  This paper considers a nonconvex optimization problem that evolves over time,
and addresses the synthesis and analysis of regularized primal-dual gradient
methods to track a Karush-Kuhn-Tucker (KKT) trajectory. The proposed
regularized primal-dual gradient methods are implemented in a running fashion,
in the sense that the underlying optimization problem changes during the
iterations of the algorithms. For a problem with twice continuously
differentiable cost and constraints, and under a generalization of the
Mangasarian-Fromovitz constraint qualification, sufficient conditions are
derived for the running algorithm to track a KKT trajectory. Further,
asymptotic bounds for the tracking error (as a function of the time-variability
of a KKT trajectory) are obtained. A continuous-time version of the algorithm,
framed as a system of differential inclusions, is also considered and
analytical convergence results are derived. For the continuous-time setting, a
set of sufficient conditions for the KKT trajectories not to bifurcate or merge
is proposed. Illustrative numerical results inspired by a real-world
application are provided.
