# Dynamic Optimization Using Local Collocation Methods and Improved Multiresolution Technique
## 1 Introduction
The present paper focuses on the application of direct collocation methods in conjunction with mesh refinement methods to chemical process and bioprocess optimization problems. The solutions of many practical dynamic optimization problems are nonsmooth due to path or terminal constraints in the problem or nonsmooth problem formulations [25; 26; 27; 28]. In order to solve these nonsmooth problems accurately using the direct collocation method, a globally dense mesh is not a good choice, because using very fine mesh may require large computational recourses, i.e., central processing unit (CPU) time and random-access memory (RAM). Furthermore, the large number of decision variables may make the resulting NLP problem poorly conditioned, which makes it very difficult or even impossible to solve by the NLP solvers [17]. Motivated by the desire to solve the dynamic optimization problems accurately while maintaining computational efficiency, many mesh refinement methods have been developed. The mesh refinement methods are different for local collocation methods, which use a local polynomial to approximate the state dynamics, and global collocation methods (i.e., pseudospectral methods), which use a global polynomial to approximate the state dynamics. For local collocation methods, the mesh point distribution can be of any form needed. The mesh can be refined by minimizing the maximum local integration error [29] or by minimizing the maximum discretization error [30; 31]. The multiresolution techniques (MRTs) of [25; 26] refine the mesh by using a data compression algorithm. [PERSON] et al. [27] found that the MRTs of [25; 26] cannot refine the mesh adequately in some cases, and proposed a mesh checking algorithm to avoid inadequate mesh refinement. In [28], an efficient mesh refinement method based on density functions was proposed to distribute the mesh points, but one must find an appropriate density function when using this method. There are also some other mesh refinement methods that were originally proposed for direct shooting methods but can also be used in conjunction with local collocation methods, e.g., the methods based on the wavelet analysis of the Lagrange functional gradients [18] or the control profiles [14; 17], the method that analyzes the slope of control profiles [21], and the method that analyzes the sensitivity of the objective function with respect to discretized control variables [22]. For global collocation methods, the mesh points are determined by the roots of orthogonal polynomials, which means the mesh points cannot be freely distributed as those of local collocation methods. As a result, mesh refinement methods for global collocation methods are less flexible. The most used routine is to divide the dynamic optimization problems into many segments by introducing soft knots in the nonsmooth regions. The resulting mesh points become denser around the knots due to the special distribution of the mesh point of global collocation methods in each segment (denser towards the ends and sparse towards the middle). Furthermore, the control variables are not necessary continuous across the knots. The pseudospectral knotting method [32; 33] solves nonsmooth dynamic optimization problems by either specifying the number and the location of the knots or automatically detecting the knots based on the control derivatives. The _hp_ method [34; 35] iteratively determines the width of each segment and number of mesh points in each segment by analyzing the midpoint residual or the state curvature. The _ph_ method [36] is similar to the _hp_ method [34; 35] except that it tries to achieve the required accuracy by increasing the number of mesh point in each segment, or, if this fails, by increasing the number of segments. The method of [37] refines the mesh by analyzing the decay rates of Legendre polynomial. The capability to reduce the mesh size has been included in the latest mesh refinement methods [37; 38].