Michael Ulbrich

1. We investigate non-convex generalized Nash equilibrium problems (GNEPs) in infinite-dimensional spaces. Nash equilibrium problems were pioneered by Nash [23] and extended by Arrow and Debreu [1]...

The sample average approximation (SAA) approach is applied to risk-neutral optimization problems governed by semilinear elliptic partial differential equations with random inputs. After constructing a compact set that contains the SAA critical points, we derive nonasymptotic sample size estimates for SAA critical points using the covering number approach. Thereby, we derive upper bounds on the number of samples needed to obtain accurate critical points of the risk-neutral PDE-constrained optimization problem through SAA critical points. We quantify accuracy using expectation and exponential tail bounds. Numerical illustrations are presented.

We derive a generic system that constructs an optimization model for an emergency stop scenario on the highway, based on map data from high definition maps that are used in Advanced Driver Assistance Systems (ADAS) and in Highly Automated Driving (HAD). New additional situative and scenario-based information is computed by applying a global maximization approach to the model. For this purpose, we develop two new rigorous and deterministic branch-and-bound algorithms that both determine the certified global optimal value up to a predefined tolerance. The underlying interval optimization algorithm, which uses first-order techniques, is enhanced by one of two second-order methods that are applied for specifically selected intervals. We investigate two approaches that either compute a concave overestimator for the objective function or approximate the function with a quadratic polynomial using Taylor expansion. We show the limits of interval arithmetic in our problem, especially for the interval versions of the derivatives, and present a local linearization of the curve data that improves the results significantly. The presented novel method for deriving secondary information is compared to state of the art methods on two exemplary and for the automotive context representative scenarios to show the advantages of our approach.