Efficient numerical methods for hierarchical dynamic optimization with application to cerebral palsy gait modeling
Efficient numerical methods for hierarchical dynamic optimization with application to cerebral palsy gait modeling
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DOI：
10.11588/heidok.00016803
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发表时间：
2014
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影响因子：
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通讯作者：
K. Hatz
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文献类型：
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作者：
K. Hatz
本论文旨在发展求解递阶动态优化问题的有效数学方法。其主要动机是模拟自然界中的过程，有证据表明它们运行最佳。我们通过最优控制问题（称为最优控制模型（OCMs））来描述这些过程的模型。然而，OCM通常包括不能完全基于理论导出的未知参数，对于成本函数尤其如此。因此，我们开发的参数估计技术来估计未知的OCM从观测数据的过程。从数学上讲，这导致了一个分层的动态优化问题的参数估计问题的上层和最优控制问题的下层。本文主要研究基于非线性常微分方程的多阶段等式和不等式约束最优控制问题。本论文的主要目标是推导出求解递阶动态优化问题的数值有效的数学方法，并利用这些方法从实际测量数据中估计高维OCM中的参数。我们为脑瘫患者和健全受试者的步态开发了参数依赖的OCM。然后，通过使用本工作中开发的数学方法，从海德堡整形外科大学诊所的海德堡运动实验室提供的真实运动捕捉数据中估计OCM中的未知参数。本文总结了本文在递阶动态优化领域的主要创新和贡献。- 基于直接多重打靶法和一阶最优性条件，建立了求解递阶动态优化问题的一种新的数学方法，即直接一次法. - 此外，我们提出了一个有效的数值算法的大规模分层动态优化问题，充分利用继承的结构从分层设置和离散化。- 利用庞特里亚金极大值原理分析了递阶动态优化问题解的性质，如下层问题的二阶最优性条件。- 此外，我们提出并讨论了分层动态优化的替代方法是基于无导数优化和捆绑的方法。这些方法保持分层问题设置，不使用一阶最优性条件重新制定较低级别的问题。- 我们建立了一个新的提升方法正则化的数学规划互补约束，这是讨论和数值研究通过一个著名的基准问题的集合。- 证明了序列二次规划方法应用于具有互补约束的提升数学规划的正则性和收敛性。- 本论文中所推导的所有数学方法的最先进的有效实现，以及分层动态优化问题的基准集合。- 建立了脑性瘫痪患者和正常人的高维最优控制步态模型。数学方法推导出在这篇论文中被用来估计未知的模型参数，从现实世界的运动捕捉数据提供的海德堡整形外科大学诊所海德堡运动实验室。本文提出的理论和实践结果可以被认为是回答当前医学研究领域中的开放性问题的初始激励步骤，如治疗计划，步态分类或通过分层动态优化评估手术。
This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.