  We consider the problem of solving nonlinear optimization programs with
stochastic objective and deterministic equality constraints. We assume for the
objective that the function evaluation, the gradient, and the Hessian are
inaccessible, while one can compute their stochastic estimates by, for example,
subsampling. We propose a stochastic algorithm based on sequential quadratic
programming (SQP) that uses a differentiable exact augmented Lagrangian as the
merit function. To motivate our algorithm, we revisit an old SQP method
\citep{Lucidi1990Recursive} developed for deterministic programs. We simplify
that method and derive an adaptive SQP, which serves as the skeleton of our
stochastic algorithm. Based on the derived algorithm, we then propose a
non-adaptive SQP for optimizing stochastic objectives, where the gradient and
the Hessian are replaced by stochastic estimates but the stepsize is
deterministic and prespecified. Finally, we incorporate a recent stochastic
line search procedure \citep{Paquette2020Stochastic} into our non-adaptive
stochastic SQP to arrive at an adaptive stochastic SQP. To our knowledge, the
proposed algorithm is the first stochastic SQP that allows a line search
procedure and the first stochastic line search procedure that allows the
constraints. The global convergence for all proposed SQP methods is
established, while numerical experiments on nonlinear problems in the CUTEst
test set demonstrate the superiority of the proposed algorithm.
