An Analytical Hessian and Parallel-Computing Approach for Efficient Dynamic Optimization Based on Control-Variable Correlation Analysis

Abstract

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.

An Adaptive Pseudospectral Method for Constrained Dynamic Optimization Problems in Chemical Engineering

“…1 summarizes the results obtained by AGPM and other methods and results given in the literature. The reported best result is 21.82414 obtained by the method combining multiple‐shooting with collocation (MSC) in 41. It is obvious that the performance index 21.8256 achieved by the proposed AGPM approach is slightly better, reflecting the effectiveness of the proposed method.…”

Symbolic elimination in dynamic optimization based on block-triangular ordering

“…This problem is the closest thing to a standard problem of dynamic optimization in the Modelica community, having been used several times for benchmark purposes [5,6,40,51,59]. …”

“…The considered scenario is a short reflux breakdown during steady state, with the objective to steer back to the desired steady state, using quadratic costs on the deviation of two tray temperatures and input signals from the high-purity steady state. The Modelica implementation of this scenario and model has been previously used for benchmarking dynamic optimization algorithms [40,45]. However, the model was then incorrectly implemented, leading to 125 differential variables, instead of the correct 40.…”

A novel analytical second‐order sensitivity calculation approach using the finite element method for chemical engineering problems

“…It is well‐known that the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) method is an effective approximation with its low computational cost for solving NLP problems . The BFGS method uses an approximate Hessian instead of a real Hessian, which reduces the computational complexity of the algorithm . However, the accuracy of the solution with a real Hessian is higher than that of applying the BFGS method, and thus providing an analytical second order sensitivity information for the solver is beneficial for the solution of the problems with a high accuracy requirement .…”

“…[31] The BFGS method uses an approximate Hessian instead of a real Hessian, which reduces the computational complexity of the algorithm. [32,33] However, the accuracy of the solution with a real Hessian is higher than that of applying the BFGS method, and thus providing an analytical second order sensitivity information for the solver is beneficial for the solution of the problems with a high accuracy requirement. [34] Moreover, the introduction of analytical second-order sensitivities can also improve the efficiency of the method for some cases.…”