A Framework for Scalable Bilevel Optimization: Identifying and Utilizing the Interactions Between Upper-Level and Lower-Level Variables

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.