Solving Non-smooth Constrained Programs with Lower Complexity than O(1/ε): A Primal-Dual Homotopy Smoothing Approach
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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(ε^-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(ε^-2/(2+β)_2(ε^-1)), where β∈(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 β=1/2, therefore enjoying a convergence time of O(ε^-4/5_2(ε^-1)). This result improves upon the O(ε^-1) convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm.
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