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In this paper a family of trust{region interior{point SQP algorithms for the solution of a class 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 v ariables. They are designed to take a d v antage 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 di erential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an a ne scaling method proposed for a di erent 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 rst{order KKT limit point i s p r o ved 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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It is common engineering practice to use response surface approximations as surrogates for an expensive objective function in engineering design. The rationale is to reduce the number of detailed, costly analyses required during optimization. In earlier work, we developed a rigorous and e ective scheme for managing the interplay between the use of surrogates in the optimization and scheduled progress checks with the expensive analysis so that the process converges to a solution of the original design problem. In this paper, we will report our latest numerical tests with a helicopter rotor design problem which has proved to be a fruitful laboratory for experimentation. The results given here support the use of an ANOVA decomposition on a DACE model to identify the most important optimization variables in an optimal design problem.
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