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

在这个项目中，我们研究了层次优化问题，上层是参数估计问题，下层是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）。这个项目的目标是推导出数值解决这类问题的数学方法。特别是，必须确保在较低层次问题中可靠地处理状态和控制约束。因此，我们首先将最优控制问题适当地离散化。然后，我们导出了离散化OCP的一阶最优性条件，并用它们代替了较低层次的问题。这导致了一个具有平衡约束的结构化数学规划（MPEC），它需要一个特殊的处理，因为它违反了数学优化中的标准规则假设（如“线性独立约束条件”）在每个可行点。为了求解MPEC，我们推导出了一种针对这类问题的数学方法结构。该方法将序列线性规划与二次规划相结合，以保证解具有理想的平稳性。我们在这个项目中推导出的方法必须被构建成这样的双级优化问题与最优控制问题在较低层次上的潜在应用，在医学、机器人或生物力学等领域，这种问题设置通常被称为“逆最优控制”，可以解决。

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.