On an Approximation of the Hessian of the Lagrangian

In the context of SQP methods or, more recently, of sequential semidefinite programming methods, it is common practice to construct a positive semidefinite approximation of the Hessian of the Lagrangian. The Hessian of the augmented Lagrangian is a suitable approximation as it maintains local superlinear convergence under appropriate assumptions. In this note we give a simple example that the orthogonal projection of the Hessian of the Lagrangian onto the cone of semidefinite matrices may lead to arbitrarily slow local convergence, and is thus not a suitable approximation.