  Before a car-following model can be applied in practice, it must first be
validated against real data in a process known as calibration. This paper
discusses the formulation of calibration as an optimization problem, and
compares different algorithms for its solution. The optimization consists of an
arbitrary car following model, posed as either an ordinary or delay
differential equation, being calibrated to an arbitrary source of trajectory
data which may include lane changes. Typically, the calibration problem is
solved using gradient free optimization. In this work, the gradient of the
optimization problem is derived analytically using the adjoint method. The
computational cost of the adjoint method does not scale with the number of
model parameters, which makes it more efficient than evaluating the gradient
numerically using finite differences. Numerical results are presented which
show that quasi-newton algorithms using the adjoint method are significantly
faster than a genetic algorithm, and also achieve slightly better accuracy of
the calibrated model.
