Abstract
We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.
Highlights
We consider the quasi-variational inequality (QVI)Find y ∈ Q(y) such that A(y) − f, v − y ≥ 0 ∀v ∈ Q(y). (1.1)Here, V is a Hilbert space, A : V → V is a mapping, and f ∈ V
We will not cover the general situation of a set-valued mapping Q : V ⇒ V, but we restrict the treatment of (1.1) to the case in which Q(y) is a moving set, i.e., Q(y) = K + Φ(y) for some non-empty, closed and convex subset K ⊂ V and Φ : V → V
It is well-known that QVIs have many important real-world applications, we refer exemplarily to [1,3,5,16] and the references therein
Summary
V is a Hilbert space, A : V → V is a (possibly nonlinear) mapping, and f ∈ V. We will not cover the general situation of a set-valued mapping Q : V ⇒ V , but we restrict the treatment of (1.1) to the case in which Q(y) is a moving set, i.e., Q(y) = K + Φ(y). For some non-empty, closed and convex subset K ⊂ V and Φ : V → V. for some non-empty, closed and convex subset K ⊂ V and Φ : V → V It is well-known that QVIs have many important real-world applications, we refer exemplarily to [1,3,5,16] and the references therein. The main contributions of this paper are the following
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