Numerical algorithms for hierarchical optimization for estimating parameters in state and control constrained optimal control problems.

在这个项目中，我们研究了具有上层参数估计问题和下层具有边界和内点条件的非线性控制和状态约束最优控制问题(OCP)的递阶优化问题，目的是推导出数值求解这类问题的数学方法。特别是要保证下层问题中状态约束和控制约束的可靠处理，因此我们首先在下层对最优控制问题进行适当的离散化。然后，我们推导了离散化的最优控制问题的一阶最优性条件，并用它们代替了低层问题。这导致了一个带有平衡约束的结构化数学规划(MPEC)，它需要特殊处理，因为它在每个可行点都违反了数学优化中的标准正则性假设(如“线性独立约束限定”)。对于MPEC的求解，我们推导出了一种利用数学方法的结构，该结构适合于这类问题。这种方法将序列线性规划和二次规划相结合，以确保解的平稳性。我们在本项目中推导的方法必须被构造为能够解决在医学、机器人或生物力学等领域中具有较低级别的最优控制问题的潜在应用，在这些领域中，这种问题设置通常被称为“逆最优控制”。

In this project, we investigate hierarchical optimization problems with a parameter estimation problem on the upper level and a nonlinear control and state constrained optimal control problem (OCP) with boundary and interior point conditions on the lower level.The goal of this project is to derive mathematical methods for numerically solving this class of problems. In particular, the reliable treatment of state and control constraints in the lower level problem has to be ensured.Therefore, we firstly discretize the optimal control problem on the lower level appropriately. Afterwards, we derive first order optimality conditions of the discretized OCP, and replace the lower level problem by them. This leads to a structured mathematical program with equilibrium constraints (MPEC), which requires a special treatment since it violates standard regularity assumptions in mathematical optimization (like the "linear independence constraint qualification") at every feasible point. For solving the MPEC, we derive a structure exploiting mathematical method which is tailored to this problem class. This method combines sequential linear programing with quadratic programing in order to ensure the desired stationarity properties in the solution.The methods we derive in this project have to be constructed such that potential applications of bi-level optimization problems with optimal control problems on the lower level in fields like medicine, robotics or biomechanics, where this problem setting is often called "inverse optimal control", can be solved.