Bilevel Optimal Control: Theory, Algorithms, and Applications

• Applicants: Professor Dr. Stephan Dempe; Professor Dr. Gerd Wachsmuth
• Subject Area: Mathematics
• Term: from 2016 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 313963978

Hierarchical optimization problems with two decision levels where at least one decision maker has to solve an optimal control problem are referred to as bilevel optimal control problems (BOCPs). Models of this structure typically arise from real-world applications which are related to e.g. pricing in energy markets, parameter estimation in process control, or data compression. BOCPs are inherently nonsmooth, infinite-dimensional programs with implicit constraints that suffer from inherent irregularity. This makes this problem class rather challenging.In this follow-up project, we plan to deepen our analysis on BOCPs of partial differential equations with a focus on potential optimality conditions and solution algorithms. Therefore, we are going to exploit two different single-level surrogates of the hierarchical model: The optimal-value-transformation which exploits the optimal value function of the lower level parametric optimization problem and the Karush-Kuhn-Tucker- (KKT-) transformation which replaces the lower level problem by first-order optimality conditions. Using appropriate upper estimates of the optimal value function which are available whenever this function is convex or concave, algorithms which iteratively refine the feasible set of relaxed surrogate problems of the optimal-value-transformation are imaginable. In the SPP's first stage, we derived a solution method of this type in the case of fully convex data. Here, the resulting convexity of the optimal value function was essential. Now, we want to investigate the situation where the optimal value function is concave which is natural in the context of parameter reconstruction. Furthermore, we are going to perform some numerical analysis in order to show that the solutions computed by our algorithms on the discretized stage converge to solutions in the function space setting under appropriate assumptions.Noting that the KKT-transformation of a BOCP with lower level inequality constraints is a complementarity-constrained optimization problem (MPCC) in function spaces, the latter problem class will be investigated carefully. We plan to derive new problem-tailored stationarity notions based on pointwise characterizations as well as associated constraint qualifications. Furthermore, second-order sufficient optimality conditions for the latter problem class will be derived. We aim for the construction of an active set method for the numerical solution of MPCCs in finite and infinite dimensions. Finally, we want to apply all our findings to prototypical applications from bilevel programming, namely parameter estimation problems as well as a problem of data compression using a bilevel measure approach. These models will be investigated from the viewpoint of optimality conditions and solution algorithms. We want to set up a corresponding collection of benchmark problems which allows a comparison of the derived theoretical results and numerical methods.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1962: Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization
• Co-Investigators: Professor Patrick Mehlitz, Ph.D.; Privatdozent Dr. Uwe Prüfert