Stable Decomposition for Dynamic Optimization

Abstract

Dynamic optimization problems are usually solved by transforming them to nonlinear programming (NLP) problems with either sequential or simultaneous approaches. However, both approaches can still be inefficient to tackle complex problems. In addition, many problems in chemical engineering have unstable components which lead to unstable intermediate profiles during the solution procedure. If the numerical algorithm chosen utilizes an initial value formulation, the error from decomposition or integration can accumulate and the Newton iterations then fail as a result of ill-conditioned constraint matrix. On the other hand, by using suitable decomposition, through multiple shooting or simultaneous collocation, the author's algorithm has favorable numerical characteristics for both stable and unstable problems; by exploiting the structure of the resulting system, a stable and efficient decomposition algorithm results. In the paper, the new algorithm for solving dynamic optimization is developed. This algorithm is based on the nonlinear programming (NLP) formulation coupled with a stable decomposition of the collocation equations. Solution of this NLP formulation is considered through a reduced Hessian successive quadratic programming (SQP) approach. The routine chosen for the decomposition of the system equations is COLDAE, in which the stable collocation scheme is implemented. To address the mesh selection, we will introduce a new bilevel framework that will decouple the element placement from the optimal control procedure. We also provide a proof for the connection of the algorithm and the calculus of variations