决策空间定向搜索的高维多目标优化策略

传统的多目标进化算法（MOEA）对于低维连续的多目标优化问题已经具有良好的性能，但是随着优化问题目标维数的增加，优化难度也将剧增，主要原因是算法本身搜索能力不足，维数增加时选择压力变小，收敛性和分布性冲突难以平衡。利用连续多目标优化问题的特性，针对高维多目标优化的难点所在，提出了一种在决策空间的定向搜索策略（decision space，简称DS），该策略可与基于支配关系的MOEA相结合。DS首先对优化问题进行采样分析，对问题特性进行解析，得到收敛性子空间控制向量和分布性子空间控制向量。将算法搜索过程分为收敛性搜索阶段和分布性搜索阶段，分别对应收敛性子空间和分布性子空间，在不同阶段搜索时，利用采样分析结果，对生成子代个体的区域进行宏观的影响。将收敛性和分布性分阶段考虑，避免了收敛性和分布性难以平衡的难点，同时，具体在某一阶段内搜索资源相对集中，一定程度上增加了算法的搜索能力。实验结合了DS策略的NSGA-Ⅱ，SPEA2算法与原NSGA-Ⅱ，SPEA2算法进行实验对比，并以DS-NSGA-Ⅱ为例，与其他高维算法MOEAD-PBI，NSGA-ⅡI，Hype，MSOPS，LMEA进行对比实验。实验结果表明，DS策略的引入，使得NSGA-Ⅱ，SPEA2算法在高维多目标优化问题上的性能有了显著提高，DS-NSGAⅡ与现有的经典高维多目标算法相比有较强的竞争力。

Traditional multi-objective evolutionary algorithm (MOEA) have sound performance when solving low dimensional continuous multi-objective optimization problems. However, as the optimization problems' dimensions increase, the difficulty of optimization will also increase dramatically. The main reasons are the lack of algorithms' search ability, and the smaller selection pressure when the dimension increases as well as the difficulty to balance convergence and distribution conflicts. In this study, after analyzing the characteristics of the continuous multi-objective optimization problem, a directional search strategy based on decision space (DS) is proposed to solve high dimensional multi-objective optimization problems. This strategy can be combined with the MOEAs based on the dominating relationship. DS first samples solutions from the population and analyzes them, and obtains the controlling vectors of convergence subspace and distribution subspace by analyzing the problem characteristics. The algorithm is divided into convergence search stage and distribution search stage, which correspond to convergent subspace and distributive subspace respectively. In different stages of search, sampling analysis are used results to macroscopically control the region of offspring generation. The convergence and distribution are divided and emphasized in different stages to avoid the difficulty of balancing them. Additionally, it can also relatively focuses the search resources on certain aspect in certain stages, which facilitates the searching ability of the algorithm. In the experiment, NSGA-II and SPEA2 algorithms are compared combining DS strategy with original NSGA-II and SPEA2 algorithms, and DS-NSGA-II is used as an example to compare it with other state-of-the-art high-dimensional algorithms, such as MOEAD-PBI, NSGA-III, Hype, MSOPS, and LMEA. The experimental results show that the introduction of the DS strategy greatly improves the performance of NSGA-II and SPEA2 when addressing high dimensional multi-objective optimization problems. It is also shown that DS-NSGA-II is more competitive when compared the existing classical high dimensional multi-objective algorithms.