Zeng Sanyou Y, Kang Lishan S, Ding Lixin X
Dept. of Computer Science and Technology, China University of GeoSciences, Wuhan 430074, Hubei, P. R. China.
Evol Comput. 2004 Spring;12(1):77-98. doi: 10.1162/evco.2004.12.1.77.
In this paper, an orthogonal multi-objective evolutionary algorithm (OMOEA) is proposed for multi-objective optimization problems (MOPs) with constraints. Firstly, these constraints are taken into account when determining Pareto dominance. As a result, a strict partial-ordered relation is obtained, and feasibility is not considered later in the selection process. Then, the orthogonal design and the statistical optimal method are generalized to MOPs, and a new type of multi-objective evolutionary algorithm (MOEA) is constructed. In this framework, an original niche evolves first, and splits into a group of sub-niches. Then every sub-niche repeats the above process. Due to the uniformity of the search, the optimality of the statistics, and the exponential increase of the splitting frequency of the niches, OMOEA uses a deterministic search without blindness or stochasticity. It can soon yield a large set of solutions which converges to the Pareto-optimal set with high precision and uniform distribution. We take six test problems designed by Deb, Zitzler et al., and an engineering problem (W) with constraints provided by Ray et al. to test the new technique. The numerical experiments show that our algorithm is superior to other MOGAS and MOEAs, such as FFGA, NSGAII, SPEA2, and so on, in terms of the precision, quantity and distribution of solutions. Notably, for the engineering problem W, it finds the Pareto-optimal set, which was previously unknown.
本文针对带约束的多目标优化问题（MOPs）提出了一种正交多目标进化算法（OMOEA）。首先，在确定帕累托支配关系时考虑这些约束。结果，得到了一个严格的偏序关系，并且在选择过程中不再考虑可行性。然后，将正交设计和统计优化方法推广到多目标优化问题，构建了一种新型的多目标进化算法（MOEA）。在此框架下，一个原始小生境首先进化，然后分裂成一组子小生境。接着每个子小生境重复上述过程。由于搜索的均匀性、统计的最优性以及小生境分裂频率的指数增长，OMOEA采用确定性搜索，无盲目性或随机性。它能很快产生大量解，这些解高精度且均匀分布地收敛到帕累托最优集。我们采用由Deb、Zitzler等人设计的六个测试问题，以及由Ray等人提供的一个带约束的工程问题（W）来测试这项新技术。数值实验表明，我们的算法在解的精度、数量和分布方面优于其他多目标遗传算法（MOGAS）和多目标进化算法，如FFGA、NSGAII、SPEA2等。值得注意的是，对于工程问题W，它找到了之前未知的帕累托最优集。
