  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.
