  We introduce adaptive sampling methods for stochastic programs with
deterministic constraints. First, we propose and analyze a variant of the
stochastic projected gradient method where the sample size used to approximate
the reduced gradient is determined on-the-fly and updated adaptively. This
method is applicable to a broad class of expectation-based risk measures and
leads to a significant reduction in the individual gradient evaluations used to
estimate the objective function gradient. Numerical experiments with expected
risk minimization and conditional value-at-risk minimization support this
conclusion and demonstrate practical performance and efficacy for both
risk-neutral and risk-averse problems. Second, we propose an SQP-type method
based on similar adaptive sampling principles. The benefits of this method are
demonstrated in a simplified engineering design application featuring
risk-averse shape optimization of a steel shell structure subject to uncertain
loading conditions and model uncertainty.
