Solving Non-smooth Constrained Programs with Lower Complexity than \mathcal{O}(1/\varepsilon): A Primal-Dual Homotopy Smoothing Approach

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 $\mathcal{O}\l(\varepsilon {-2/(2 \beta)}\log_2(\varepsilon {-1})\r)$, where $\beta\in(0,1]$ is a local error bound parameter. As an example application, 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 $\mathcal{O}\l(\varepsilon {-4/5}\log_2(\varepsilon {-1})\r)$. This result improves upon the $\mathcal{O}(\varepsilon {-1})$ convergence time bound achieved by existing distributed optimization algorithms.