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


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

ω = ∈ flimitsx ∈ X\suplimitsu ∈ Q(x),f(x,u)    
and    \varphi,=,
, ∈ flimitsx ∈ X ∈ flimitsu ∈ Q(x),f(x,u),


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

S(x,u),=, {w ,:, hj(x,u,w) , ≤ ,0,, j=, 1, … ,n }.
Using suitable regularizations for quasi-variational inequalities, we determine classes of functions f, gi, hj 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


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.


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