When solving bilevel optimization problems (BOPs) by evolutionary algorithms (EAs), it is necessary to obtain the lower-level optimal solution for each upper-level solution, which gives rise to a large number of lower-level fitness evaluations, especially for large-scale BOPs. It is interesting to note that some upper-level variables may not interact with some lower-level variables. Under this condition, if the value(s) of one/several upper-level variables change(s), we only need to focus on the optimization of the interacting lower-level variables, thus reducing the dimension of the search space and saving the number of lower-level fitness evaluations. This article proposes a new framework (called GO) to identify and utilize the interactions between upper-level and lower-level variables for scalable BOPs. GO includes two phases: 1) the grouping phase and 2) the optimization phase. In the grouping phase, after identifying the interactions between upper-level and lower-level variables, they are divided into three types of subgroups (denoted as types I–III), which contain only upper-level variables, only lower-level variables, and both upper-level and lower-level variables, respectively. In the optimization phase, if type-I and type-II subgroups only include one variable, a multistart sequential quadratic programming is designed; otherwise, a single-level EA is applied. In addition, a criterion is proposed to judge whether a type-II subgroup has multiple optima. If multiple optima exist, by incorporating the information of the upper level, we design new objective function and degree of constraint violation to locate the optimistic solution. As for type-III subgroups, they are optimized by a bilevel EA (BLEA). The effectiveness of GO is demonstrated on a set of scalable test problems by applying it to five representative BLEAs. Moreover, GO is applied to the resource pricing in mobile edge computing.

中文翻译：

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可扩展双层优化的框架：识别和利用上层和下层变量之间的相互作用

在通过进化算法（EA）解决双层优化问题（BOP）时，需要为每个上层解求出下层最优解，这就产生了大量的下层适应度评估，特别是对于大- 规模的 BOP。有趣的是，一些上层变量可能不会与一些下层变量相互作用。在这种情况下，如果一个/几个上层变量的值发生变化，我们只需要关注相互作用的下层变量的优化，从而减少搜索空间的维数，节省较低级别的适合度评估的数量。本文提出了一个新框架（称为 GO）来识别和利用可扩展 BOP 的上级和下级变量之间的相互作用。GO 包括两个阶段：1) 分组阶段和 2) 优化阶段。在分组阶段，识别出上层变量和下层变量之间的交互作用后，将它们分为三类子组（记为I-III类），其中只包含上层变量，只包含下层变量，以及上层和下层变量，分别。在优化阶段，如果I类和II类子群只包含一个变量，则设计多起点序贯二次规划；否则，将应用单级 EA。此外，提出了判断II型子群是否具有多个最优值的标准。如果存在多个最优解，则通过结合上层的信息，设计新的目标函数和违反约束的程度来定位乐观解。对于 III 型亚组，它们由双层 EA (BLEA) 优化。通过将 GO 应用于五个有代表性的 BLEA，在一组可扩展的测试问题上证明了 GO 的有效性。此外，GO应用于移动边缘计算中的资源定价。