## Invited Session Tue.2.MA 313

#### Tuesday, 13:15 - 14:45 h, Room: MA 313

**Cluster 3: Complementarity & variational inequalities** [...]

### MPECs in function space I

**Chair: Michael Hintermüller and Christian Meyer**

**Tuesday, 13:15 - 13:40 h, Room: MA 313, Talk 1**

**Daniel Wachsmuth**

Convergence analysis of smoothing methods for optimal control of stationary variational inequalities

**Coauthors: Karl Kunisch, Anton Schiela**

**Abstract:**

In the talk an optimal control problem subject to a stationary variational inequality

is investigated. The optimal control problem is complemented with pointwise control constraints.

The convergence of a smoothing scheme is analyzed. There, the variational inequality

is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal

control problem converge to solutions of the original one. Passing to the limit in the

optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem.

Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained.

These rates coincide with rates obtained by numerical experiments.

**Tuesday, 13:45 - 14:10 h, Room: MA 313, Talk 2**

**Thomas Michael Surowiec**

A PDE-constrained generalized Nash equilibrium problem with pointwise control and state constraints

**Coauthor: Michael Hintermüller**

**Abstract:**

We formulate a class of generalized Nash equilibrium problems (GNEP) in which the feasible sets of each player's game are partially governed by the solutions of a linear elliptic partial differential equation (PDE). In addition, the controls (strategies) of each player are assumed to be bounded pointwise almost everywhere and the state of the entire system (the solution of the PDE) must satisfy a unilateral lower bound pointwise almost everywhere. Under certain regularity assumptions (constraint qualifications), we prove the existence of a pure strategy Nash equilibrium. After deriving multiplier-based necessary and sufficient optimality conditions for an equilibrium, we develop a numerical method based on a non-linear Gauss-Seidel iteration, in which each respective player's game is solved via a nonsmooth Newton step. Convergence of stationary points is demonstrated and the theoretical results are illustrated by numerical experiments.

**Tuesday, 14:15 - 14:40 h, Room: MA 313, Talk 3**

**Carlos Nicolas Rautenberg**

Hyperbolic quasi-variational inequalities with gradient-type constraints

**Coauthor: Michael Hintermüller**

**Abstract:**

The paper addresses a class of hyperbolic quasi-variational inequality (QVI) problems of first order and with constraints of the gradient-type. We study existence and approximation of solutions based on recent results of appropriate parabolic regularization, monotone operator theory and *C*_{0}-semigroup methods. Numerical tests, where the subproblems are solved using semismooth Newton methods, with several nonlinear constraints are provided.