Evolutionary algorithms are popular algorithms for multiobjective optimisation (also called Pareto optimisation) as they use a population to store trade-offs between different objectives. Despite their popularity, the theoretical foundation of multiobjective evolutionary optimisation (EMO) is still in its early development. Fundamental questions such as the benefits of the crossover operator are still not fully understood. We provide a theoretical analysis of well-known EMO algorithms GSEMO and NSGA-II to showcase the possible advantages of crossover. We propose a class of problems on which these EMO algorithms using crossover find the Pareto set in expected polynomial time. In sharp contrast, they and many other EMO algorithms without crossover require exponential time to even find a single Pareto-optimal point. This is the first example of an exponential performance gap through the use of crossover for the widely used NSGA-II algorithm.

翻译：进化算法因利用种群存储不同目标间的权衡关系，已成为多目标优化（亦称帕累托优化）领域的主流算法。然而，尽管应用广泛，多目标进化优化（EMO）的理论基础仍处于早期发展阶段。交叉算子等核心机制的优势尚未得到充分认知。本文对经典EMO算法GSEMO与NSGA-II进行理论分析，以揭示交叉操作的潜在优势。我们提出一类问题，在此类问题上，采用交叉的EMO算法能在期望多项式时间内找到帕累托前沿解集。与之形成鲜明对比的是，若不含交叉操作，这些算法乃至其他众多EMO算法在寻找单个帕累托最优解时都需指数级时间。这是首次针对广泛使用的NSGA-II算法，通过理论证明揭示交叉操作可带来指数级性能差距的案例。