Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion

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The subject of this proposal is generalized Nash equilibrium problems (GNEPs) with partial differential equation (PDE) constraints under uncertainty and their extension to new classes of equilibrium problems in function space. The latter includes so-called equilibrium problems with equilibrium constraints (EPECs) and multiple optimization problems with equilibrium constraints (MOPECs) in infinite dimensions. The new classes of PDE-constrained problems under consideration offer a wide range of mathematical novelty. From a theoretical perspective, the derivation of existence and stationarity results requires techniques from non-smooth optimization and set-valued analysis, which includes fixed point theorems and generalized differentiation for set-valued mappings. On the other hand, the variational form of these problems typically lack the necessary monotonicity properties that would allow the direct application of known function-space based numerical methods from the study of non-smooth variational problems and standard PDE-constrained optimization. The proposed framework extends the classical mathematical programming paradigm beyond a single-objective optimization problem and thus, allows one to consider a broader array of equilibrium problems in the sciences and economics. In particular, problems with hierarchical solution concepts such as Nash equilibria or Nash-Stackelberg equilibria (multi-leader-follower equilibria) can be considered. The existence proofs and derivation of stationarity conditions are to be carried out in conjunction with the derivation of fast numerical algorithms, which take into account the subtleties of partial differential operators and the distributed parameter setting. In order to model risk-aversion in a competitive setting with a complex system subject to uncertainties, coherent risk measures are employed. A four part research program is proposed for the rigorous development of an existence and stationarity theory, algorithms and numerical analysis, and handling uncertainties via risk measures.