  In this paper, we address a coordination problem for connected and autonomous
vehicles (CAVs) in mixed traffic settings with human-driven vehicles (HDVs).
The main objective is to have a safe and optimal crossing order for vehicles
approaching unsignalized intersections. This problem results in a mixed-integer
quadratic programming (MIQP) formulation which is unsuitable for real-time
applications. Therefore, we propose a computationally tractable
optimization-based heuristic that monitors platoons of CAVs and HDVs to
evaluate whether alternative crossing orders can perform better. It first
checks the future constraint violation that consistently occurs between pairs
of platoons to determine a potential swap. Next, the costs of quadratic
programming (QP) formulations associated with the current and alternative
orders are compared in a depth-first branching fashion. In simulations, we show
that the heuristic can be a hundred times faster than the original and
simplified MIQPs and yields solutions that are close to optimal and have better
order consistency.
