We cons...

This paper considers the connection between the intrinsic Riemannian Newton method and other more classically inspired optimization algo-rithms for equality-constrained optimization problems. We consider the feasibly-projected sequential quadratic programming (FP-SQP) method and show that it yields the same update step as the Riemannian Newton, subject to a minor assumption on the choice of multiplier vector. We also consider Newton update steps computed in various ‘natural ’ local coordinate systems on the constraint manifold and find simple condi-tions that guarantee that the update step is the Riemannian Newton update. In particular, we show that this is the case for projective local coordinates, one of the most natural choices that have been proposed in the literature. Finally we consider the case where the full constraints are approximated to simplify the computation of the update step. We show that if this approximation is good at least to second-order then the resulting update step is the Riemannian Newton update. The conclu-sion of this study is that the intrinsic Riemannian Newton algorithm is the archetypal feasible second order update for non-degenerate equality constrained optimisation problems. Key words. Feasibly-projected sequential quadratic programming (FP-SQP); equality-constrained optimization; Riemannian manifold; Riemannian Newton method; sequential Newton method; retraction; osculating paraboloid; second-order correction; second funda