Achieving Hierarchy-Free Approximation for Bilevel Programs with Equilibrium Constraints

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.