Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion
- Gahururu, Deborah
- Hintermüller, Michael
- Stengl, Steven-Marian
- Surowiec, Thomas M.
2010 Mathematics Subject Classification
- 49J20 49J55 49K20 49K45 49M99 65K10 65K15 90C15 91A10
- Generalized Nash equilibrium problems, PDE-constrained optimization, L-convexity, set-valued analysis, fixed-point theory, risk averse optimization, coherent risk measures, stochastic optimization, method of multipliers
PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs.
- Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., vol. 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 145--181.