A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) REPRODUCTION OPERATORS <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> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) REPRODUCTION OPERATORS <s> Combining ant colony optimization (ACO) and the multiobjective evolutionary algorithm (EA) based on decomposition (MOEA/D), this paper proposes a multiobjective EA, i.e., MOEA/D-ACO. Following other MOEA/D-like algorithms, MOEA/D-ACO decomposes a multiobjective optimization problem into a number of single-objective optimization problems. Each ant (i.e., agent) is responsible for solving one subproblem. All the ants are divided into a few groups, and each ant has several neighboring ants. An ant group maintains a pheromone matrix, and an individual ant has a heuristic information matrix. During the search, each ant also records the best solution found so far for its subproblem. To construct a new solution, an ant combines information from its group's pheromone matrix, its own heuristic information matrix, and its current solution. An ant checks the new solutions constructed by itself and its neighbors, and updates its current solution if it has found a better one in terms of its own objective. Extensive experiments have been conducted in this paper to study and compare MOEA/D-ACO with other algorithms on two sets of test problems. On the multiobjective 0-1 knapsack problem, MOEA/D-ACO outperforms the MOEA/D with conventional genetic operators and local search on all the nine test instances. We also demonstrate that the heuristic information matrices in MOEA/D-ACO are crucial to the good performance of MOEA/D-ACO for the knapsack problem. On the biobjective traveling salesman problem, MOEA/D-ACO performs much better than the BicriterionAnt on all the 12 test instances. We also evaluate the effects of grouping, neighborhood, and the location information of current solutions on the performance of MOEA/D-ACO. The work in this paper shows that reactive search optimization scheme, i.e., the “learning while optimizing” principle, is effective in improving multiobjective optimization algorithms. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) REPRODUCTION OPERATORS <s> Adaptive operator selection (AOS) is used to determine the application rates of different operators in an online manner based on their recent performances within an optimization process. This paper proposes a bandit-based AOS method, fitness-rate-rank-based multiarmed bandit (FRRMAB). In order to track the dynamics of the search process, it uses a sliding window to record the recent fitness improvement rates achieved by the operators, while employing a decaying mechanism to increase the selection probability of the best operator. Not much work has been done on AOS in multiobjective evolutionary computation since it is very difficult to measure the fitness improvements quantitatively in most Pareto-dominance-based multiobjective evolutionary algorithms. Multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. Thus, it is natural and feasible to use AOS in MOEA/D. We investigate several important issues in using FRRMAB in MOEA/D. Our experimental results demonstrate that FRRMAB is robust and its operator selection is reasonable. Comparison experiments also indicate that FRRMAB can significantly improve the performance of MOEA/D. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) REPRODUCTION OPERATORS <s> Abstract The traditional reproduction operators, which are originally designed for single-objective optimization, are directly adopted in most state-of-the-art multi-objective evolutionary algorithms (MOEAs). However, these reproduction operators might not be suitable for multiobjective optimization problems (MOPs) due to the regularity property of continuous MOP, and that is to say its Pareto optimal set in the decision space is generally piecewise continuous manifold rather than a set of independent points. Few researches have used this regularity property of continuous MOP to help design their algorithms. In this paper, based on the regularity property, a Baldwinian learning operator is incorporated into the framework of MOEA/D (multi-objective evolutionary algorithm based on decomposition) and thereby we propose MOEA/D/BL. The Baldwinian learning operator obtains the evolving information based on the learned distribution model of a current population. It constructs a candidate descent direction based on the learned distribution model and the evolving history of the parent individuals. Experimental results on twenty-three popular test problems show that the proposed algorithm performs better than or as well as four other compared algorithms. It also experimentally demonstrates that the proposed Baldwinian learning operator can accelerate the convergence of solutions. <s> BIB004 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) REPRODUCTION OPERATORS <s> This paper studies how to use two reproduction operators with different characteristics for balancing the convergence and the diversity in MOEA/D. We consider two operators. One is a differential evolution and polynomial mutation, and the other is a neighbor learning and inversion mutation. We show that these two operators have different search abilities. Then we propose a scheme to use these two operators in our recently proposed MOEA/D-GR framework. We test the proposed algorithm on some benchmark problems to demonstrate its effectiveness. <s> BIB005 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 4) REPRODUCTION OPERATORS <s> The bias feature is a major factor that makes a multiobjective optimization problem (MOP) difficult for multiobjective evolutionary algorithms (MOEAs). To deal with this problem feature, an algorithm should carefully balance between exploration and exploitation. The decomposition-based MOEA decomposes an MOP into a number of single objective subproblems and solves them in a collaborative manner. Single objective optimizers can be easily used in this algorithm framework. Covariance matrix adaptation evolution strategy (CMA-ES) has proven to be able to strike good balance between the exploration and the exploitation of search space. This paper proposes a scheme to use both differential evolution (DE) and covariance matrix adaptation in the MOEA based on decomposition. In this scheme, single objective optimization problems are clustered into several groups. To reduce the computational overhead, only one subproblem from each group is selected to optimize by CMA-ES while other subproblems are optimized by DE. When an evolution strategy procedure meets some stopping criteria, it will be reinitialized and used for solving another subproblem in the same group. A set of new multiobjective test problems with bias features are constructed in this paper. Extensive experimental studies show that our proposed algorithm is suitable for dealing with problems with biases. <s> BIB006
Reproduction is a direct and effective way to generate new solutions. Li and Zhang  adopted DE as a crossover operator and proposed an MOEA/D-DE to deal with continuous MOPs with complex PSs. According to the characteristics of the DE operation, three parent solutions should be selected from the mating pool, which can produce a wide range of child solutions to effectively maintain population diversity. Zamuda et al. BIB001 extended the work on DE in  and introduced adaptive DE and local search into MOEA/D for solving constrained multiobjective optimization problems (CMOPs). Each individual has two parameters, self-adaptive F (amplification factor of the difference vector) and CR (crossover control parameter). The algorithm adjusts both to the appropriate values during an evolution, resulting in faster convergence than the original DE. Local search only performs on partial individuals that have improved since the last local search. The combination of global search and local search could enable the improved algorithm to maintain a good balance between exploration and exploitation. As one of swarm intelligence algorithms, ant colony optimization (ACO) can achieve segmentation estimation of a PF by exploiting cooperative evolution among different ant groups. To avoid heavy computational overhead and accelerate algorithm convergence, Ke et al. BIB002 proposed an MOEA/D-ACO algorithm in which an ant is considered as a basic evolution unit. The N ants are divided into K groups by clustering their corresponding weight vectors, and each group is designed to approximate the small range of a PF. Different ant groups co-evolve with the POSs via sharing a global solution pool. Each ant records the best solution found so far for its subproblem and constructs a new solution with the pheromone matrix about itself and its neighbors. Then the algorithm performs solution updates like the original MOEA/D. This work has implied that many intelligent algorithms, such as particle swarm optimization, simulated annealing and artificial immune, can be combined with MOEA/D to further improve the algorithm performance. Li et al. BIB003 proposed a bandit-based adaptive operator selection method to determine which reproduction operator should be selected in terms of operators' recent performances. To be more specific, this method uses the fitness improvement rate that refers to the solution quality difference between the parent and its offspring to calculate the reward of each operator, and sums all these rewards caused by its generated offspring solution replacing several parent solutions up as the final reward. The best operator is therefore selected from a group of operators for use at the next point time based on these VOLUME 8, 2020 reward values. The idea of using cumulative improvement rates to evaluate the quality of operators in this literature is useful for some online selection problems, such as the allocation of different neighborhood sizes or population sizes to different search stages of an algorithm. Ma et al. BIB004 incorporated an improved Baldwinian learning operator into the MOEA/D framework (MOEA/D-BL). The algorithm divides the current parent population into K disjoint clusters, and then obtains the evolution information by building a local distribution model for each cluster. Next, the Baldwinian learning operator constructs an offspring descent direction by combining the evolution history of the parent individuals with the distribution model. The combination of clustering and learning can make full use of the distribution information in the local region, which enables MOEA/D-BL to converge faster than MOEA/D. Different operators have different search characteristics, and it is promising to make full use of these operators to balance convergence and diversity. Wang et al. BIB005 first explained the search behaviors of two different operators, namely, differential evolution operator and polynomial mutation, as well as neighbor learning operator and inversion mutation. Specifically, the former is more likely to find a set of solutions distributed widely and uniformly, while the latter has a good searching ability for higher quality solutions. Then the two operators were employed jointly in the MOEA/D-GR framework proposed by the authors to improve algorithm performance. Similarly, Li et al. BIB006 embedded both DE and covariance matrix adaptation evolution strategy (CMA-ES) in MOEA/D. To overcome the high computational costs of CMA-ES and to use the simplicity of DE, all subproblems are clustered into several groups, and each group selects only one subproblem to be solved with CMA-ES, while the others are solved with DE.