For equality constrained smooth optimization problems the Hessian of the Lagrangian is often known only partially, i.e. in form of an analytic part and a numerical approximation. Proceeding from this situation, multilevel least-change updates are defined. A q-superlinear convergence statement is derived from existing results and applied to a simple isoperimetrical problem. Similar results are obtained for updates approximating the inverse Hessian, and for methods without derivatives, i.e. which approximate them numerically. In addition, methods are investigated that employ projections of fragments of the augmented Hessian matrix of the Lagrangian.