A nested, simultaneous approach for dynamic optimization problems - II: the outer problem

In a previous paper (Tanartkit, P. and Biegler, L. T (1996) A nested, simultaneous approach for dynamic optimization problems - I. Comput. Chem. Eng. 20(6/7), 735-741), we introduced and demonstrated a general framework for solving dynamic optimization using bilevel programming. This framework decouples the element placement from the optimal control procedure and leads to a more robust algorithm. The optimization problem is replaced by two connected but simpler formulations, the inner and outer problems. The inner problem is essentially a dynamic optimization with fixed time steps. On the other hand, the outer problem adjusts the time step given the gradient information from the inner counterpart. By coupling a well-implemented collocation solver with reduced Hessian successive quadratic programming (SQP), we are able to tackle the inner part of the system in an efficient and stable fashion for both initial value and boundary value problems. However, the overall success of the algorithm still depends on robustness and performance of the outer problem. In this article this is achieved by combining a bundle underestimator with SQP Also included in this article are different options of obtaining subgradients for the outer problem via sensitivity analysis and finite difference schemes. Here a decomposition is presented by taking advantage of the inner problem structure to reduce computational expense of the sensitivity evaluation. We will also address the limitations and properties involved in both schemes. In the final segment of the paper the focus is shifted to the issue of finite element addition. By utilizing insight from optimal control theory, we develop a systematic procedure for element addition with a rigorous stopping criterion. Finally, examples are given to illustrate the effectiveness and potential of the algorithm. (C) 1997 Elsevier Science Ltd.