Complexity of an inexact stochastic SQP algorithm for equality constrained optimization

In this paper, we consider nonlinear optimization problems with a stochastic objective function and deterministic equality constraints. We propose an inexact two-stepsize stochastic sequential quadratic programming (SQP) algorithm and analyze its worst-case complexity under mild assumptions. The method utilizes a step decomposition strategy and handles stochastic gradient estimates by assigning different stepsizes to different components of the search direction. We establish the first known $\mathcal{O}(\epsilon_c^{-2})$ worst-case complexity with respect to the infeasibility measure when no constraint qualification is assumed and a worst-case complexity of $\mathcal{O}(\epsilon_c^{-1})$ when LICQ holds, matching the best known result in the literature. In addition, under mild conditions, our method achieves the optimal $\mathcal{O}(\epsilon_L^{-4})$ complexity with respect to the gradient of the Lagrangian regardless of constraint qualifications. Our results provide the first complexity guarantees for the popular Byrd-Omojukun step decomposition strategy and verify its theoretical efficacy. Numerical experiments show that our algorithm has a superior infeasibility convergence performance and a competitive KKT convergence rate compared to the state-of-the-art stochastic SQP method.