# Dynamic Optimization Using Local Collocation Methods and Improved Multiresolution Technique
## 1 Introduction
However, it is not easy to determine the number and the locations of the knots needed to solve a nonsmooth dynamic optimization problem. In most cases, the abovementioned mesh refinement methods for global collocation methods generate a larger number of segments and mesh points [34; 35; 36; 37; 38]. Among the various mesh refinement methods, MRTs [25; 26; 27] are quite attractive because they only require simple analysis of the local interpolation error to refine the mesh, and avoid the complex analysis that is needed in other mesh refinement methods [14; 17; 18; 28; 29; 30; 31; 34; 35; 36; 38]. Moreover, the mesh point distribution of a local collocation method can be freely displaced according to the solution of the dynamic optimization problem, which is more flexible for accurately capturing the irregularities in the solution. In contrast, the mesh point distribution of a global collocation method is not so flexible because it is dictated by the roots of the first derivative of a certain global polynomial in each mesh interval. However, the MRTs of [25; 26; 27] still suffer from two disadvantages. First of all, the MRTs of [25; 26] are based on dyadic meshes. Dyadic meshes consist of a hierarchy of meshes obtained from successively bisection of the unit interval [0, 1]. As a result of the dyadic bisection, the MRTs of [25; 26] must start with an initial mesh consisting of \(2^{j}+1\) mesh points (where \(j\) is an integer). Although a generalized dyadic mesh has been proposed in [27] to overcome this limitation, the method of [27] must start with an initial mesh with a mesh size twice that of the coarsest mesh. This inconsistency between the initial mesh and the coarsest mesh makes it inconvenient to use in practical applications. Secondly, although the MRTs of [25; 26; 27] are quite efficient mesh refinement methods, they still generate some redundant mesh points in the solution. The efficiency of MRTs can be further improved if the redundant mesh points are removed. It is quite challenging to remove all the redundant mesh points, but it is possible to remove most of the redundant mesh points in the constant-control regions. Therefore, the other purpose of this paper is to improve the MRTs of [25; 26; 27] to overcome the abovementioned disadvantages. We first propose new generalized dyadic meshes so that the MRT can start with an initial mesh consisting of an odd number of mesh points rather than being limited to \(2^{j}+1\) mesh points. Furthermore, the MRT developed based on the new generalized dyadic meshes ensures that the initial optimization starts with the coarsest mesh. Finally, a method to remove the redundant points generated by MRTs in the constant control regions is developed by analyzing the control slopes. We currently only remove the redundant mesh points in the constant control regions because it is very common to have some constant control regions in the control solution for constrained dynamic optimization problems, e.g., the bang-bang control problem. The rest of this paper is organized as follows. In Section 2, the general continuous dynamic optimization problem is stated and the NLP formulations from the local collocation methods based on Runge-Kutta (RK) discretizations are described. In Section 3, the improved MRT, which avoids the disadvantages of the MRTs of [25; 26; 27], is presented. In Section 4, the effectiveness of the method is demonstrated on three chemical process and bioprocess optimization problems and compared with other mesh refinement methods. In Section 5, the conclusion is summarized.