On Lipschitzian Properties of Implicit Multifunctions

Abstract

Abstract. The paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool one employs the directional limiting coderivative which, together with the graphical derivative, enable us a fine analysis of the local behavior of the investigated multifunction along relevant directions. For verification of the calmness property, in addition, a new condition has been discovered which parallels the missing implicit function paradigm and permits us to replace the original multifunction by a substantially simpler one. Moreover, as an auxiliary tool, a handy formula for the computation of the directional limiting coderivative of the normal-cone map with a polyhedral set has been derived which perfectly matches the framework of [11]. All important statements are illustrated by examples.

Stability Analysis for Parameterized Variational Systems with Implicit Constraints

“…In particular, in [16] one finds rather weak (non-restrictive) conditions ensuring the calmness and the Aubin property of general implicitly defined multifunctions. The criterion for the Aubin property has then been worked out in [17] for a class of parametric variational systems with fixed (non-perturbed) constraint sets and in [18] for systems with implicit parameter-dependent constraints.…”

“…The last preliminary Section 2.4 is then devoted to the directional metric subregularity of a particular multifunction, which arises later as a qualification condition, and to the new notion of directional non-degeneracy of a constraint system, playing a central role in the subsequent development. Section 3 concerns the general model of an implicitly defined multifunction considered in [16]. 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.…”

“…Section 3 concerns the general model of an implicitly defined multifunction considered in [16]. 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. In the rest of the paper these statements are worked out for the considered variational system with implicit constraints.…”

“…Given a set-valued mapping M : R l × R n ⇒ R m , the general implicitly defined multifunction analyzed in [16] is given by the relation 0 ∈ M(p, x).…”

“…[16, Theorem 4.4]). Assume that M has locally closed graph around the reference point (p,x, 0) ∈ gph M and assume that…”