  Devising efficient algorithms that track the optimizers of continuously
varying convex optimization problems is key in many applications. A possible
strategy is to sample the time-varying problem at constant rate and solve the
resulting time-invariant problem. This can be too computationally burdensome in
many scenarios. An alternative strategy is to set up an iterative algorithm
that generates a sequence of approximate optimizers, which are refined every
time a new sampled time-invariant problem is available by one iteration of the
algorithm. These types of algorithms are called running. A major limitation of
current running algorithms is their key assumption of strong convexity and
strong smoothness of the time-varying convex function. In addition, constraints
are only handled in simple cases. This limits the current capability for
running algorithms to tackle relevant problems, such as $\ell_1$-regularized
optimization programs. In this paper, these assumptions are lifted by
leveraging averaged operator theory and a fairly comprehensive framework for
time-varying convex optimization is presented. In doing so, new results
characterizing the convergence of running versions of a number of widely used
algorithms are derived.
