  This paper considers an online proximal-gradient method to track the
minimizers of a composite convex function that may continuously evolve over
time. The online proximal-gradient method is inexact, in the sense that: (i) it
relies on an approximate first-order information of the smooth component of the
cost; and, (ii) the proximal operator (with respect to the non-smooth term) may
be computed only up to a certain precision. Under suitable assumptions,
convergence of the error iterates is established for strongly convex cost
functions. On the other hand, the dynamic regret is investigated when the cost
is not strongly convex, under the additional assumption that the problem
includes feasibility sets that are compact. Bounds are expressed in terms of
the cumulative error and the path length of the optimal solutions. This
suggests how to allocate resources to strike a balance between performance and
precision in the gradient computation and in the proximal operator.
