Bilevel optimization is a field of significant theoretical and practical interest, yet solving such optimization problems remains challenging. Evolutionary methods have been employed to address these problems in the black-box setting; however, they incur high computational cost due to the nested nature of bilevel optimization. Although previous methods have attempted to reduce this cost through various heuristic techniques, such approaches limit versatility on challenging optimization landscapes, such as those with multimodality and significant interaction between upper- and lower-level decision variables. In this study, we propose an efficient framework that exploits the invariance of rank-based evolutionary algorithms to monotonic transformations, thereby reducing the computational burden of the lower-level optimization loop. Specifically, our method directly approximates the rankings of the upper-level value function, bypassing the need to run the lower-level optimizer until convergence for each upper-level iteration. We apply this framework to the setting where both levels are continuous, adopting CMA-ES as the optimizer. We demonstrate that our method achieves competitive performance on standard bilevel optimization benchmarks and can solve problems that are intractable with previously proposed methods, particularly those with multimodality and strong inter-variable interactions.

翻译：双层优化在理论和实践中具有重要意义，但求解此类优化问题仍具挑战性。进化方法已被用于解决黑箱设置下的此类问题，但由于双层优化的嵌套特性，其计算成本高昂。尽管先前的方法尝试通过多种启发式技术降低这一成本，但此类方法在具有多模态性及上下层决策变量间显著相互作用的复杂优化景观中限制了通用性。本研究提出一种高效框架，利用基于秩的进化算法对单调变换的不变性，以降低下层优化循环的计算负担。具体而言，我们的方法直接逼近上层值函数的排序，从而避免每次上层迭代时运行下层优化器直至收敛。我们将该框架应用于上下层均为连续变量的场景，并采用CMA-ES作为优化器。实验表明，该方法在标准双层优化基准测试中取得了具有竞争力的性能，并能解决先前方法难以处理的复杂问题，尤其是具有多模态性和强变量间相互作用的问题。