  We consider solving nonlinear optimization problems with a stochastic
objective and deterministic equality constraints. We assume for the objective
that its evaluation, gradient, and 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 design, we first revisit and simplify an old SQP method
\citep{Lucidi1990Recursive} developed for solving deterministic problems, which
serves as the skeleton of our stochastic algorithm. Based on the simplified
deterministic algorithm, we then propose a non-adaptive SQP for dealing with
stochastic objective, where the gradient and Hessian are replaced by stochastic
estimates but the stepsizes are deterministic and prespecified. Finally, we
incorporate a recent stochastic line search procedure
\citep{Paquette2020Stochastic} into the non-adaptive stochastic SQP to
adaptively select the random stepsizes, which leads to an adaptive stochastic
SQP. The global "almost sure" convergence for both non-adaptive and adaptive
SQP methods is established. Numerical experiments on nonlinear problems in
CUTEst test set demonstrate the superiority of the adaptive algorithm.
