A Smoothed Augmented Lagrangian Framework for Convex Optimization with Nonsmooth Stochastic Constraints

We consider a convex stochastic optimization problem where both the objective and constraints are convex but possibly complicated by uncertainty and nonsmoothness. We present a smoothed sampling-enabled augmented Lagrangian (AL) framework that relies on inexact solutions to the AL subproblem, obtainable via a stochastic approximation framework. Under a constant penalty parameter, the dual suboptimality is shown to diminish at a sublinear rate while primal infeasibility and suboptimality both diminish at a slower sublinear rate.