Incremental Constraint Projection-Proximal Methods for Nonsmooth Convex Optimization

We consider convex optimization problems with structures that are suitable for stochastic sampling. In particular, we focus on problems where the objective function is an expected value or is a sum of a large number of component functions, and the constraint set is the intersection of a large number of simpler sets. We propose an algorithmic framework for projection-proximal methods using random subgradient/function updates and random constraint updates, which contain as special cases several known algorithms as well as new algorithms. To analyze the convergence of these algorithms in a unied manner, we prove a general coupled convergence theorem. It states that the convergence is obtained from an interplay between two coupled processes: progress towards feasibility and progress towards optimality. Moreover, we consider a number of typical sampling/randomization schemes for the subgradients/component functions and the constraints, and analyze their performance using our unied convergence framework.

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