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

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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.