## Contributed Session Wed.2.MA 313

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

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

### Advances in the theory of complementarity and related problems II

**Chair: Joachim Gwinner**

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

**Maria Beatrice Lignola**

Mathematical programs with quasi-variational inequality constraints

**Coauthor: Jacqueline Morgan**

**Abstract:**

\noindent We illustrate how to approximate the following values for mathematical programs with quasi-variational inequality constraints

** **

ω = ∈ flimits_{x ∈ X}\suplimits_{u ∈ Q}**(x)**,f(x,u) and* \varphi,=,*

, ∈ flimits_{x ∈ X} ∈ flimits_{u ∈ Q}**(x)**,f(x,u),

where

** **

**Q**(x) = { u ∈ S(x,u),: ⟨ A(x,u),u - w ⟩ ≤ 0 ∀ w ∈ S(x,u) },

via the values of appropriate regularized programs under or without perturbations.

In particular, we consider the case where the constraint set *X* and the constraint set-valued mapping *S* are defined by inequalities

\begin{gather*}

X ,=, {x ,:, g_{i}(x) , ≤ ,0,, i=, 1, … , m }

\

S(x,u),=, {w ,:, h_{j}(x,u,w) , ≤ ,0,, j=, 1, … ,n }.

\end{gather*}

Using suitable regularizations for quasi-variational inequalities, we determine classes of functions *f*, *g*_{i}, *h*_{j} allowing to obtain one-sided (from above and below) approximation of *\varphi* and *ω*, and classes of functions providing a global approximation.

* *

*
*

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

**Joachim Gwinner**

On linear differential variational inequalities

**Abstract:**

Recently Pang and Stewart introduced and investigated a new class

of differential variational inequalities in finite dimensions as a new modeling paradigm of variational analysis. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. In this contribution we lift this class of nonsmooth dynamical systems to the level of a Hilbert space, but in contrast to recent work of the author we focus to linear input/output systems. This covers in particular linear complementarity systems studied by Heemels, Schumacher and Weiland.

Firstly, we provide an existence result based on maximal monotone

operator theory. Secondly we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated linear maps and the constraint set.