Browsing by Author "Vicente, Luís N."

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    Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems
    (1994-12) Dennis, J.E.; Heinkenschloss, Matthias; Vicente, Luís N.
    In this paper a family of trust-region interior-point SQP algorithms for the solution of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using a primal-dual affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trust-region techniques for equality-constrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques. Global convergence of these algorithms to a first-order KKT limit point is proved under very mild conditions on the trial steps. Under reasonable, but more stringent conditions on the quadratic model and on the trial steps, the sequence of iterates generated by the algorithms is shown to have a limit point satisfying the second-order necessary KKT conditions. The local rate of convergence to a nondegenerate strict local minimizer is q-quadratic. The results given here include as special cases current results for only equality constraints and for only simple bounds. Numerical results for the solution of an optimal control problem governed by a nonlinear heat equation are reported.
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    What Happens When We Trust a Region That is a Line?
    (1995-04) Vicente, Luís N.
    Line searches and trust regions are two techniques to globalize nonlinear optimization algorithms. We claim that the trust-region technique has built-in an appropriate regularization of ill-conditioned second-order approximation. The question we ask and then answer in this short paper supports this claim. We force the trust-region technique to act like a line search and we accomplish this by always choosing the step along the quasi-Newton direction. We obtain global first-order convergence as long as the condition number of the second-order approximation is uniformly bounded, a condition that is required in line searches but not in trust regions.