  This paper concerns model reduction of dynamical systems using the nuclear
norm of the Hankel matrix to make a trade-off between model fit and model
complexity. This results in a convex optimization problem where this trade-off
is determined by one crucial design parameter. The main contribution is a
methodology to approximately calculate all solutions up to a certain tolerance
to the model reduction problem as a function of the design parameter. This is
called the regularization path in sparse estimation and is a very important
tool in order to find the appropriate balance between fit and complexity. We
extend this to the more complicated nuclear norm case. The key idea is to
determine when to exactly calculate the optimal solution using an upper bound
based on the so-called duality gap. Hence, by solving a fixed number of
optimization problems the whole regularization path up to a given tolerance can
be efficiently computed. We illustrate this approach on some numerical
examples.
