  We propose a new primal-dual homotopy smoothing algorithm for a linearly
constrained convex program, where neither the primal nor the dual function has
to be smooth or strongly convex. The best known iteration complexity solving
such a non-smooth problem is $\mathcal{O}(\varepsilon^{-1})$. In this paper, we
show that by leveraging a local error bound condition on the dual function, the
proposed algorithm can achieve a better primal convergence time of
$\mathcal{O}\left(\varepsilon^{-2/(2+\beta)}\log_2(\varepsilon^{-1})\right)$,
where $\beta\in(0,1]$ is a local error bound parameter. As an example
application of the general algorithm, we show that the distributed geometric
median problem, which can be formulated as a constrained convex program, has
its dual function non-smooth but satisfying the aforementioned local error
bound condition with $\beta=1/2$, therefore enjoying a convergence time of
$\mathcal{O}\left(\varepsilon^{-4/5}\log_2(\varepsilon^{-1})\right)$. This
result improves upon the $\mathcal{O}(\varepsilon^{-1})$ convergence time bound
achieved by existing distributed optimization algorithms. Simulation
experiments also demonstrate the performance of our proposed algorithm.
