Symbolic elimination in dynamic optimization based on block-triangular ordering

Abstract

We consider dynamic optimization problems for systems described by differential-algebraic equations (DAEs). Such problems are usually solved by discretizing the full DAE. We propose techniques to symbolically eliminate many of the algebraic variables in a preprocessing step before discretization. These techniques are inspired by the causalization and tearing techniques often used when solving DAE initial value problems. Since sparsity is crucial for some dynamic optimization methods, we also propose a novel approach to preserving sparsity during this procedure.The proposed methods have been implemented in the open-source JModelica.org platform. We evaluate the performance of the methods on a suite of optimal control problems solved using direct collocation. We consider both computational time and probability of solving the problem in a timely manner. We demonstrate that the proposed methods often are an order of magnitude faster than the standard way of discretizing the full DAE, and also significantly increase probability of successful convergence.

Framework for dynamic optimization of district heating systems using Optimica Compiler Toolkit

“…As demonstrated in (Magnusson and Åkesson, 2016), and as we will also see is the case in this work, the symbolic elimination not only reduces the solution time, but also improves convergence robustness, that is, probability of successfully solving an optimization problem in a timely manner.…”

“…There has been recent work carried out (Magnusson and Åkesson, 2016) to address this problem in general by applying a block-lower triangular (BLT) transformation of the DAE to identify algebraic variables that only depend affinely on the corresponding block variables. This allows symbolic elimination of such variables by forward substitution.…”

Failure Modes of Tearing and a Novel Robust Approach

“…These enhancements share that the computation of y for a given z only involves fast and numerically stable algorithms such as solving implicit univariate equations or small systems of equations. Another recent approach tries to balance between minimizing the border width and preserving the sparsity during the elimination (Magnusson and Åkesson, 2017). The reader is referred to (Baharev et al, 2017a) for an in-depth discussion of the variations on tearing.…”