  We consider online fractional covering problems with a convex objective,
where the covering constraints arrive over time. Formally, we want to solve
$\min\,\{f(x) \mid Ax\ge \mathbf{1},\, x\ge 0\},$ where the objective function
$f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex, and the constraint matrix
$A_{m\times n}$ is non-negative. The rows of $A$ arrive online over time, and
we wish to maintain a feasible solution $x$ at all times while only increasing
coordinates of $x$. We also consider "dual" packing problems of the form
$\max\,\{c^\intercal y - g(\mu) \mid A^\intercal y \le \mu,\, y\ge 0\}$, where
$g$ is a convex function. In the online setting, variables $y$ and columns of
$A^\intercal$ arrive over time, and we wish to maintain a non-decreasing
solution $(y,\mu)$.
  We provide an online primal-dual framework for both classes of problems with
competitive ratio depending on certain "monotonicity" and "smoothness"
parameters of $f$; our results match or improve on guarantees for some special
classes of functions $f$ considered previously.
  Using this fractional solver with problem-dependent randomized rounding
procedures, we obtain competitive algorithms for the following problems: online
covering LPs minimizing $\ell_p$-norms of arbitrary packing constraints, set
cover with multiple cost functions, capacity constrained facility location,
capacitated multicast problem, set cover with set requests, and profit
maximization with non-separable production costs. Some of these results are new
and others provide a unified view of previous results, with matching or
slightly worse competitive ratios.
