Complexity of an inexact stochastic SQP algorithm for equality constrained optimization

TL;DR

This paper introduces an inexact stochastic SQP algorithm for nonlinear optimization with stochastic objectives and equality constraints, providing the first complexity guarantees and demonstrating superior convergence in experiments.

Contribution

It presents a novel inexact two-stepsize stochastic SQP method with proven worst-case complexity bounds and validates its effectiveness through numerical experiments.

Findings

Achieves $ ext{O}(rac{1}{\e_c^2})$ complexity for infeasibility without constraint qualification.

Attains $ ext{O}(rac{1}{\e_c})$ complexity under LICQ.

Demonstrates superior infeasibility convergence and competitive KKT rates in experiments.

Abstract

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.