  The motivation for this paper stems from the desire to develop an adaptive
sampling method for solving constrained optimization problems in which the
objective function is stochastic and the constraints are deterministic. The
method proposed in this paper
  is a proximal gradient method that can also be applied to the composite
optimization problem min f(x) + h(x), where f is stochastic and h is convex
(but not necessarily differentiable). Adaptive sampling methods employ a
mechanism for gradually improving the quality of the gradient approximation so
as to keep computational cost to a minimum. The mechanism commonly employed in
unconstrained optimization is no longer reliable in the constrained or
composite optimization settings because it is based on pointwise decisions that
cannot correctly predict the quality of the proximal gradient step. The method
proposed in this paper measures the result of a complete step to determine if
the gradient approximation is accurate enough; otherwise a more accurate
gradient is generated and a new step is computed. Convergence results are
established both for strongly convex and general convex f. Numerical
experiments are presented to illustrate the practical behavior of the method.
