Simultaneous Framework for Dynamic Optimization Based on Density Functions

Regarding the discretization issue in the process of a simultaneous approach for dynamic optimization problems, a configuration strategy based on the density function has been proposed for the finite element distribution of dynamic optimization problems. By utilizing the error at the non-collocation points, a bilevel problem has been constructed and solved, and the number and distribution of the finite elements have been evaluated. For the inner problem in the bilevel problem, a new smoothing function has been introduced to improve the solution accuracy of the inner problem. The grid density function has been constructed using the error at the non-collocation points on each finite element. Finally, the grid density function has been used to update the positions of the finite element endpoints, with the aim of reallocating the finite elements. Finally, two case studies have been provided to specifically demonstrate the role of the proposed method in reducing the optimization problem scale with required solution accuracy.