An algorithm for bilevel optimization with traffic equilibrium constraints: convergence rate analysis

Bilevel optimization with traffic equilibrium constraints plays an important role in transportation planning and management problems such as traffic control, transport network design, and congestion pricing. In this paper, we consider a double-loop gradient-based algorithm to solve such bilevel problems and provide a non-asymptotic convergence guarantee of $\mathcal{O}(K^{-1})+\mathcal{O}(\lambda^D)$ where $K$, $D$ are respectively the number of upper- and lower-level iterations, and $0<\lambda<1$ is a constant. Compared to existing literature, which either provides asymptotic convergence or makes strong assumptions and requires a complex design of step sizes, we establish convergence for choice of simple constant step sizes and considering fewer assumptions. The analysis techniques in this paper use concepts from the field of robust control and can potentially serve as a guiding framework for analyzing more general bilevel optimization algorithms.