  Inspired by classical sensitivity results for nonlinear optimization, we
derive and discuss new quantitative bounds to characterize the solution map and
dual variables of a parametrized nonlinear program. In particular, we derive
explicit expressions for the local and global Lipschitz constants of the
solution map of non-convex or convex optimization problems, respectively. Our
results are geared towards the study of time-varying optimization problems
which are commonplace in various applications of online optimization, including
power systems, robotics, signal processing and more. In this context, our
results can be used to bound the rate of change of the optimizer. To illustrate
the use of our sensitivity bounds we generalize existing arguments to quantify
the tracking performance of continuous-time, monotone running algorithms.
Further, we introduce a new continuous-time running algorithm for time-varying
constrained optimization which we model as a so-called perturbed sweeping
process. For this discontinuous scheme, we establish an explicit bound on the
asymptotic solution tracking for a class of convex problems.
