Abstract
In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples.
Highlights
A great effort has been devoted to the study of stability and sensitivity of solution maps to parameter-dependent optimization and equilibrium problems
In this framework we find there a directional variant of the Levy-Rockafellar characterization of the isolated calmness property and a counterpart of [16, Theorem 4.4] corresponding to the Aubin property relative to a set of feasible parameters
DG ((p, x), −f (p, x)) (0, u) = D ((p, x, x), −f (p, x)) (0, u, u) and the mapping M = f + G is metrically subregular in direction (0, u) at ((p, x), 0), second-order condition for isolated calmness (SOCIC) is necessary for the isolated calmness property of S at (p, x)
Summary
A great effort has been devoted to the study of stability and sensitivity of solution maps to parameter-dependent optimization and equilibrium problems. In this framework we find there a directional variant of the Levy-Rockafellar characterization of the isolated calmness property and a counterpart of [16, Theorem 4.4] corresponding to the Aubin property relative to a set of feasible parameters. Both these final results as well as some other important statements are illustrated by examples. Given a setvalued map F : Rn ⇒ Rm, gph F := {(x, y) ∈ Rn × Rm | y ∈ F (x)} stands for the graph of F and Lim supx→x F (x) denotes the outer set limit in the sense of Painleve-Kuratowski
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