme, which assigns a new solution to a subproblem, plays a key role in balancing diversity and convergence in MOEA/D. This paper proposes a global replacement scheme which assigns a new solution to its most suitable subproblems. We demonstrate that the replacement neighborhood size is critical for population diversity and convergence, and develop an approach for adjusting this size dynamically. A steady-state algorithm and a generational one with this approach have been designed and experimentally studied. The experimental results on a number of test problems have shown that the proposed algorithms have some advantages. <s> BIB004
In the original MOEA/D, a superior offspring solution could replace several inferior neighboring solutions, leading to the deterioration in population diversity. Wang et al. BIB003 proposed a constrained decomposition approach that imposes some constraints on unconstrained subproblems to reduce the replacement range, and further developed a new strategy to adaptively adjust constraint factors by using information collected from the search, which is helpful to balancing population diversity and convergence at different search stages for an algorithm. It is uncertain whether the new solution x i new of the subproblem i is most suitable for its neighboring subproblems B(i), i.e., x i new may be bad for B(i), while good for other subproblems. It is thus unreasonable to directly replace the solutions of B(i) with x i new . Wang et al. BIB001 investigated a global replacement (GR) strategy. It finds the most suitable subproblem j for the new solution x i new by comparing the objective function value of each subproblem, and then selects T closest subproblems to subproblem j to constitute the replacement neighborhood B (j), which decreases the probability of replacing multiple current individuals with one generated newly. However, the replacement neighborhood is set to a fixed size in the experiment, which is not suitable for complex MOPs. Ref. BIB004 extended the GR strategy and presented an approach to dynamically adjust the replacement neighborhood size, which can effectively control the tradeoff between convergence and diversity at different search stages of an algorithm. The mutual selection between promising solutions and subproblems during the replacement process can be regarded as a match between solutions and subproblems. However, the solution does not explicitly express a preference for subproblems in MOEA/D. Li et al. BIB002 suggested the use of a simple and effective stable matching model to coordinate the selection process. Each subproblem ranks all the solutions including its parent and offspring according to its aggregation function and prefers the solutions with better function values. Besides, each solution ranks all subproblems by calculating its distance to the direction vectors of these subproblems and prefers the subproblems whose direction vectors are close to it. Finally, the model matches each subproblem with a single solution.