Solving Non-smooth Constrained Programs with Lower Complexity than 𝒪(1/ε): 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 O(e−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 O\l(e−2/(2+β)log2(e−1)\r), where β∈(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 β=1/2, therefore enjoying a convergence time of O\l(e−4/5log2(e−1)\r). This result improves upon the O(e−1) convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm.