On the convergence theory of trust-region-based algorithms for equality-constrained optimization

In a recent paper, Dennis, El-Alem, and Maciel proved global convergence to a stationary point for a general trust-region-based algorithm for equality-constrained optimization. This general algorithm is based on appropriate choices of trust-region subproblems and seems particularly suitable for large problems. This paper shows global convergence to a point satisfying the second-order necessary optimality conditions for the same general trust-region-based algorithm. The results given here can be seen as a generalization of the convergence results for trust-regions methods for unconstrained optimization obtained by More and Sorensen. The behavior of the trust radius and the local rate of convergence are analyzed. Some interesting facts concerning the trust-region subproblem for the linearized constraints, the quasi-normal component of the step, and the hard case are presented. It is shown how these results can be applied to a class of discretized optimal control problems.