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.