Using a standard first-order optimality condition for nonsmooth optimization prob-
lems, a general framework for a descent method is developed. This setting is applied to
a class of mathematical programs with equilibrium constraints in function space from
which a new algorithm is derived. Global convergence of the algorithm is demonstrated
in function space and the results are then illustrated by numerical experiments.
The directional differentiability of the solution mapping for a class of variational inequali-
ties of the second kind inspired by applications in fluid mechanics and moving free boundary
problems is investigated. The result is particularly relevant for the model predictive control
or optimal control of such variational inequalities in that it can be used to derive stationarity
conditions and efficient numerical methods.
A class of risk-neutral PDE-constrained generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player’s actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples.