JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics Stability Analysis for Parameterized Variational Systems with Implicit Constraints

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 (nonrestrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples.


Introduction
In variational analysis, a great effort has been devoted to the study of stability and sensitivity of solution maps to parameter-dependent optimization and equilibrium problems.In particular, the researchers have investigated various Lipschitzian properties of these maps around given reference points.To obtain useful results, one employs typically some efficient tools of generalized differentiation discussed in a detailed way in the monographs [5,24,27,29,32].Starting from 2011, the available arsenal of these tools includes also the calculus of directional limiting normal cones and coderivatives which enables us in some cases a finer analysis of parametric equilibria than its nondirectional counterpart.This new theory has been initiated in [19] and then thoroughly developed in a number of papers authored and co-authored by H. Gfrerer [1,9,10,11,12,13,15,16,18].
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 model from [18] was investigated already in [28] by using the (classical) generalized differential calculus of B. Mordukhovich.It encompasses quasi-variational inequalities (QVIs), implicit complementarity problems and also standard variational inequalities of the first kind with parameterdependent constraints.
In this paper we consider the same model as in [28] and [18] but remove the (rather severe) nondegeneracy-type assumption imposed in [18] on the constraint system.Instead of it, we make use of a (much weaker) metric inequality stated in Assumption 1. Further, we analyze now not just the standard Aubin property of the considered solution map, denoted by S, but the Aubin property relative to a given set of feasible parameters.Clearly, S may enjoy this type of Lipschitzian stability even when the standard Aubin property is violated.Finally, we provide in this paper also a new condition, ensuring the isolated calmness of S.
The structure of the considered constraint system has enabled us to employ some strong results from [4,5] and [16] concerning tangents and normals to the graph of the normal-cone mapping associated with a convex polyhedral set.More precisely, these tangents and normals can be expressed via some faces of an associated critical cone.This representation substantially contributes to the workability of final conditions ensuring the Aubin property of S. In addition, also some other statements in connection with directional non-degeneracy and directional metric regularity could be formulated in terms of these faces.
The plan of the paper is as follows.Sections 2.1 and 2.2 provide the reader with basic notions of the standard and directional generalized differential calculus and with some basic facts about those Lipschitzian stability properties which are extensively used throughout the whole paper.Section 2.3 contains the necessary background from the theory of convex polyhedral sets and polyhedral multifunctions.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.In the rest of the paper these statements are worked out for the considered variational system with implicit constraints.So, in Section 4 the respective graphical derivative is computed, which is a basis for the formulation of the final condition ensuring the isolated calmness property of S presented in Section 5. Therafter, in Section 6 one finds a new workable sufficient condition guaranteeing the Aubin property of S relative to a given set of feasible parameters.Both these final results as well as some other important statements are illustrated by examples.
There are well-known equilibria in economy and mechanics modeled by QVIs and implicit complementarity problems, cf.[2].As an example, let us mention the generalized Nash equilibrium problems (GNEPs) which describe, e.g., the behavior of agents acting on markets with a limited amount of resources.Very often, these equilibria depend on some uncertain data which can be viewed as parameters.The results of this paper can then be used in post-optimal analysis of such equilibria, where the stability issues are of ultimate importance.
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). (1.1) We are going to analyze the associated solution mapping S : R l ⇒ R n defined by 2) The variational system investigated in [28] and [18] attains the form where The following notation is employed.Given a set A ⊂ R n , sp A stands for the linear hull of A, ri A is the relative interior of A and A • is the (negative) polar of A. We denote by dist(x, A) := inf y∈A x − y the usual point to set distance with the convention dist(x, / 0) = ∞.For a sequence x k , x k A → x stands for x k → x with x k ∈ A. For a convex cone K, lin K denotes the lineality space of K, i.e., the set K ∩ (−K).Further, B R n , S R n is the unit ball and the unit sphere in R n , respectively.Given a vector a ∈ R n , [a] is the linear space generated by a and [a] ⊥ stands for the orthogonal complement to [a].Finally, given a set-valued map } stands for the graph of F and Lim sup x→ x F(x) denotes the outer set limit in the sense of Painlevé-Kuratowski.

Variational geometry and generalized differentiation
We start by recalling several definitions and results from variational analysis.Let Ω ⊂ R n be an arbitrary closed set and x ∈ Ω.The contingent (also called Bouligand or tangent) cone to Ω at x, denoted by T Ω ( x), is given by We denote by N Ω ( x) := T Ω ( x) the Fréchet (regular) normal cone to Ω at x.The limiting (Mordukhovich) normal cone to Ω at x is defined by amounts to the classical normal cone in the sense of convex analysis and we will write N Ω ( x).
Given a pair ( x, x * ) ∈ gph N Ω we denote by the critical cone to Ω at x with respect to x * .
The following generalized derivatives of set-valued mappings are defined by means of the tangent cone and the (directional) limiting normal cone to the graph of the mapping.Definition 2.1.Let F : R n ⇒ R m be a set-valued mapping having locally closed graph around ( x, ȳ) ∈ gph F.
(i) The set-valued map DF( x, ȳ) : R n ⇒ R m , defined by

Regularity and Lipschitzian properties of set-valued mappings
First we recall some well-known definitions.
Definition 2.2.Let F : R n ⇒ R m be a mapping and let ( x, ȳ) ∈ gph F. We say that F is metrically regular around ( x, ȳ) if there are neighborhoods U of x and V of ȳ along with some real κ ≥ 0 such that dist(x, When fixing y = ȳ in this condition, F is said to be metrically subregular at ( x, ȳ), i.e., we require dist(x, A well-known coderivative characterization of metric regularity is known as "Mordukhovich criterion" and reads as follows. (2.6) One can find numerous sufficient conditions for metric subregularity in the literature, see, e.g., [7,8,9,10,11,12,20,23,25,33].However, these sufficient conditions are often very difficult to verify.The following sufficient condition for metric subregularity is not as week as possible but it is stable with respect to certain perturbations, cf.[6].
In order to define a directional version of metric (sub)regularity, consider for a direction u ∈ R n and positive reals ρ, δ the set We say that V is a directional neighborhood of u if V ρ,δ (u) ⊂ V for some ρ, δ > 0.
Definition 2.5.Let F : R n ⇒ R m be a mapping and let ( x, ȳ) ∈ gph F.
1. Given a direction u ∈ R n we say that F is metrically subregular in direction u at ( x, ȳ) if (2.5) holds with x + U in place of U, where U is a directional neighborhood of u.
Theorem 2.6.Assume that the set-valued mapping F : R n ⇒ R m has locally closed graph around ( x, ȳ) ∈ gph F and let u ∈ R n be given.Then F is metrically regular in direction (u, 0) at ( x, ȳ) if and only if Proof.Follows from [10,Theorem 5].
Comparing Definition 2.5 with Definition 2.2 we see that metric regularity around ( x, ȳ) is equivalent with metric regularity in direction (0, 0) at ( x, ȳ).This is reflected also in conditions (2.6) and (2.7) with u = 0. Further note that the sufficient condition for metric subregularity of Theorem 2.4 says that mapping F is metrically regular at ( x, ȳ) in every direction (u, 0) with u = 0.
The following notion of stability was introduced by Robinson [ Comparing the definition of Robinson stability with that of metric regularity we see that in case when P = R l and h is of the form h(p, x) = h(x) − p, the property of Robinson stability of (2.8) at ( p, x) is equivalent to metric regularity of the mapping h(•) −C around ( x, p).For sufficient conditions for Robinson stability we refer to the recent paper [14].Here we mention only the following result.
Theorem 2.8.Let ( p, x) ∈ h −1 (C) be given and assume that h is differentiable with respect to the second component and both h and ∇ 2 h are continuous, whereas C is closed.If then the system (2.8) enjoys the Robinson stability property at ( p, x).
We now turn to Lipschitzian properties of set-valued mappings.Definition 2.9.Let S : R m ⇒ R n be a set-valued map and let ( ȳ, x) ∈ gph S.
1. S is called to be calm at ( ȳ, x) if there is a neighborhood U of x together with a real L ≥ 0 such that S(y) ∩U ⊂ S( ȳ) If, in addition, S( ȳ) = { x} is a singleton we say that S has the isolated calmness property at ( ȳ, x).This condition with V in place of Y ∩V is simply the Aubin propery around ( ȳ, x).
It is well-known [4] that F is metrically subregular at ( x, ȳ) if and only if its inverse mapping F −1 is calm at ( ȳ, x).Further, metric regularity is equivalent with the Aubin property of the inverse mapping.

Polyhedral sets
Recall that a set D ⊂ R s is said to be convex polyhedral if it can be represented as the intersection of finitely many halfspaces.We say that a set E ⊂ R s is polyhedral if it is the union of finitely many convex polyhedral sets.If a set E is polyhedral, then for every z ∈ E there is some neighborhood W of z such that Given a convex polyhedral set D and a point z ∈ D, then the tangent cone T D (z) and the normal cone N D (z) are convex polyhedral cones and there is a neighborhood W of z such that The graph of the normal cone mapping to D is a polyhedral set and for every pair (z, z * ) ∈ gph N D we have see, e.g., [5,Lemma 2E.4].
For two convex polyhedral cones K 1 , K 2 ⊂ R s their polars as well as their sum K 1 + K 2 and their intersection K 1 ∩ K 2 are again convex polyhedral cones and For a convex polyhedral cone K ⊂ R s and a point z ∈ K we have A face F of K can always be written in the form for some z * ∈ K • .The cone K has the representation where J ⊂ Ī are two finite index sets and a i ∈ R s , i ∈ Ī.By enlarging J when necessary we can assume that there exists some z 0 such that By possibly enlarging J we can find a unique index set, denoted by J F , such that ri

Directional non-degeneracy
In what follows the property of directional metric (sub)regularity of a particular mapping will play an important role.Let D ⊂ R s be a convex polyhedral set, let g : R m → R s be continuously differentiable and consider the mapping Given some point ( ȳ, λ ) ∈ F −1 (0) and some direction (v, η) ∈ R m × R s , we we want to investigate metric subregularity of F in direction (v, η) at ( ȳ, λ ), in particular when v = 0. We denote Recall that F is by definition metrically subregular in direction (v, η) at ( ȳ, λ ) whenever i.e., taking into account (2.9), whenever (λ , η) ∈ Θ( ȳ, v).
In our analysis we restrict ourselves to the characterization of metric regularity of F in directions (v, η), (0, 0) which implies metric subregularity of F in direction (v, η).The following lemma is a slight generalization of [18,Proposition 2].Lemma 2.10.Let ȳ ∈ g−1 (D), v ∈ R m and (λ , η) ∈ Θ( ȳ, v) be given.Then the mapping F defined in (2.12) is metrically regular in direction (v, η), (0, 0) at ( ȳ, λ ), (0, 0) if and only if for every face Proof.The characterization (2.7) reads in our special case as Thus, by Theorem 2.6 the claimed directional metric regularity is equivalent to the condition that the implication • , the statement of the lemma follows.
This characterization of directional metric regularity can be considerably simplified.

∇ g( ȳ)
Proof.Assume that the tangent cone T D ( g( ȳ)) has the representation (2.10) and consider any ) and therefore where Since F v depends neither on λ nor on η, the equivalence between (i) and (ii) is established.To show the equivalence of (2.14) with (2.13) just observe that • and the proof is complete.
From the proof of Theorem 2.11 we also obtain the following corollary.
Definition 2.13.Let ȳ ∈ g−1 (D) and v ∈ R m be given.We say that the system g(•) ∈ D is nondegenerate in direction v at ȳ if condition (2.13) is fulfilled.In case when v = 0 we simply say that the system g( Clearly, for v = 0 we obtain the standard definition of non-degeneracy from [3, Formula 4.17].
We now state some properties of directional non-degeneracy.
Proposition 2.14.Let ȳ ∈ g−1 (D) and v ∈ R n such that the system g(•) ∈ D is non-degenerate in direction v at ȳ. Then there is a directional neighborhood V of v and a constant β > 0 such that for all y ∈ ( ȳ + V ) ∩ g−1 (D) , y = ȳ, one has In particular, for all y ∈ ( ȳ + V ) ∩ g−1 (D) , y = ȳ, the system g(•) ∈ D in non-degenerate at y.
Proof.By contraposition.Assume on the contrary that there are sequences Since for all k sufficiently large we have ) and, by passing to some subsequence if necessary, we can assume that µ k converges to some µ ∈ sp N T D ( g(y)) (∇ g(y)v) ∩S R s .Obviously we also have ∇g(y) T µ = 0, a contradiction to the assumed directional non-degeneracy and (2.16) is proved.The additional statement concerning the non-degeneracy is an immediate consequence of (2.16).
It turns out that the directional non-degeneracy can be fulfilled in all non-zero directions even if the (standard) non-degeneracy is violated.
Thus, non-degeneracy in direction v is equivalent to the linear independence of the gradients ∇ gi ( ȳ), i ∈ J (v) whereas non-degeneracy amounts to the so-called linear independence constraint qualification (LICQ), i.e., to the linear independence of all gradients ∇ gi ( ȳ), i = 1, . . ., s.
Consider the system Obviously LICQ is violated at ȳ = 0.However, it is not difficult to verify that the system is nondegenerate in every direction v = 0. Further note that in this example also the so-called constant rank constraint qualification is violated at ȳ.

Stability properties through generalized differentiation
Throughout this section we consider the solution mapping S given by (1.2).Given some reference point ( p, x) ∈ gph S, we will provide point-based sufficient conditions for the isolated calmness property, the Aubin property and the Aubin property relative to some set P ⊂ R l , respectively, in terms of generalized derivatives of the mapping M.
We start with the Levy-Rockafellar characterization of isolated calmness [26], who showed that then S has the isolated calmness property at ( p, x).Conversely, if there is some u = 0 such that 0 ∈ DM( p, x, 0)(0, u) and M is metrically subregular in direction (0, u) then S is not isolatedly calm at ( p, x).
Proof.Note that the closedness of gph M readily implies that gph S = M −1 (0) is locally closed around ( p, x).The sufficiency of (3.18) for the isolated calmness property of S is due to (3.17) together with the inclusion following from the definition of the graphical derivative, see also [26,Theorem 3.1].In order to show the second statement, consider u = 0 verifying 0 ∈ DM( p, x, 0)(0, u) and assume that M is metrically subregular in direction (0, u) at ( p, x, 0).By [16,Proposition 4.1] we obtain (0, u) ∈ T M −1 (0) ( p, x) = T gph S ( p, x) and consequently u ∈ DS( p, x)(0).Thus mapping S is not isolatedly calm at ( p, x) by (3.17).
Since metric subregularity of M implies metric subregularity in any direction, we obtain the following corollary.
The above assertions remain true provided assumptions (ii), (iii) are replaced by (iv) For every nonzero (q, u) ∈ R l × R n verifying 0 ∈ DM( p, x, 0)(q, u) one has the implication Sufficient conditions for the Aubin property of S relative to some set P are based on the following statement, where h : (i) for every q ∈ T P ( p) and every sequence t k ↓ 0 there exists some u ∈ R n satisfying lim inf k→∞ dist(( p + t k q, x + t k u, 0), gph M)/t k = 0 (3.20) (ii) For every nonzero (q, u) ∈ T P ( p) × R n verifying 0 ∈ DM( p, x, 0)(q, u) one has the implication Then S has the Aubin property relative to P around ( p, x) and for any q ∈ T P ( p)  [14] as the closed cone generated by 0 and those v ∈ R l × R n × R m for which there is a sequence p k ⊂ P with is exactly the set {(q, 0, 0) | q ∈ T P ( p)}.Further for every u ∈ R n we have ∇ x h( p, x)u = (0, u, 0) and thus [14, Condition 3.10] is fulfilled by (3.20).Next we have to verify that for every pair (0, 0) = (q, u) ∈ T P ( p) × R n satisfying (q, u, 0) ∈ T gph M ( p, x, 0) the implication λ ∈ N gph M ( p, x, 0), (q, u, 0) , ∇ x h( p, x) T λ = 0 ⇒ λ = 0 is fulfilled.Setting λ := (q * , u * , −v * ) this amounts to which is obviously equivalent to (3.21).By taking into account that the condition (q, u, 0) ∈ T gph M ( p, x, 0) is the same as requiring 0 ∈ DM( p, x, 0)(q, u), all assumption of [14, Corollary 3.6] are fulfilled and the claimed Robinson stability property of the system (3.19) at ( p, x) follows.By virtue of Proposition 3.4 this implies the Aubin property of S relative to P around ( p, x).There remains to show (3.22).Since {u | 0 ∈ DM( p, x, 0)(q, u)} ⊃ DS( p, x)(q) always holds by [26, Theorem 3.1], we only have to show {u | 0 ∈ DM( p, x, 0)(q, u)} ⊂ DS( p, x)(q).Consider u satisfying 0 ∈ DM( p, x, 0)(q, u) for some q ∈ T P ( p).By Theorem 2.6, condition (3.21) implies that M is metrically subregular in direction (q, u) at ( p, x, 0) and hence we can invoke [16, Proposition 4.1] to obtain (q, u) and consequently u ∈ DS( p, x)(q).Thus {u | 0 ∈ DM( p, x, 0)(q, u)} ⊂ DS( p, x)(q) and the proof of the theorem is complete.
Remark 3.6.Assumption (i) of Theorem 3.5 is fulfilled in particular if for every q ∈ T P ( p) there is some u ∈ R n satisfying 0 ∈ DM( p, x, 0)(q, u) and the tangent (q, u, 0) to gph M is derivable.We see that in this case Theorem 3.5 is a generalization of Theorem 3.3.

Graphical derivative of the normal cone mapping
This section deals with computation of the graphical derivative of M given by (1.3).Throughout the rest of the paper we assume that we are given a reference solution ( p, x) of (1.3) fulfilling the following assumption.
Note that by Theorem 2.8 Assumption 1 is fulfilled, e.g., in the case when which is equivalent to Robinson's constraint qualification As a consequence of Assumption 1 we obtain that for all (p, x, z) ∈ gph Γ sufficiently close to ( p, x, x) the mapping g(p, x, •) − D is metrically subregular at (z, 0) with modulus κ and therefore In order to unburden the notation we introduce the mappings b(p, x) := ∇ 3 g(p, x, x), g(p, x) := g(p, x, x) and denote the set-valued part of M(p, x) as G(p, x) := N Γ(p,x) (x).For (p, x) close to ( p, x) one has The graphical derivative of G is closely related with the graphical derivative of the mapping Ψ : In order to give a formula for the graphical derivative of ψ we employ the following notation.Given Proof.The first equality is an immediate consequence of [15,Theorem 5.3].By ⊥ we have y * = ∇g(y) T µ for some µ ∈ N D g(y) with µ T ∇g(y)v = 0 and due to λ ∈ Λ(y, ∇g(y) T µ; v) we also have ∇g(y) T λ = y * .Since [31,Corollary 16.3.2]and by taking into account that the set ∇g(y) T K D (g(y), λ ) • is a convex polyhedral cone and therefore closed.Thus showing π 3 (N K gph Γ (y,y * ) (v)) = ∇ 3 g(y) T N K D (g(y),λ ) (∇g(y)v) and the proof is complete.
In what follows we will also use the following multiplier sets Theorem 4.2.Assume that Assumption 1 is fulfilled.Then for all (p, x) ∈ dom G sufficiently close to ( p, x), all x * ∈ G(p, x) and all (q, u) ∈ R l × R n we have λ ∈ Λ (p, x), x * ; (q, u) .
Proof.The inclusion (4.24) follows immediately from the definition of the graphical derivative, whereas (4.25) is a consequence of Proposition 4.1.Consider now (q, u) ∈ R l ×R n , λ ∈ Λ (p, x), x * ; (q, u)) and η ∈ N K D ( g(p,x),λ ) (∇ g(p, x)(q, u)) such that the mapping (4.26) is directionally metrically subregular.We conclude that and thus g(p, x), λ + t ∇ g(p, x)(q, u), η ∈ gph N D for all t ≥ 0 sufficiently small, because gph N D is a polyhedral set.Consequently we have dist(( g(p and by the assumed directional metric subregularity of F we can find for every t > 0 some (q t , u t , η t ) with lim t ↓ 0 (q t , u t , η t ) = (q, u, η) and 0 ∈ F(p On the other hand, by Taylor expansion we obtain b(p + tq t , x + tu t ) T (λ showing (4.27) and the derivability of the tangent q, u, ∇(b(•) T λ )(p, x)(q, u) + b(p, x) T η .

Isolated calmness of the solution mapping
In what follows we define for every Definition 5.1.We say that the second-order condition for isolated calmness (SOCIC) holds at ( p, x) if for every u = 0 and every λ ∈ Λ ( p, x), − f ( p, x); (0, u) with (5.29) Theorem 5.2.Assume that Assumption 1 is fulfilled.If SOCIC holds at ( p, x), then the solution map S to the variational system (1.3) has the isolated calmness property at ( p, x).
Conversely, if for every u = 0 there holds and the mapping M = f + G is metrically subregular in direction (0, u) at (( p, x), 0), SOCIC is also necessary for the isolated calmness property of S at ( p, x).
By (4.25) this is equivalent to for some λ ∈ Λ(( p, x), − f ( x); (0, u)).In particular, ∇ 2 g( p, x)u ∈ K D ( g( p, x), λ ) follows.Next observe that This follows from [31,Corollary 16.3.2]because the set on the left hand side is a convex polyhedral set and therefore closed.Thus (5.32) is equivalent to • which in turn is equivalent to contradicting (5.29).Thus the claimed equivalence between SOCIC and (5.31) holds true.Combining Theorem 3.1 and (4.24) we see that the condition (5.31) and consequently SOCIC as well are sufficient for the isolated calmness property of S at ( p, x).In order to show the second statement of the theorem, just note that condition (5.30) ensures that (5.31) and SOCIC are equivalent to the condition 0 ∈ ∇ f (p, x)(0, u) + DG(( p, x, x), − f ( p, x))(0, u) ⇒ u = 0 and thus by Theorem 3.1 the necessity of SOCIC for the isolated calmness property of S follows.By Theorem 4.3(ii), a sufficient condition for (5.30) is that the system g(•) ∈ D is non-degenerate in every direction (0, u), u = 0 at ( p, x).We now state a sufficient condition for the metric regularity of the mapping M = f + G in some direction (q, u).Theorem 5.3.Let (q, u) ∈ R l × R n and assume that the system g(•) ∈ D is non-degenerate in direction (q, u) at ( p, x).Further assume that for every λ ∈ Ξ(( p, x), − f ( p, x); (q, u)), every η ∈ Then the mapping M is metrically regular in direction ((q, u), 0) at (( p, x), 0).Proof.By contraposition.Assume on the contrary that M = f +G is not metrically regular in direction ((q, u), 0) at (( p, x), 0).By virtue of Theorem 2.6 there is some w = 0 such that (0, 0) ∈ D * ( f + G) (( p, x), 0); ((q, u), 0) (−w).In particular, this implies By the definition of the directional limiting coderivative there are sequences t k ↓ 0, (q k , u k , w * k ) → (q, u, 0) and (q * k , u * k , w k ) → (0, 0, w) such that where (5.33)By Proposition 2.14, the system g(•) ∈ D is non-degenerate at (p k , x k ) and we deduce from Theorem 4.
By passing to a subsequence if necessary we can assume that λ k converges to some λ .Obviously we have λ ∈ N D ( g( p, x)) and b( p, x) T λ = − f ( p, x).By [5, Lemma 4H.2], for each k sufficiently large there are two closed faces and a close look at the proof of [5,Lemma 4H.2] tells us that we also have g(p k , x k ) − g( p, x) ∈ ri F k 2 .Since K D ( g( p, x), λ ) is a closed convex cone, it has only finitely many faces and by passing to a subsequence once more we can assume F k 1 = F 1 and F k 2 = F 2 for all k.A face of a closed convex cone is again a cone and thus ( g(p k , x k ) − g( p, x))/t k ∈ ri F 2 ∀k.This yields by passing to the limit that ∇ g( p, x)(q, u) ∈ F 2 ⊂ K D ( ḡ( p, x), λ ), and consequently λ ∈ Ξ ( p, x), − f ( p, x), (q, u) .Further we have x), λ ), we have and consequently [λ k − λ ] ⊂ (sp F 1 ) ⊥ .We can now invoke Hoffman's lemma [3,Theorem 2.200] to find for every k some η k ∈ N K D ( g( p, x), λ ) (∇ g( p, x)(q, u)) ∩ (sp for some constant β > 0 not depending on k.Since the right hand side of (5.35) is bounded, so is η k and by possibly passing to a subsequence we can assume that η k converges to some η ∈ N K D ( g( p, x), λ ) (∇ g( p, x)(q, u)) Moreover, by passing k to infinity in (5.34) it follows that By the assumption of the theorem there is some ( q, ũ) with ∇ g( p, x)( q, ũ) ∈ F 1 − F 2 and w T ∇L λ ( p, x)( q, ũ) > 0. Applying Corollary 2.12 we obtain From Theorem 2.8 we can deduce that for every k there is some ( qk , ũk ) satisfying for some constant β ≥ 0 not depending on k.
In case when (q, u) = (0, 0) Theorem 5.3 constitutes a sufficient condition for the metric regularity of M around (( p, x), 0).This is an interesting result for its own sake.On the other hand, when applying Theorem 5.3 for directions (0, u), u = 0, we have an efficient tool for verifying the necessity of SOCIC for the isolated calmness property of S.

On the Aubin property of the solution map
In the following theorem we state our main result concerning the Aubin property of the solution map S relative to some set P. Theorem 6.1.Assume that Assumption 1 is fulfilled and we are given a closed set P ⊂ R l containing p such that the following assumptions are fulfilled: (i) For every q ∈ T P ( p) there is some u ∈ R n such that (ii) For every (0, 0) = (q, u) verifying (6.38) the (partial) directional non-degeneracy condition ∇ 2 g( p, x) T µ = 0, µ ∈ sp N T D ( g( p, x)) (∇ g( p, x)(q, u)) ⇒ µ = 0 (6.39) is fulfilled and for every λ ∈ Ξ(( p, x), − f ( p, x); (q, u)), every η ∈ N K D ( g( p, x), λ ) (∇ g( p, x)(q, u)) satisfying 0 = ∇L λ ( p, x)(q, u) + b( p, x) T η, every pair of faces F 1 , F 2 of the critical cone Then the solution mapping S to the variational system (1.3) has the Aubin property relative to P around ( p, x) and for every q ∈ T P ( p) there holds Proof.We will invoke Theorem 3.5 in order to prove the proposition.Observe that (6.39) implies the non-degeneracy of the system g(•) ∈ D in direction (q, u) at ( p, x) and by Theorem 4.3 we have that DΨ ( p, x, x), − f ( p, x) (q, u, u) = DG ( p, x), − f ( p, x) (q, u) and all tangents (q, u, u * ) to gph G at (( p, x), − f ( p, x)) are derivable.Since DM(( p, x), 0)(q, u) = ∇ f ( p, x)(q, u)+DG(( p, x), − f ( p, x))(q, u) and taking into account Remark 3.6, assumption (i) of Theorem 3.5 is satisfied due to the first assumption.
By the definition of the directional limiting coderivative there are sequences t k ↓ 0, (q k , u k , w * k ) → (q, u, 0) and (q * k , u * k , w k ) → (q * , 0, w) such that , where p k := p + t k q k , x k := x + t k u k .We can now proceed as in the proof of Theorem 5.3 to find the sequences x * k and λ k as well as λ = lim As in the proof of Theorem 5.3 we can also deduce b( p, x)w ∈ F 1 − F 2 .By assumption (ii) of the theorem there is some w with ∇ 2 g( p, x) w ∈ F 1 − F 2 and w T ∇ 2 L λ ( p, x) w > 0, provided w = 0, which we now assume.
Next observe that the implication follows from (6.39) by virtue of Corollary 2.12.By condition (6.40) and Theorem 2.8, there is some real β > 0 such that for every k sufficiently large there are some wk satisfying  (5.33).By passing to the limit we obtain the contradiction w T ∇ 2 L λ w ≤ 0 and thus w = 0.It remains to show that q * = 0. Observe that (6.40) is equivalent to Hence there is some ū ∈ R n with ∇ 1 g( p, x)q * + ∇ 2 g( p, x) ū ∈ F 1 − F 2 .Further, by assumption (6.40) and Theorem 2.8, there is some real β > 0 such that for every k sufficiently large there exist some vectors ūk satisfying ∇ by means of (5.33).By passing k to infinity we obtain q * T q * ≤ 0 implying q * = 0. Thus all assumptions of Theorem 3.5 are fulfilled and the statement is established.

Conclusion
In most rules of generalized differentiation one associates with the data a certain mapping and requires, as a qualification condition, the metric subregularity of this mapping at the considered point, see, e.g., [22,21,20,23].Correspondingly, in the directional limiting calculus the qualification conditions amount typically to the directional metric subregularity of the respective associated mappings, cf.[1].In both cases, however, the required property should be verifiable via suitable conditions expressed solely in terms of problem data.In this paper we construct such conditions on the basis of the (stronger) property of directional metric regularity, see Theorems 2.11, 3.5, 5.3 and 6.1.
In general, the principal questions related to metric subregularity, calmness and the associated areas of error bounds and subtransversality have been thoroughly investigated by many notable researchers including A. Y. Kruger ([6,7,8,25] and many other works on this subject).Via the research, discussed in this paper, the authors would like to give credit to their friend Alex on the occasion of his 65 th birthday.

Theorem 2 . 3 ([ 29 ,
Theorem 3.3]).Assume that the set-valued mapping F : R n ⇒ R m has locally closed graph around ( x, ȳ) ∈ gph F. Then F is metrically regular around ( x, ȳ) if and only if

2 .
Given a set Y ⊂ R m containing ȳ, the mapping S is said to have the Aubin property relative to Y around ( ȳ, x) if there are neighborhoods V of ȳ, U of x and a real L ≥ 0 such that S(y) ∩U ⊂ S(y ) + L y − y B R n ∀y, y ∈ Y ∩V.

Corollary 3 . 2 .
Assume that M has locally closed graph around and is metrically subregular at ( p, x, 0) ∈ gph M. Then S is isolatedly calm at ( p, x) if and only if (3.18) holds.A sufficient condition for the Aubin property of S around ( p, x) is constituted by the following theorem.Theorem 3.3 ([16, Theorem 4.4]).Assume that M has locally closed graph around the reference point ( p, x, 0) ∈ gph M and assume that h(p, x) := (p, x, 0).Proposition 3.4.Let ( p, x, 0) ∈ gph M and consider a subset P ⊂ R l containing p.If the system h(p, x) ∈ gph M (3.19) enjoys the Robinson stability property at ( p, x), where P is equipped with the induced norm topology of R l , then S has the Aubin property relative to P around ( p, x).Proof.Obviously S is also the solution mapping of the inclusion (p, x, 0) ∈ gph M. By the definition of the Robinson stability together with the assumption on the topology of P, there are neighborhoods Q of p in R l , U of x and a constant κ ≥ 0 such that dist(x, S(p)) ≤ κdist((p, x, 0), gph M) ∀(p, x) ∈ (Q ∩ P) ×U.Next consider p, p ∈ Q ∩ P and x ∈ S(p) ∩U.Then dist(x, S(p )) ≤ κdist((p , x, 0), gph M) ≤ κ dist((p, x, 0), gph M) + p − p = κ p − p and thus x ∈ S(p ) + (κ + 1) p − p B R n .It follows that S(p) ∩ U ⊂ S(p ) + (κ + 1) p − p B R n showing the Aubin property of S relative to P. Theorem 3.5.Assume that M has a locally closed graph around the reference point ( p, x, 0) ∈ gph M and consider a closed set P ⊂ R l containing p. Further assume that