  The primary goal of this paper is to provide an efficient solution algorithm
based on the augmented Lagrangian framework for optimization problems with a
stochastic objective function and deterministic constraints. Our main
contribution is combining the augmented Lagrangian framework with adaptive
sampling, resulting in an efficient optimization methodology validated with
practical examples. To achieve the presented efficiency, we consider inexact
solutions for the augmented Lagrangian subproblems, and through an adaptive
sampling mechanism, we control the variance in the gradient estimates.
Furthermore, we analyze the theoretical performance of the proposed scheme by
showing equivalence to a gradient descent algorithm on a Moreau envelope
function, and we prove sublinear convergence for convex objectives and linear
convergence for strongly convex objectives with affine equality constraints.
The worst-case sample complexity of the resulting algorithm, for an arbitrary
choice of penalty parameter in the augmented Lagrangian function, is
$O(\epsilon^{-3-\delta})$, where $\epsilon > 0$ is the expected error of the
solution and $\delta > 0$ is a user-defined parameter. If the penalty parameter
is chosen to be $O(\epsilon^{-1})$, we demonstrate that the result can be
improved to $O(\epsilon^{-2})$, which is competitive with the other methods
employed in the literature. Moreover, if the objective function is strongly
convex with affine equality constraints, we obtain
$O(\epsilon^{-1}\log(1/\epsilon))$ complexity. Finally, we empirically verify
the performance of our adaptive sampling augmented Lagrangian framework in
machine learning optimization and engineering design problems, including
topology optimization of a heat sink with environmental uncertainty.
