  In this paper, we investigate regrets of an online semi-proximal alternating
direction method of multiplier (Online-spADMM) for solving online linearly
constrained convex composite optimization problems. Under mild conditions, we
establish ${\rm O}(\sqrt{N})$ objective regret and ${\rm O}(\sqrt{N})$
constraint violation regret at round $N$ when the dual step-length is taken in
$(0,(1 +\sqrt{5})/2)$ and penalty parameter $\sigma$ is taken as $\sqrt{N}$. We
explain that the optimal value of parameter $\sigma$ is of order ${\rm
O}(\sqrt{N})$. Like the semi-proximal alternating direction method of
multiplier (spADMM), Online-spADMM has the advantage to resolve the potentially
non-solvability issue of the subproblems efficiently. We show the usefulness of
the obtained results when applied to different types of online optimization
problems and verify the theoretical result by numerical experiments}. The
inequalities established for Online-spADMM are also used to develop iteration
complexity of the average update of spADMM for solving linearly constrained
convex composite optimization problems.
