A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) ADAPTATION OF THE ORIGINAL DECOMPOSITION METHODS <s> It is well-known that multiobjective problems with many objectives are difficult for Pareto dominance-based algorithms such as NSGA-II and SPEA. This is because almost all individuals in a population are non-dominated with each other in the presence of many objectives. In such a population, the Pareto dominance relation can generate no strong selection pressure toward the Pareto front. This leads to poor search ability of Pareto dominance-based algorithms for many-objective problems. Recently it has been reported that better results can be obtained for many-objective problems by the use of scalarizing functions. The weighted sum usually works well in scalarizing function-based algorithms when the Pareto front is convex. However, we need other functions such as the weighted Tchebycheff when the Pareto front is non-convex. In this paper, we propose an idea of automatically choosing between the weighted sum and the weighted Tchebycheff for each individual in each generation. The characteristic feature of the proposed idea is to use the weighted Tchebycheff only when it is needed for individuals along non-convex regions of the Pareto front. The weighted sum is used for the other individuals in each generation. The proposed idea is combined with a high-performance scalarizing function-based algorithm called MOEA/D (multiobjective evolutionary algorithm based on decomposition) of Zhang and Li (2007). Effectiveness of the proposed idea is demonstrated through computational experiments on modified multiobjective knapsack problems with non-convex Pareto fronts. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) ADAPTATION OF THE ORIGINAL DECOMPOSITION METHODS <s> Decomposition-based algorithms have become increasingly popular for evolutionary multiobjective optimization. However, the effect of scalarizing methods used in these algorithms is still far from being well understood. This paper analyzes a family of frequently used scalarizing methods, the ${L} _{ {p}}$ methods, and shows that the ${p}$ value is crucial to balance the selective pressure toward the Pareto optimal and the algorithm robustness to Pareto optimal front (PF) geometries. It demonstrates that an ${L} _{ {p}}$ method that can maximize the search ability of a decomposition-based algorithm exists and guarantees that, given some weight, any solution along the PF can be found. Moreover, a simple yet effective method called Pareto adaptive scalarizing (PaS) approximation is proposed to approximate the optimal ${p}$ value. In order to demonstrate the effectiveness of PaS, we incorporate PaS into a state-of-the-art decomposition-based algorithm, i.e., multiobjective evolutionary algorithm based on decomposition (MOEA/D), and compare the resultant MOEA/D-PaS with some other MOEA/D variants on a set of problems with different PF geometries and up to seven conflicting objectives. Experimental results demonstrate that the PaS is effective. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) ADAPTATION OF THE ORIGINAL DECOMPOSITION METHODS <s> The decomposition-based evolutionary multiobjective optimization (EMO) algorithm has become an increasingly popular choice for a posteriori multiobjective optimization. However, recent studies have shown that their performance strongly depends on the Pareto front (PF) shapes. This can be attributed to the decomposition method, of which the reference points and subproblem formulation settings are not well adaptable to various problem characteristics. In this paper, we develop a learning-to-decompose (LTD) paradigm that adaptively sets the decomposition method by learning the characteristics of the estimated PF. Specifically, it consists of two interdependent parts, i.e., a learning module and an optimization module. Given the current nondominated solutions from the optimization module, the learning module periodically learns an analytical model of the estimated PF. Thereafter, useful information is extracted from the learned model to set the decomposition method for the optimization module: 1) reference points compliant with the PF shape and 2) subproblem formulations whose contours and search directions are appropriate for the current status. Accordingly, the optimization module, which can be any decomposition-based EMO algorithm in principle, decomposes the multiobjective optimization problem into a number of subproblems and optimizes them simultaneously. To validate our proposed LTD paradigm, we integrate it with two decomposition-based EMO algorithms, and compare them with four state-of-the-art algorithms on a series of benchmark problems with various PF shapes. <s> BIB003
The introduction of adaptive strategies or learning mechanisms into MOEA/D makes it more suitable for solving complex problems. Adaptively selecting decomposition methods according to the problem characteristics is helpful to accelerating algorithm convergence and to obtaining more solutions dispersed widely. Ishibuchi et al. BIB001 designed a mechanism for automatic selection between WS and TCH, which uses TCH only when the current solution mapping vector is close to the nonconvex regions of the PF, i.e., a current individual and at least T neighbors in its neighborhood have the same objective vector. This mechanism employs WS during the MOEA/D execution for the other solution mapping vectors around the convex regions. Since the PF is usually unknown before we find POSs, a method of detecting the nonconvex regions of the PF was proposed to determine whether TCH is required. TCH will be used when a single individual becomes the best with respect to the weighted sum under different weight vectors. It is helpful for an algorithm to combine modules with different functions together. Wu et al. BIB003 proposed a learning-to-decomposition (LTD) paradigm including a learning module and an optimization module to adaptively set the decomposition method. Specifically, the learning module takes nondominated solutions from the optimization module as the training data, and then uses the Gaussian process regression to learn the characteristics of the estimated PF. According to the analytical model extracted from the learning module, the optimization module that adopts MOEA/D adaptively sets the decomposition method, including effective reference points and appropriate subproblem formulations. The validity of the LTD model was proved by a series of experiments on the benchmark problems with different PF shapes. This research provides an inspiration that the organic combination of MOEA/D and machine learning algorithms can make better use of the information generated in evolution, making the learning and optimization processes interact and benefit each other. Wang et al. BIB002 systematically analyzed the searching ability of a family of scalarizing methods, the L p scalarizing methods, where L p=1 represents WS and L p=∞ corresponds to TCH. Simulation results showed that the probability of finding a better solution decreases with an increase of the p value, and the difference in the improvement region size for various L p scalarizing methods becomes smaller when the solution mapping vectors approach PF. Therefore, the p value is crucial to the balance between convergence and diversity for a specific problem. A Pareto adaptive scalarizing strategy was then proposed to assign an appropriate L p scalarizing method for each weight vector during the search. To be more specific, an appropriate p value is found from the set of different p values when using the L p scalarizing method, i.e., the optimal solution obtained with this p value is closer to the search direction compared with other p values.