A Two Stepsize SQP Method for Nonlinear Equality Constrained Stochastic Optimization

Abstract: We develop a Sequential Quadratic Optimization (SQP) algorithm for minimizing a stochastic objective function subject to deterministic equality constraints. The method utilizes two different stepsizes, one which exclusively scales the component of the step corrupted by the variance of the stochastic gradient estimates and a second which scales the entire step. We prove that this stepsize splitting scheme has a worst-case complexity result which improves over the best known result for this class of problems. In terms of approximately satisfying the constraint violation, this complexity result matches that of deterministic SQP methods, up to constant factors, while matching the known optimal rate for stochastic SQP methods to approximately minimize the norm of the gradient of the Lagrangian. We also propose and analyze multiple variants of our algorithm. One of these variants is based upon popular adaptive gradient methods for unconstrained stochastic optimization while another incorporates a safeguarded line search along the constraint violation. Preliminary numerical experiments show competitive performance against a state of the art stochastic SQP method. In addition, in these experiments, we observe an improved rate of convergence in terms of the constraint violation, as predicted by the theoretical results.