An Eecient Trust Region Method for Unconstrained Discrete-time Optimal Control Problems 1

Discrete-time optimal control (DTOC) problems are large-scale optimization problems with a dynamic structure. In previous work this structure has been exploited to provide very fast and eecient local procedures. Two examples are the diierential dynamic programming algorithm (DDP) and the stagewise Newton procedure { both require only O(N) operations per iteration, where N is the number of timesteps. Both exhibit a quadratic convergence rate. However, most algorithms in this category do not have a satisfactory global convergence strategy. The most popular global strategy is shifting: this sometimes works poorly due to the lack of automatic adjustment to the shifting element. In this paper we propose a method that incorporates the trust region idea with the local stagewise Newton's method. This method possesses advantages of both the trust region idea and the stagewise Newton's method, i.e., our proposed method has strong global and local convergence properties yet remains economical. Preliminary numerical results are presented to illustrate the behavior of the proposed algorithm. We also collect in the Appendix some DTOC problems that have appeared in the literature.