The approach of combined multiple-shooting with collocation is efficient for solving large-scale dynamic optimization problems. The aim of this work was to further improve its computational performance by providing an analytical Hessian and realizing a parallel-computing scheme. First, we derived the formulas for computing the second-order sensitivities for the combined approach. Second, a correlation analysis of control variables was introduced to determine the necessity of employing the analytical Hessian to solve an optimization problem. Third, parallel computing was implemented thanks to the nature of the combined approach, because the solutions of model equations and evaluations of both first-order and second-order sensitivities for individual time intervals are decoupled. Because these computations are expensive, a high speedup factor was gained through the parallelization. The performance of the proposed analytical Hessian, correlation analysis, and parallel computing is demonstrated in this article by benchmark problems including optimal control of a distillation column containing more than 1000 dynamic variables.

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To numerically solve a dynamic optimization problem, the model equations need to be discretized over a time horizon. The very first step therefore is to decide the number of time intervals. In principle, the decision is made to achieve a compromise between the numerical accuracy of the discretization and the computation load for solving the discretized optimization problem. However, there have been no comprehensive rules for this purpose. In the context of collocation on finite elements, we propose a novel bilevel approach to evaluate an upper limit of the approximation error by formulating and solving an error maximization problem. In this way, a minimum number of time intervals can be determined a priori, which guarantees a userdefined error tolerance. In addition, the impact of the initial conditions on the maximum approximation error is taken into account so that the determined number of intervals is valid for varying initial conditions and thus can be applied to nonlinear model predictive control. Several case studies are used to demonstrate the efficacy of the proposed approach.

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Significant progresses in developing approaches to dynamic optimization have been made. However, its practical implementation poses a difficult task and its realtime application such as in nonlinear model predictive control (NMPC) remains challenging. A toolchain is developed in this work to relieve the implementation burden and, meanwhile, to speed up the computations for solving the dynamic optimization problem. To achieve these targets, symbolic computing is utilized for calculating the first and second order sensitivities on the one hand and parallel computing is used for separately accomplishing the computations for the individual time intervals on the other hand. Two optimal control problems are solved to demonstrate the efficiency of the developed toolchain which solves one of the problems with approximately 25,000 variables within a reasonable CPU time.