Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower's best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer linear bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.

翻译：双层优化处理的是嵌套问题，其中领导者首先做出决策以最小化其目标函数，同时考虑跟随者的最优响应反应。包含整数变量的约束双层问题因其求解难度而尤为棘手。虽然针对混合整数线性双层优化已提出精确求解器，但它们往往随问题规模增大而扩展性不佳，且难以推广至非线性情形。另一方面，针对特定问题的算法（精确算法与启发式算法）适用范围有限。在数据驱动的背景下，当需要常规性地求解双层问题的相似实例时，我们提出的框架Neur2BiLO将领导者或跟随者价值函数的神经网络近似（通过监督回归训练得到）嵌入到一个易于求解的混合整数规划中。Neur2BiLO作为一种启发式方法，在四个具有线性和非线性目标函数、纯整数和混合整数变量的应用案例中，能够极快地生成高质量解。