Large-scale DAE optimization using a simultaneous NLP formulation

The differential-algebraic equation (DAE) optimization problem is transformed to a nonlinear programming problem by applying collocation on finite elements. The resulting problem is solved using a reduced space successive quadratic programming (rSQP) algorithm. Here, the variable space is partitioned into range and null spaces. Partitioning by choosing a pivot sequence for an LU factorization with partial pivoting allows us to detect unstable modes in the DAE system, which can now be stabilized without imposing new boundary conditions. As a result, the range space is decomposed in a single step by exploiting the overall sparsity of the collocation matrix, which performs better than the two-step condensation method used in standard collocation solvers. To deal with ill-conditioned constraints, we also extend the rSQP algorithm to include dogleg steps for the range space step that solves the collocation equations. The performance of this algorithm was tested on two well known unstable problems and on three chemical engineering examples including two reactive distillation columns and a plug-flow reactor with free radicals. One of these is a batch column where an equilibrium reaction takes place. The second reactive distillation problem is the startup of a continuous column with competitive reactions. These optimization problems, which include more than 150 DAEs, are solved in less than 7 CPU minutes on workstation class computers.