Convergence of approximate and incremental subgradient methods for convex optimization

We present a unified convergence framework for approximate subgradient methods that covers various stepsize rules (including both diminishing and nonvanishing stepsizes), convergence in objective values, and convergence to a neighborhood of the optimal set. We discuss ways of ensuring the boundedness of the iterates and give efficiency estimates. Our results are extended to incremental subgradient methods for minimizing a sum of convex functions, which have recently been shown to be promising for various large-scale problems, including those arising from Lagrangian relaxation.