  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.
