Boundary value approach for dynamic optimization

Abstract: "Process engineering provides a wealth of applications for dynamic optimization problems. This problem is usually solved by transforming it to a nonlinear programming (NLP) problem with either sequential or simultaneous approaches. However, both approaches can still be inefficient to tackle large problems. In addition, many problems in chemical engineering are naturally boundary value problems (BVP) which suffer from instability if we utilize a decomposition based on single shooting solvers. In this paper, we will introduce a simple extension to the simultaneous approach that will alleviate the dimensionality problem as well as ensure stability for BVP's. Many numerical aspects of the problem will be discussed, especially the discretization of the differential equations and the index problem. By using Radau collocation, the algorithm has favorable stability properties for high index problems and by exploiting the structure of the resulting system, a stable and efficient decomposition algorithm results. Here solution of this NLP formulation is considered through a reduced Hessian Successive Quadratic Programming (SQP) approach, where linearized state variables are eliminated and reduced quadratic programming (QP) subproblems update the control variables. Although this study primarily addresses fixed element problems, another key aspect of the success of the DAE optimization is the element placement. In order to enforce accuracy in the solution profiles, highly nonlinear constraints have to be added and these further complicate the solution formulation. As a result, the formulation turns out to be very sensitive to initializations. To address these problems, we will introduce a new framework that will decouple the element placement from the optimal control procedure. This framework consists of two layers of optimization, the inner and outer problems. The inner problem is a traditional optimal control problem with fixed element sizes and the outer problem is then used to update them solely via error control criteria and optimality conditions. We will also present an example to illustrate the element placement via bilevel optimization."