Solving nonsmooth convex optimization with complexity O(ε)

This paper describes an algorithm for solving structured nonsmooth convex optimization problems using the optimal subgradient algorithm (OSGA), which is a first-order method with the complexity O(ε) for Lipschitz continuous nonsmooth problems and O(ε) for smooth problems with Lipschitz continuous gradient. If the nonsmoothness of the problem is manifested in a structured way, we reformulate the problem in a form that can be solved efficiently by OSGA with the complexity O(ε). To solve the reformulated problem, we equip OSGA by an appropriate prox-function for which the OSGA subproblem can be solved either in a closed form or by a simple iterative scheme, which decreases the computational cost of applying the algorithm, especially for large-scale problems. We show that applying the new scheme is feasible for many problems arising in applications. Some numerical results are reported.