Grid Refinement in Multiscale Dynamic Optimization

In the present work we explore an adaptive discretization scheme for dynamic optimization problems applied to input and state estimation. The proposed method is embedded into a solution methodology where the dynamic optimization problem is approximated by a hierarchy of successively refined finite dimensional problems. Information on the solution of the coarser approximations is used to construct a fully adaptive, problem dependent discretization where the finite dimensional spaces are spanned by biorthogonal wavelets arising from B-splines. We demonstrate exemplarily that the proposed strategy is capable to identify accurate discretization meshes which are more economical than uniform meshes with respect to the ratio of approximation quality vs. number of used trial functions

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. On the other hand, by using suitable decomposition, either through Multiple Shooting or collocation, our 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. Here solution of this NLP formulation is considered through a reduced Hessian Successive Quadratic Programming (SPQ) approach. The routine chosen for the decomposition of the system equations is COLDAE, in which the stable multiple shooting 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 will also provide a proof for the connection of our algorithm and the calculus of variations.

Reformulating ill-conditioned DAE optimization problems

Abstract: "This paper discusses numerical issues in Differential- Algebraic Equation (DAE) optimization concerning the stability and accuracy of the discretized Nonlinear Programming Problems (NLP). First, a brief description of the solution strategy based on reduced-Hessian Successive Quadratic Programming (rSQP) is described, focusing on the decomposition step of the DAE constraints. Next, some difficulties associated with unstable DAE problem formulations are exposed via examples. A new procedure for detecting ill-conditioning and problem reformulation is then presented. Furthermore, some properties of this procedure as well as its limitations are also discussed. Numerical examples are provided, including a flowsheet optimization problem with an unstable reactor.

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.