Convergence of Projected Hessian Approximations in Quasi-Newton Methods for the Nonlinear Programming Problem

Two recently proposed algorithms for the problem of minimizationsubject to nonlinear equality constraints are examined. Bothmaintain quasi-Newton approximations to the projection of theHessian of the Lagrangian, onto the (linearized) manifold ofactive constraints at each iteration. We show that both algorithmscan be used with a more general quasi-Newton updating rule,and, using the analysis of Stoer (1984), that the sequence ofprojected Hessian approximations is convergent.     

Convergence of an Inexact Algorithm for Composite Nonsmooth Optimization

We describe an inexact version of Fletcher's second-order correctionalgorithm for minimizing composite nondifferentiable functions,and add a test which allows global convergence to be demonstratedwithout the assumption that a global minimum of the model functionis found at each iteration. Implementable criteria for acceptinginexact solutions of the subproblem, while retaining local convergenceproperties, are also given.