Analysis of Sequential Quadratic Programming through the Lens of Riemannian Optimization

TL;DR

This paper analyzes the convergence properties of a first-order Sequential Quadratic Programming algorithm for equality constrained optimization, revealing its local linear convergence rate linked to the Riemannian Hessian's condition number.

Contribution

It establishes a novel connection between SQP and Riemannian optimization, providing new insights into their similar behavior near the constraint manifold.

Findings

Local linear convergence rate depends on Riemannian Hessian condition number

Global convergence rate is approximately $k^{-1/4}$

SQP and Riemannian gradient methods exhibit similar behavior near constraints

Abstract

We prove that a "first-order" Sequential Quadratic Programming (SQP) algorithm for equality constrained optimization has local linear convergence with rate $(1-1/\kappa_R)^k$, where $\kappa_R$ is the condition number of the Riemannian Hessian, and global convergence with rate $k^{-1/4}$. Our analysis builds on insights from Riemannian optimization -- we show that the SQP and Riemannian gradient methods have nearly identical behavior near the constraint manifold, which could be of broader interest for understanding constrained optimization.