TRUST-REGION INTERIOR-POINT SQP ALGORITHMS FOR A CLASS OF NONLINEAR-PROGRAMMING PROBLEMS

In this paper, a family of trust-region interior-point sequential quad ratic programming (SQP) algorithms for the solution of a class of mini mization problems with nonlinear equality constraints and simple bound s on some of the variables is described and analyzed. Such nonlinear p rograms arise, e.g., from the discretization of optimal control proble ms. 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 lineari zed constraints but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems a rising from optimal control problems governed by partial differential equations.The algorithms keep strict feasibility with respect to the b ound constraints by using an affine scaling method proposed, for a dif ferent class of problems, by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418-445] and they exploit trust-region techniques for equality-co nstrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques. Globa l convergence of these algorithms to a first-order Karush-Kuhn-Tucker (KKT) limit point is proved under very mild conditions on the trial st eps. Under reasonable, but more stringent, conditions on the quadratic model and on the trial steps, the sequence of iterates generated by t he algorithms is shown to have a limit point satisfying the second-ord er necessary KKT conditions. The local rate of convergence to a nondeg enerate strict local minimizer is q-quadratic. The results given here include, as special cases, current results for only equality constrain ts and for only simple bounds. Numerical results for the solution of a n optimal control problem governed by a nonlinear heat equation are re ported.