Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques for solving the subproblems occurring in the algorithm. Second derivative information can be used, but when it is not available, limited memory quasi-Newton approximations are made. The performance of the code is studied using a set of difficult test problems from the CUTE collection.

and loving encouragement. Contents 1 Introduction 3 1.1 The mathematical problem formulation : : : : : : : : : : : : : : : : : : : : 7 1.2 Notational conventions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2 The Collocation Discretization 11 2.1 Collocation for two point BVP in ODE : : : : : : : : : : : : : : : : : : : : 11 2.1.1 Choice of collocation points : : : : : : : : : : : : : : : : : : : : : : 14 2.1.2 The polynomial representation : : : : : : : : : : : : : : : : : : : : : 14 2.1.3 A tempting combination : : : : : : : : : : : : : : : : : : : : : : : : 15 2.2 Collocation for BVP in DAE with invariants : : : : : : : : : : : : : : : : : 17 2.2.1 DAE models from mechanics : : : : : : : : : : : : : : : : : : : : : : 17 2.2.2 Collocation discretization of two point BVP in DAE : : : : : : :

This work presents a convergence theory for a general class of trust-region-based algorithms for solving the smooth nonlinear programming problem with equality constraints. The results are proved under very mild conditions on the quasi-normal and tangential components of the trial steps. The Lagrange multiplier estimates and the Hessian estimates are assumed to be bounded. In addition, the regularity assumption is not made. In particular, the linear independence of the gradients of the constraints is not assumed. The theory proves global convergence for the class. In particular, it shows that a subsequence of the iteration sequence satisfies one of four types of Mayer-Bliss stationary conditions in the limit. This theory holds for Dennis, El-Alem, and Maciel's class of trust-region-based algorithms. Key Words: Nonlinear programming, equality constrained problems, constrained optimization, global convergence, regularity assumption, augmented Lagrangian, Mayer-Bliss points, stationary p...

In a recent paper, the author (Ref. 1) proposed a trust-region algorithm for solving the problem of minimizing a non-linear function subject to a set of equality constraints. The main feature of the algorithm is that the penalty parameter in the merit function can be decreased whenever it is warranted. He studied the behavior of the penalty parameter and proved several global and local convergence results. One of these results is that there exists a subsequence of the iterates generated by the algorithm, that converges to a point that satisfies the first-order necessary conditions. In the current paper, we show that, for this algorithm, there exists a subsequence of iterates that converges to a point that satisfies both the first-order and the second-order necessary conditions. Key Words : Constrained optimization, equality constrained, penalty parameter, nonmonotonic penalty parameter, convergence, trust-region methods, first-order point, secondorder point, necessary conditions. B 1...