In this paper, we develop an approximation scheme for solving bilevel programs with equilibrium constraints, which are generally difficult to solve. Among other things, calculating the first-order derivative in such a problem requires differentiation across the hierarchy, which is computationally intensive, if not prohibitive. To bypass the hierarchy, we propose to bound such bilevel programs, equivalent to multiple-followers Stackelberg games, with two new hierarchy-free problems: a $T$-step Cournot game and a $T$-step monopoly model. Since they are standard equilibrium or optimization problems, both can be efficiently solved via first-order methods. Importantly, we show that the bounds provided by these problems -- the upper bound by the $T$-step Cournot game and the lower bound by the $T$-step monopoly model -- can be made arbitrarily tight by increasing the step parameter $T$ for a wide range of problems. We prove that a small $T$ usually suffices under appropriate conditions to reach an approximation acceptable for most practical purposes. Eventually, the analytical insights are highlighted through numerical examples.

翻译：本文提出一种求解均衡约束双层规划的近似方案，该类问题通常难以求解。其中，计算此类问题的一阶导数需要跨越层级进行微分，即使不是不可行的，计算成本也极高。为规避层级结构，我们提出将此类等价于多追随者斯塔克尔伯格博弈的双层规划问题，用两个新的层级无关问题来界定：一个$T$步古诺博弈和一个$T$步垄断模型。由于它们属于标准均衡或优化问题，均可通过一阶方法高效求解。重要的是，我们证明这些问题给出的界——$T$步古诺博弈的上界与$T$步垄断模型的下界——可通过增加步参数$T$在广泛问题范围内任意收紧。我们证明在适当条件下，较小的$T$通常足以达到大多数实际应用可接受的近似精度。最后，通过数值算例突出分析性见解。