Directional Hölder Metric Regularity

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Directional Hölder Metric Regularity

van Ngai Huynh, Huu Tron Nguyen, Michel Théra

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van Ngai Huynh, Huu Tron Nguyen, Michel Théra. Directional Hölder Metric Regularity. 2015.

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Directional H¨older Metric Regularity

Huynh Van Ngai · Nguyen Huu Tron · Michel Th´era

Received: date / Accepted: date

Abstract This paper sheds new light on regularity of multifunctions through various characterizations of directional H¨older/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these charac- terizations, we show that directional H¨older/Lipschitz metric regularity is stable, when the multifunction under consid- eration is perturbed suitably. Applications of directional H¨older/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

Keywords Slope · Metric regularity · H¨older metric regularity · Generalized equation · Fr´echet subdifferential · Asplund spaces · Ekeland variational principle, Hadamard directional differentiability.

Mathematics Subject Classification (2000) 49J52; 49J53; 58C06; 47H04; 54C60; 90C30

1 Introduction

Throughout the last three decades, metric regularity has been significantly developed and has become one of the central concepts of modern variational analysis.The terminology “metric regularity” was coined by Borwein [1], but the roots of this notion can be traced back to the classical open mapping theorem and its subsequent generaliza- tion to nonlinear mappings known as the Lyusternik-Graves theorem. The theory of metric regularity is extraordi-

Research partially supported by Ministerio de Economıa y Competitividad under grant MTM2011-29064-C03(03), by LIA “ FormathVietnam ” and by NAFOSTED.

Van Ngai Huynh

Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Vietnam E-mail: ngaivn@yahoo.com

Huu Tron Nguyen

Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Vietnam E-mail: nguyenhuutron@qnu.edu.vn

Michel Th´era ( )

Laboratoire XLIM, Universit´e de Limoges and Federation University Australia

E-mail: michel.thera@unilim.fr, m.thera@federation.edu.au

nary useful for investigating the behavior of solutions of a nonlinear equation under small perturbations of the data, or more generally the behavior of the solution set of generalized equations associated with a set-valued mapping.

As a result, metric regularity plays an important role in many aspects of optimization, differential inclusions, con- trol theory, numerical methods and in many problems of analysis. According to the long history of metric regular- ity there is an abundant literature on conditions ensuring this property. We refer the reader to the basic monographs [2, 3, 4, 5, 6], to the excellent survey of A. Ioffe [7] (in preparation) and to some (non exhaustives) references [1, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

Apart from the study of the usual (Lipschitz) metric regularity, H¨older metric regularity or more generally nonlinear metric regularity have been studied over the years 1980 − 1990s by several authors, including for example Borwein and Zhuang [1], Frankowska [16], Penot [25], and recently, for instance, Frankowska and Quincampoix [17], Ioffe [26], Li and Mordukhovich [27], Oyang and Mordukovhich [28].

Recently, several directional versions of metric regularity notions were considered. In [29, 30], Arutyunov et al have introduced and studied a notion of directional metric regularity. This notion is an extension of an earlier notion used by Bonnans and Shapiro [10] to study sensitivity analysis. Later, Ioffe [31] has introduced and investigated an extension called relative metric regularity which covers many notions of metric regularity in the literature. In particular, another version of directional metric regularity/subregularity has been introduced and extensively studied by Gfrerer in [32, 33] where some variational characterizations of this concept have been established and successfully applied to study optimality conditions for mathematical programs. In fact, this directional regularity property has been earlier used by Penot [34] to study second order optimality conditions. In the line of the directional version of metric subregularity considered by Gfrerer [32], Huynh, Nguyen and Tinh [35] have studied directional H¨older metric subregularity in order to investigate tangent cones to zero sets in degenerate cases.

It is our aim in the present article to study a directional version of H¨older metric regularity. The structure of the

article is as follows. In the Section 2, we establish slope-based characterizations of directional H¨older/Lipschitz met-

ric regularity. In Section 3, we present a stability property for directional Lipschitz metric regularity. In Section 4, a

sufficient condition for directional H¨older/Lipschitz metric regularity based on the Fr´echet coderivative is established

in Asplund spaces. This condition becomes necessary when, either the multifunction under consideration is convex, or

when considering directional Lipschitz metric regularity. It was silmultaneously showed that under this condition, di-

rectional H¨older/Lipschitz metric regularity persists when the multifunction is perturbed by a Hadamard differentiable

mapping. Applications to the study of the stability and the sensitivity analysis of parameterized optimization problems

are discussed in Section 4. The last section contains concluding remarks.

2 Notations and Preliminaries

Throughout we let X and Y denote metric spaces endowed with metrics both denoted by d( · , · ). We denote the open and closed balls with center x and radius r > 0 by B(x, r) and ¯ B(x,r), respectively. For a given set C, we write intC for its topological interior. A set-valued mapping (also called multifunction) F : X ⇉ Y is a mapping assigning, to each point x ∈ X , a subset (possibly empty) F(x) of Y . We use the notations

gph F := { (x, y) ∈ X × Y : y ∈ F(x) } and Dom F := { x ∈ X : F (x) 6 = /0 }

for the graph of and the domain of F, respectively. For each set-valued mapping F : X ⇉ Y , we define the inverse of F, as the mapping F ^{− 1} : Y ⇉ X defined by F ^{− 1} (y) := { x ∈ X : y ∈ F (x) } , y ∈ Y } and satisfying

(x, y) ∈ gph F ⇐⇒ (y, x) ∈ gph F ^{− 1} .

We use the standard notation d (x,C) to denote the distance from x to a set C ; it is defined by d (x,C) = inf z ∈ C d(x,z), with the convention that d (x, S) = +∞ whenever S is the empty set. As pointed out in the introduction, main attention in this contribution is paid to the study of the concept of metric regularity. Recall that a mapping F is said to be metrically regular at (¯ x, y) ¯ ∈ gph F with modulus τ > 0, if there exists a neighborhood U × V of ( x, ¯ y) ¯ such that

d(x, F ^{− 1} (y)) ≤ τ d(y, F(x)) for all (x,y) ∈ U × V. (1) In other words, metric regularity allows to estimate the dependence of the distance of a trial point x ∈ X from the solution set F ^{− 1} (y) in terms of the residual quantity d(y , F (x)) for all pairs (x, y) around the reference pair (¯ x, y) ¯ ∈ gph F . The infinum of all moduli τ is denoted by reg F( x, ¯ y). ¯

If in the above definition we fix y = y ¯ in (1), then we obtain a weaker notion called metric subregularity, see e.g.

[4, 6, 36]. Observe that this latter property is equivalent to the existence of some neighborhood U of ¯ x such that d (x, F ^{− 1} ( y)) ¯ ≤ τ d( y, ¯ F(x)) for all x ∈ U.

It is also well known that metric subregularity can be treated in the framework of the theory of error bounds of extended- real-valued functions, see e.g. [37, 38].

An H¨older version of metric regularity is defined as follows (Frankowska and Quincampoix [17], Ioffe [26]). Let q ∈ (0, 1] be given. A mapping F is said to be metrically q-regular or H¨older metrically regular of order q at ( x, ¯ y) ¯ ∈ gph F with modulus τ > 0 if there exists a neighborhood U × V of ( x, ¯ y) ¯ such that

d(x,F ^{− 1} (y)) ≤ τ [d(y, F(x))] ^q for all (x, y) ∈ U × V. (2) The infimum of all moduli τ satisfying (2) is denoted by reg ^q F( x, ¯ y), ¯ i.e.,

reg _q F ( x, ¯ y) = ¯ inf { τ > 0 : ∃ δ > 0 s.t . d(x,F ^{− 1} (y)) ≤ τ (d(y , F (x))) ^q for all (x, y) ∈ B( x, ¯ δ ) × B(¯ y, δ ) } .

Fixing y = y ¯ in the above definition, gives the concept of q-H¨older metric subregularity of the set-valued mapping F at ( x, ¯ y). ¯

In the present paper, we are interested in a directional version of H¨older metric regularity, defined as follows.

Definition 2.1 Let X ,Y be normed linear spaces. Let a real γ _∈ ]0, 1] and (u, v) ∈ X × Y be given. A multifunction F is said to be (directionally) metrically γ -regular at ( x, ¯ y) ¯ ∈ gph F in the direction (u, v) with a modulus τ > 0 iff there exists δ , ε , η > 0 such that

d(x, F ^{− 1} (y)) ≤ τ [d(y , F (x))] ^γ (3)

for all (x,y) ∈ B((¯ x, y), ¯ δ ) with (x, y) ∈ (¯ x, y)+cone ¯ B((u, v), ε ); d(y, F(x)) ≤ η _k (x, y) − ( x, ¯ y) ¯ k ^1/γ . Here cone B((u, v), ε ) stands for the conic hull of B((u, v), ε ), i.e., cone B((u, v), ε ) = ∪ λ ≥ 0 λ B((u, v), ε ).

Remark 2.1 In Definition 2.1, the triple δ , ε , η > 0 may be replaced by just a single positive number.

If (3) is required to be verified only at y = y, ¯ and x ∈ B(¯ x, δ ) with x ∈ x ¯ + cone B(u, ε ), we say that F is directionally H¨older metrically subregular at (¯ x, y) ¯ in the direction u. When γ = 1, one refers to the (Lipschitz) directional metric regularity, as equivalently introduced by Gfrerer [32].

Since in the definition above, the gauge condition d(y , F (x)) ≤ η _k (x,y) − ( x, ¯ y) ¯ k ^1/γ is added, when (u, v) = (0,0), the version of H¨older/Lipschitz metric regularity in Definition 2.1 is even weaker than the usual ones defined by (2).

For example, obviously, the function f (x) = x ² , x ∈ R is H¨older metrically regular of order 1/2 at (0, 0) in the direction (0, 0) in the sense of Definition 2.1, but is not in the usual sense of (2). The added gauge conditions in concepts of metric regularity are really needed when the usual regularity is not satisfied (see, e.g., Ioffe [26]). Let us mention that directional H¨older/Lipschitz metric regularity is obviously stronger than H¨older/Lipschitz metric subregularity. The main purpose of the present paper is to show that the tools of variational analysis such as the f and the concept of coderivative can be used to efficiently characterize directional H¨older/Lipschitz metric regularity. Our aim is to show that this directional version of H¨older/Lipschitz metric regularity, although weaker than the usual metric regularity, possesses the suitable stability properties which are lost in the case of metric subregularity.

3 Slope Characterizations of Directional H¨older Metric Regularity

Let X be a metric space. Let f : X → R ∪ { +∞ } be a given extended-real-valued function. As usual, dom f := { x ∈ X : f (x) < +∞ } denotes the domain of f .

Recall from [39], (see also [40] and the discussion in [38]) that the local slope and the nonlocal slope (see, e.g., [41]) of the function f at x ∈ dom f , are the quantities denoted respectively by | ∇ f | (x) and | Γ f | (x). Using the notation [a] + for max { a,0 } , they are defined by | ∇ f | (x) = | Γ f | (x) = 0 if x is a local minimum of f and otherwise by

| ∇ f | (x) = lim sup

y → x, y 6 =x

f (x) − f (y)

d(x, y) (4)

and

| Γ f | (x) = sup

y 6 =x

[ f (x) − f (y)] +

d(x, y) . (5)

For x ∈ / dom f , we set | ∇ f | (x) = | Γ f | (x) = +∞. Obviously, | ∇ f | (x) ≤ | Γ f | (x) for all x ∈ X .

Let X, Y be normed spaces. If not specified otherwise, we assume that the norm on X × Y is defined by k (x, y) k = k x k + k y k , (x, y) ∈ X × Y.

For a closed multifunction F : X ⇉ Y , (i.e., when the graph of F is closed in X × Y ), the lower semicontinuous envelope of the distance function (x,y) → d(y , F (x)) is defined, for a given (x, y) ∈ X × Y , by

ϕ (x,y) := lim inf

(u,v) → (x,y) d(v , F(u)) = lim inf

u → x d (y , F(u)).

In what follows, for u ∈ X , we shall use the notation x → _u x ¯ to mean x → x ¯ if u = 0,

x − x ¯

k x − x ¯ k − _k ^u _{u k} → 0

x → x ¯ if u 6 = 0.

Obviously, given a sequence { x _n } ⊆ X, u ∈ X the two facts are equivalent:

(C 1 ) : { x _n } →

u x; ¯

(C ₂ ) : { x _n } → x ¯ and there is a sequence of nonnegative reals { δ n } → 0 such that x _n ∈ x ¯ + cone B(u, δ n ), ∀ n ∈ N . We need the following series of useful lemmas whose proofs are straightforward.

Lemma 3.1 Let X be a Banach space and Y be a normed space. Suppose a closed multifunction F : X ⇉ Y and a point (¯ x, y) ¯ ∈ gph F are given. Given (u, v) ∈ X × Y , and γ _∈ ]0,1], then F is metrically γ ₋ regular at ( x, ¯ y) ¯ in the direction (u, v) with modulus τ > 0, if and only if, there exist real numbers τ , δ > 0 such that

d(x, F ^{− 1} (y)) ≤ τϕ ^γ (x, y)

for all (x,y) ∈ B((¯ x, y), ¯ δ ) ∩ (( x, ¯ y) + ¯ cone B((u, v), δ )) with d(y,F (x)) ≤ δ _k (x, y) − (¯ x, y) ¯ k ^{1 γ} .

Lemma 3.2 Let u ∈ X and x ¯ ∈ X be given as well as a sequence { x n } such that x n → _u x. ¯ Then, for any sequence { δ n } ↓ 0 of nonnegative reals and any sequence { z _n } ⊆ X with

k z _n − x _n k ≤ δ n k x _n − x ¯ k , n ∈ N , (6)

one has z n → _u x. ¯

Proof It suffices to prove the result when k u k = 1. If x n = x, ¯ then from (6) we have z n = x ¯ and we are done. Otherwise, one has

(1 − δ n ) k x _n − x ¯ k ≤ k z _n − x ¯ k , and since

k z _n − x ¯ − u k z _n − x ¯ kk ≤ 2 k z _n − x _n k + k x _n − x ¯ − u k x _n − x ¯ kk

≤ 2 δ n k x _n − x ¯ k + k x _n − x ¯ − u k x _n − x ¯ kk , it follows that

z n − x ¯ k z _n − x ¯ k − u

= k z n − x ¯ − u k z n − x ¯ kk

k z _n − x ¯ k ≤ (1 − δ n ) ^{− 1}

2 δ n +

x n − x ¯ k x _n − x ¯ k − u

. As

x n − x ¯ k x n − x ¯ k − u

→ 0 and { δ n } → 0, one obtains

z _n − x ¯ k z _n − x ¯ k − u

→ 0

as n tends to infinity. Moreover, it is easy to see that z n → x. So, one has that ¯ z n → _u x. ¯

Theorem 3.1 Let X be a Banach space and Y be a normed space. Suppose a closed multifunction F : X ⇉ Y and a point ( x, ¯ y) ¯ ∈ gph F are given. Given (u, v) ∈ X × Y , and γ _∈ ]0, 1], F is directional metrically γ -regular at ( x, ¯ y) ¯ in the direction (u, v), if and only if,

lim inf

(x,y) →

(u,v) ( x,¯ ¯ y), ϕ(x,y)>0 ϕγ (x,y) k(x,y)−( x, ¯ y)k → ¯ 0

| Γ ϕ ^γ ( · , y) | (x) > 0. (7)

Proof For the sufficiency, assume that (7) holds and assume on the contrary that F fails to be directional metrically γ -regular in the direction (u,v). Then for every n ∈ N , there exists (x n , y n ) with

0 < k x _n − x ¯ k + k y _n − y ¯ k < 1

n ; d(y n , F(x n )) ^γ ≤ 1

n ² k (x n , y _n ) − ( x, ¯ y) ¯ k ; (x n , y n ) →

(u,v) ( x, ¯ y) ¯ and such that

d(x n , F ^{− 1} (y n )) > n ² [d(y n , F(x n ))] ^γ ( ≥ n ² ϕ ^γ (x n , y _n )).

From the relation d(y n , F(x n )) ^γ ≤ _n ¹ 2 k (x n ,y n ) − ( x, ¯ y) ¯ k , it follows that ϕ ^γ (x n , y n ) ≤ 1

n ² k (x n , y n ) − (¯ x, y) ¯ k . (8) Applying the Ekeland variational principle to the lower semicontinuous function ϕ ^γ ( · , y n ) on the Banach space X , one gets a point z _n satisfying the conditions:

k z _n − x _n k ≤ 1

n k (x n , y _n ) − (¯ x, y) ¯ k , ϕ ^γ (z n , y _n ) ≤ ϕ ^γ (x n , y _n ), and

ϕ ^γ (z n , y _n ) ≤ ϕ ^γ (x, y _n ) + 1

n k x − z _n k , ∀ x ∈ X.

Consequently,

| Γ ϕ ^γ ( · ,y n ) | (z n ) ≤ 1

n , (9)

and by

k (x n , y _n ) − (¯ x, y) ¯ k ≤ k z _n − x _n k + k (z n ,y n ) − ( x, ¯ y) ¯ k ≤ 1

n k (x n , y _n ) − ( x, ¯ y) ¯ k + k (z n , y _n ) − (¯ x, y) ¯ k , one obtains

k (x n , y _n ) − ( x, ¯ y) ¯ k ≤ n

n − 1 k (z n , y _n ) − ( x, ¯ y) ¯ k . (10) Hence, combining (8) and (10) we obtain:

ϕ ^γ (z n , y _n ) ≤ ϕ ^γ (x n , y _n ) ≤ 1

n ² k (x n , y _n ) − (¯ x, y) ¯ k ≤ 1

n(n − 1) k (z n ,y n ) − ( x, ¯ y) ¯ k . The latter relation implies that

n lim → ∞

ϕ ^γ (z n , y n )

k (z n , y _n ) − ( x, ¯ y) ¯ k = 0.

Moreover, invoking Lemma 3.2, relations (x n , y _n ) →

(u,v) (¯ x, y) ¯ and k z _n − x _n k ≤ 1/n k (x n , y _n ) − (¯ x, y) ¯ k imply that (z n ,y n ) →

(u,v) ( x, ¯ y). ¯ Hence, by (7), z _n ∈ F ^{− 1} (y n ) for n large, say, n ≥ n ₀ . For n ≥ n ₀ , as

ϕ ^γ (x n , y n ) < 1

n ² d(x n , F ^{− 1} (y n )),

applying again the Ekeland variational principle to the lower semicontinuous ϕ ^γ ( · , y _n ), one gets a point w _n ∈ X satisfying the conditions:

k w _n − x _n k < 1

n d(x n , F ^{− 1} (y n )), ϕ ^γ (w n , y _n ) ≤ ϕ ^γ (x n , y _n ) and,

ϕ ^γ (w n , y _n ) ≤ ϕ ^γ (x, y _n ) + 1

n k x − w _n k , ∀ x ∈ X . We deduce that

| Γ ϕ ^γ ( · ,y n ) | (w n ) ≤ 1 n . Moreover we claim that

ϕ ^γ (w n , y _n )

k (w n ,y n ) − ( x, ¯ y) ¯ k ≤ 1

n ² + 1

n ² (n ² − 1) . (11)

Indeed, as

k (x n , y _n ) − (¯ x, y) ¯ k ≤

≤ k (x n , y _n ) − (w n , y _n ) k + k (w n ,y n ) − ( x, ¯ y) ¯ k

= k x _n − w _n k + k (w n , y _n ) − ( x, ¯ y) ¯ k

≤ 1

n d(x n ,F ^{− 1} (y n )) + k (w n , y _n ) − ( x, ¯ y) ¯ k

≤ 1

n k x _n − z _n k + k (w n , y _n ) − ( x, ¯ y) ¯ k

≤ 1

n ² k (x n , y _n ) − ( x, ¯ y) ¯ k + k (w n , y _n ) − ( x, ¯ y) ¯ k ,

we have

k (x n , y _n ) − (¯ x, y) ¯ k ≤ n ²

n ² − 1 k (w n , y _n ) − ( x, ¯ y) ¯ k , and

ϕ ^γ (w n , y _n ) ≤ ϕ ^γ (x n , y _n ) ≤ 1

n ² k (x n , y _n ) − ( x, ¯ y) ¯ k

≤ 1

n ² k (w n , y _n ) − ( x, ¯ y) ¯ k + 1

n ² k x _n − w _n k

≤ 1

n ² k (w n , y _n ) − ( x, ¯ y) ¯ k + 1

n ³ d(x n , F ^{− 1} (y n )) ≤ 1

n ² k (w n , y _n ) − ( x, ¯ y) ¯ k + 1

n ³ k x _n − z _n k

≤ 1

n ² k (w n , y _n ) − ( x, ¯ y) ¯ k + 1

n ⁴ k (x n , y _n ) − ( x, ¯ y) ¯ k . Therefore,

ϕ ^γ (w n , y _n ) ≤ 1

n ² + 1 n ² (n ² − 1)

k (w n , y _n ) − (¯ x, y) ¯ k , and (11) is established. By virtue of Lemma 3.2, since (x n , y n ) →

(u,v) (¯ x, y) ¯ and k w n − x n k ≤ _n ¹ 2 k (x n , y n ) − (¯ x, y) ¯ k , one has (w n ,y n ) →

(u,v) ( x, ¯ y). ¯

In conclusion, we have obtained a sequence { w _n } which satisfies w _n ∈ / F ^{− 1} (y n ); (w n , y _n ) →

(u,v) ( x, ¯ y); ¯ ϕ ^γ (w n , y _n )

k (w n , y n ) − ( x, ¯ y) ¯ k → 0 and | Γ ϕ ^γ ( · , y _n ) | (w n ) ≤ 1 n . Hence, condition (7) is violated, and the sufficiency is proved.

For the necessary part, suppose that there exist reals τ , δ > 0 such that d(x, F ^{− 1} (y)) ≤ τ [d(y , F (x))] ^γ

for all (x, y) ∈ B((¯ x, y), ¯ δ ) ∩ (( x, ¯ y) + ¯ cone B((u, v), δ )) with d(y,F (x)) ≤ δ _k (x, y) − (¯ x, y) ¯ k ^{1 γ} . According to Lemma 3.1,

d(x,F ^{− 1} (y)) ≤ τϕ ^γ (x,y) ∀ (x, y) ∈ B(( x, ¯ y), ¯ δ ) ∩ (( x, ¯ y) + ¯ cone B((u, v), δ )) with 0 < ϕ ^γ (x, y)

k (x, y) − ( x, ¯ y) ¯ k ≤ δ . Let (x, y) ∈ B((¯ x, y), ¯ δ ) ∩ ((¯ x, y) + ¯ cone B((u,v), δ )) with (x, y) 6 = ( x, ¯ y) ¯ and 0 < _{k (x,y)} ^{ϕ γ (x,y)} _{− ( x, ¯ y) ¯ k} ≤ δ . Then, for every ε > 0, there exists an element z ∈ F ^{− 1} (y) such that

k x − z k ≤ ( τ + ε ) ϕ ^γ (x, y) = ( τ + ε )[ ϕ ^γ (x, y) − ϕ ^γ (z, y)].

Consequently,

| Γ ϕ ^γ ( · ,y) | (x) ≥ 1 ( τ + ε ) . As ε > 0 is arbitrary, one obtains

lim inf

(x,y) →

(u,v) ( x,¯ ¯ y), ϕ(x,y)>0 ϕγ (x,y)

kx−¯ xk → 0

| Γ ϕ ^γ ( · , y) | (x) ≥ 1 τ ^{> 0,}

completing the proof.

The theorem above yields the following local slope characterization of the directional metric γ ₋ regularity . Theorem 3.2 Let X be a Banach space and Y be a normed space. Suppose a closed multifunction F : X ⇉ Y and a point ( x, ¯ y) ¯ ∈ X × Y are given such that y ¯ ∈ F (¯ x). Let (u, v) ∈ X × Y , and γ _∈ (0, 1] be fixed. If

lim inf

(x,y) →

(u,v) ( x,¯ ¯ y), ϕ(x,y)>0 ϕγ (x,y) k(x,y)−(¯ x, y)k → ¯ 0

| ∇ ϕ ^γ ( · , y) | (x) > 0, (12)

then there exist reals τ , δ > 0 such that

d(x, F ^{− 1} (y)) ≤ τ [d(y , F (x))] ^γ

for all (x,y) ∈ B((¯ x, y), ¯ δ ) ∩ (( x, ¯ y) + ¯ cone B((u, v), δ )) with d(y , F(x)) ≤ δ _k (x, y) − ( x, ¯ y) ¯ k ^{1 γ} . That is, F is directionally metrically γ -regular at ( x, ¯ y) ¯ in the direction (u,v) with modulus τ .

Remark 3.1 Condition (12) of Theorem 3.2 fails to be a necessary condition when γ _∈ ]0,1[. To see this, let us consider the mapping F : R ² → R defined by

F (x) = (x 1 − x ₂ ) ³ , x = (x 1 , x ₂ ) ∈ R ² .

Here, R ² is equipped with the Euclidean norm. For x = (x 1 , x ₂ ) ∈ R ² and y ∈ R , F ^{− 1} (y) = { (t + √ ³ y, t) : t ∈ R } . Therefore, d(x, F ^{− 1} (y)) = | x ₁ − x ₂ − √ ³ y | / √

2. Noticing that from the inequality 0 ≤ 3(a + b) ² , we deduce that (a − b) ² ≤ 4(a ² + ab + b ² ) and therefore that | a − b | ³ ≤ 4 | a ³ − b ³ | , for all a, b ∈ R . Using the last inequality yields

d (x, F ^{− 1} (y)) = | x ₁ − x ₂ − √ ³ y | / √

2 ≤ 2 ^1/6 | (x 1 − x ₂ ) ³ − y | ^1/3 = 2 ^1/6 | y − F(x) | ^1/3 for all x ∈ R ² , y ∈ R .

Consequently, F is metrically 1/3 − regular at (0, 0) (in the direction (0,0)). However, for any x = (x 1 , x ₁ ) ∈ R ² and y ∈ R with y 6 = 0, one has | ∇ ϕ ^1/3 ( · , y) | (x) = 0, where, ϕ (x, y) = | y − F(x) | .

As stated in the next theorem, when γ = 1, then condition (12) becomes a necessary condition.

Theorem 3.3 Let X be a Banach space and Y be a normed space. Suppose a closed multifunction F : X ⇉ Y and a point ( x, ¯ y) ¯ ∈ X × Y are given such that y ¯ ∈ F (¯ x). Let us fix (u, v) ∈ X × Y . Then, the following are equivalent:

(i)

lim inf

(x,y) →

(u,v) ( x, ¯ y), ¯ ϕ(x,y)>0 ϕ (x,y) k(x,y)−( x, ¯ y)k → ¯ 0

| ∇ ϕ ( · , y) | (x) > 0, (13)

(ii) there exist reals τ > 0 and δ > 0 such that

d(x, F ^{− 1} (y)) ≤ τ d(y,F (x))

for all (x, y) ∈ B(( x, ¯ y), ¯ δ ) ∩ ((¯ x, y) + ¯ cone B((u, v), δ )) with d(y,F (x)) ≤ δ _k (x, y) − (¯ x, y) ¯ k .

That is F is directionally metrically regular at ( x, ¯ y) ¯ in the direction (u, v) with modulus τ .

Proof According to Theorem 3.2, condition (13) is sufficient to obtain (ii). Conversely, suppose that there exist τ > 0 and δ _∈ ]0, 1[ such that

d(x, F ^{− 1} (y)) ≤ τ d(y , F (x)) (14)

for all (x, y) ∈ B((¯ x, y), ¯ 2 δ ) ∩ ((¯ x, y) +cone ¯ B((u, v), 2 δ )) with d (y, F(x)) ≤ 2 δ _k (x, y) − (¯ x, y) ¯ k . Let (x, y) ∈ B(( x, ¯ y), ¯ δ ) ∩ ((¯ x, y) + ¯ cone B((u, v), δ )) be such that

0 < d(y,F (x)) ≤ δ ²

4 k (x, y) − ( x, ¯ y) ¯ k . (15)

Since ϕ (x, y) = sup

ε>0

w ∈ inf B(x,ε) d(y,F (w)) = lim inf

w → x d (y, F(w)), we have ϕ (x, y)

k (x, y) − ( x, ¯ y) ¯ k ≤ δ ² 4 and we can write ϕ (x, y) = lim

n → +∞ d(y,F (u n )) for some sequence { u _n } in X converging to x. Without loss of generality, we can suppose that d (y, F(u n )) > (1 − _n ¹ 2 ) ϕ (x,y), and

k u _n − x k < 1

n ϕ (x, y) ≤ δ ² _k (x, y) − ( x, ¯ y) ¯ k

4n ≤ δ ³

4n < δ

n (16)

and

d(y , F (u n )) <

1 + 1

n

ϕ (x,y). (17)

Note also that for every n ∈ N , there exists y _n ∈ F(u n ) such that d (y, F(u n )) ≤ k y − y _n k <

1 + 1

n

d(y, F(u n )). (18)

Set z _n := ¹⁺ⁿ

1 2

1+n y + ^{n(1 − n −}

1 2 ) 1+n y _n . Claim 3.1

k y − z _n k <

1 − 1

n

ϕ (x, y) for large n. (19)

Indeed,

k y _n − z _n k ≤ 1 + n ^1/2

1 + n k y − y _n k , (20)

and

k y − z n k = n(1 − n ^{− 2 1} )

1 + n k y − y n k < n(1 − n ^{− 1 2} ) 1 + n

1 + 1

n

d(y , F (u n ))

<

1 − n ^{− 2 1}

d(y, F(u n )) <

1 − n ^{− 2 1} 1 + 1

n

ϕ (x,y)

≤

1 − n ^{− 2 1} 1 +n ^{− 1 2} ϕ (x, y).

Hence, k y − z n k < (1 − ¹ _n ) ϕ (x,y), as claimed.

Claim 3.2 z _n ∈ / F(u n ) for large n.

Indeed, using (19), if we suppose that z n ∈ F(u n ) (for n ≥ n ₀ ), we deduce that (1 − 1

n ² ) ϕ (x, y) < d (y, F(u n )) ≤ k y − z _n k <

1 − 1

n

ϕ (x, y),

we have a contradiction. Hence, as claimed, for n large, z _n ∈ / F(u n ).

Claim 3.3

(⋆⋆) (u n ,z n ) ∈ ( x, ¯ y) + ¯ cone B((u, v), 2 δ ) for large n.

As (x,y) ∈ ( x, ¯ y) + ¯ cone B((u,v), δ ), there is some λ > 0 such that ^{(x,y) −} _λ ^{( x,¯ ¯ y)} ∈ B((u, v), δ ). Then, λ _≥ ^{k (x, y) − ( x, ¯ y) ¯ k}

k (u, v) k + δ ^. Let us observe that

k z _n − y k λ by( ^< 19 )

(1 − 1/n)

λ ϕ (x, y) < (1 − 1/n) δ ²

4n λ ^{k (x, y) − ( x, ¯ y) ¯ k ≤}

(1 − ¹ _n ) δ ²

4n ( k (u, v) k + δ ) ≤ (1 − ¹ _n ) δ ²

4 ≤ δ

2 . Therefore, one has

(u n ,z n ) − ( x, ¯ y) ¯

λ − (u, v)

≤

(u n ,z n ) − (x,y) λ

+

(x,y) − (¯ x,¯ y)

λ − (u, v)

≤ ^{k u n − x k +} _λ ^{k z n − y k} + δ

≤ _λ ^δ _n + ^δ ₂ + δ < 2 δ . This yields, (⋆⋆).

Claim 3.4

(⋆ ⋆ ⋆) d(z n , F(u n )) < 2 δ ² _k (u n ,z n ) − ( x, ¯ y) ¯ k for large n.

We know that

( ♣ ) d(z n , F(u n )) < 1 + n ^1/2 1 + n (1 + 1

n ) ² ϕ (x, y) ≤ 1 + n ^1/2 1 + n (1 + 1

n ) ² δ ²

4 k (x,y) − (¯ x, y) ¯ k . We have:

k (x, y) − ( x, ¯ y) ¯ k ≤ k (u n , z _n ) − (x,y) k + k (u n ,z n ) − (¯ x, y) ¯ k , and

k (u n , z _n ) − (x,y) k = k u _n − x k + k z _n − y k < δ ²

4n k (x, y) − ( x, ¯ y) ¯ k + (1 − 1/n) δ ²

4n k (x, y) − ( x, ¯ y) ¯ k = δ ²

4n k (x, y) − ( x, ¯ y) ¯ k , it follows

( ♥ ) k (x, y) − ( x, ¯ y) ¯ k < 1

(1 − ^δ _4n ² ) k (u n ,z n ) − ( x, ¯ y) ¯ k . Hence combining ( ♣ ) and ( ♥ ) we derive

d(z n , F (u n )) < 1 +n ^1/2 1 + n (1 + 1

n ) ² δ ²

4 k (x,y) − ( x, ¯ y) ¯ k < 1 + n ^1/2 1 + n (1 + 1

n ) ² δ ²

4(1 − ^δ _4n ² ) k (u n ,z n ) − ( x, ¯ y) ¯ k . Observing that the quantity ¹⁺ⁿ _1+n ^{1/2 (1+}

1 n ) ²

(1 − ^δ 4n ² ) tends to 0 as n tends to +∞, we obtain (⋆ ⋆ ⋆), as desired.

Claim 3.5

k (u n , z _n ) − ( x, ¯ y) ¯ k ≤ 2 δ for large n. (21)

k (u n , z n ) − ( x, ¯ y) ¯ k ≤ k (u n ,z n ) − (x, y) k + k (x,y) − (¯ x , y) ¯ k

≤ δ

n + (1 − 1

n ) ϕ (x, y) + k ((x, y) − ( x, ¯ y) ¯ k

≤ δ

n + (2 − 1 n ) δ ²

4 k (x, y) − ( x, ¯ y) ¯ k

≤ 2 δ .

Combining relations (21), (⋆⋆) and (⋆ ⋆ ⋆) we see that the point (u n ,z n ) verifies (14). Hence by assumption we have d(u n ,F ^{− 1} (z n )) ≤ τ d(z n , F(u n ).

Next, select ˜ x _n ∈ F ^{− 1} (z n ) such that

k x ˜ _n − u _n k ≤ (1 + n ^{− 2 1} )d(u n , F ^{− 1} (z n ))

≤ τ (1 +n ^{− 1 2} )d(z n , F(u n ))

≤ τ (1 +n ^{− 1 2} ) k z _n − y _n k

= τ (1 +n ^{− 1 2} ) (1 + n ^{1 2} )

1 +n k y − y _n k

= τ ^{2 + n}

1 2 +n ^{− 1 2}

1 + n k y − y _n k . (22)

Consequently, k x ˜ _n − u _n k < τ ^{2+n 1 /2+n} _1+n ^{− 1/2} 1 + ¹ _n 2

ϕ (x, y) and therefore, lim n → ∞ k x ˜ _n − x k = 0. Next, for large n, we have the following estimate:

ϕ (x, y) − ϕ ( x ˜ _n , y) >

by( 17 ⁾ n n + 1

d (y , F (u n )) − d(y , F( x ˜ _n ))

>

by( 18 )

n ²

(n +1) ² k y − y _n k − k y − z _n k , and by the definition of z _n , we derive

ϕ (x, y) − ϕ (˜ x _n , y) > n ^3/2 − n + n ^1/2

(n + 1) ² k y − y _n k . Thus, using also (16), we obtain

ϕ (x,y) − ϕ ( x ˜ _n ,y)

k x ˜ n − x k ≥ ϕ (x,y) − ϕ ( x ˜ _n ,y) k u n − x k + k x ˜ n − u n k

>

by( 22 )

(n ^3/2 − n + n ^1/2 )(n + 1) ^{− 2} k y − y _n k δ n ^{− 1} + τ (2 + n ^{2 1} +n ^{− 1 2} )(n + 1) ^{− 1} k y − y _n k

= n ^3/2 − n + n ^1/2 (2 + n ^{1 2} + n ^{− 1 2} )(n + 1)

1 τ + ^{δ (n+1)}

n(2+n ^1/2 +n ^{− 1 2} ) k y − y _n k ^{− 1} .

Since ^{n 3/2 − n+n 1/2}

(2+n ^{1 2} +n ^{− 1 2} )(n+1) → 1, lim n → ∞ k y − y _n k = ϕ (x, y) > 0 and δ (n + 1)

n(2 +n ^{2 1} + n ^{− 1 2} ) · 1

k y − y _n k → 0, we deduce that

| ∇ ϕ ( · , y) | (x) = lim sup

u → x

ϕ (x, y) − ϕ (u, y)

k x − u k ≥ lim sup

n → ∞

ϕ (x,y) − ϕ ( x ˜ _n ,y) k x ˜ _n − x k ≥ 1

τ ^,

establishing the proof.

4 Directional Metric Regularity under Perturbations

In [42, 43, 19, 44], it was established that metric regularity is stable under Lipschitz (single-valued or set-valued) perturbations with a suitable Lipschitz modulus. We shall show that directional metric regularity (with γ = 1) is also stable under suitable Lipschitz perturbations. Recall that a mapping g : X → Y between normed spaces is Hadamard differentiable at ¯ x ∈ X with respect to the direction u ∈ X , if the following limit exists:

lim t ↓ 0

w→u

g( x ¯ +tw) − g(¯ x)

t = Dg( x)(u). ¯

Obviously, if g is locally Lipschitz around ¯ x, then g is Hadamard differentiable at ¯ x with respect to the direction 0 and Dg( x)(0) = ¯ 0.

Theorem 4.1 Let X be a Banach space, Y be a normed space. For a given (u, v) ∈ X × Y, suppose F : X ⇉ Y is a set-valued mapping which is directionally metrically regular at ( x, ¯ y) ¯ ∈ gph F in direction (u, v) with modulus τ > 0. If g : X → Y is locally Lipschitz around x with a constant ¯ λ > 0 satisfying λ τ < 1 and if g is Hadamard differentiable at x ¯ with respect to the direction u, then the set-valued mapping F + g is directionally metrically regular at (¯ x, y ¯ + g( x)) ¯ in the direction (u, v + Dg( x)(u)). ¯

Proof Since g is locally Lipschitz around ¯ x with constant λ and F is directionally metrically regular at ( x, ¯ y) ¯ ∈ gph F in direction (u, v) with modulus τ > 0, take δ _∈ (0, λ ) such that

k g(x) − g(z) k ≤ λ _k x − z k ∀ x,z ∈ B( x, ¯ δ ), and

d(x, F ^{− 1} (y)) ≤ τ d (y , F (x)) for all (x, y) ∈ B(( x, ¯ y), ¯ δ ) ∩ ((¯ x, y) +cone ¯ B((u, v), δ )), with d (y, F(x)) ≤ 2 δ _k (x,y) − ( x ¯ y) ¯ k .

(23)

Taking into account the Hadamard differentiability of g at ¯ x with respect to the direction u, pick some ε _∈ ⁱ 0, ^δ ₂ h and then some η _∈ (0,1) such that

g( x ¯ +tw) − g( x) ¯

t − Dg(¯ x)(u)

< ε for all t ∈ ]0, η [, w ∈ B(u, η ). (24)

Set

ϕ F+g (x,y) = lim inf

z → x d(y,F (z) +g(z)), (x,y) ∈ X × Y.

Take ρ > 0 such that

ρ <



 

  min n

δ

λ +3 , δ ₋ ε , ^η _1+η ^{k u k} , _2τ+1 ^δ o

, if u 6 = 0 ρ < min n

δ

λ +3 , δ ₋ ε , _2τ+1 ^δ o

, if u = 0

and fix

(x, y) ∈ B(( x, ¯ y ¯ + g(¯ x)), ρ ) ∩ ((¯ x, y+ ¯ g( x)) + ¯ cone B((u, v + Dg( x)(u)), ¯ ρ )) with

0 < d(y, F(x) +g(x)) < ρ _k (x,y) − (¯ x, y+ ¯ g( x)) ¯ k , (25) and select a sequence { x _n } converging to x such that ϕ F+g (x, y) = lim n → ∞ d(y , F (x n ) + g(x n )).

With the aim of making the proof clearer, we will establish some claims.

Claim 4.1

k (x n , y) − (¯ x, y+ ¯ g(¯ x)) k < δ , for n large. (26) Claim 4.2

(x n , y) ∈ ( x, ¯ y ¯ + g( x)) + ¯ cone B((u,v + Dg( x)(u)), ¯ ρ ) for n large. (27) Indeed, since (x, y) − ( x, ¯ y ¯ + g( x)) ¯ ∈ / cone B((u, v + Dg( x)(u)), ¯ ρ ) \ { (0, 0) } , for large n, one has

(x, y) − ( x, ¯ y ¯ + g( x)) ¯ ∈ int(cone B((u, v +Dg(¯ x)(u)), ρ )) It follows that (x n ,y) − ( x, ¯ y ¯ +g( x)) ¯ ∈ cone B((u, v + Dg( x)(u)), ¯ ρ ).

Claim 4.3

0 < d(y,F (x n ) +g(x n )) < ρ _k (x n ,y) − ( x, ¯ y ¯ + g( x)) ¯ k for n large. (28) Indeed, reasoning ad absurdum, suppose the existence of a subsequence still denoted by { x n } such that

d(y,F (x n ) +g(x n )) ≥ ρ _k (x n , y) − ( x, ¯ y ¯ + g( x)) ¯ k . Since F has closed graph then

0 < ϕ F+g (x, y) = lim

n → +∞ d (y, F(x n ) + g(x n )) ≤ d(y , F (x) + g(x)), due to the fact that { x _n } converges to x and thanks to (25), we obtain a contradiction.

Claim 4.4 For n sufficiently large,

(x n ,y − g(x n )) ∈ B(( x, ¯ y), ¯ δ ), (29) d(y − g(x n ), F(x n )) ≤ 2 δ _k (x n , y − g(x n )) − (¯ x, y) ¯ k . (30) and

(x n , y − g(x n )) ∈ ( x, ¯ y) +cone ¯ B((u, v), δ ). (31)

We know that

k (x − x, ¯ y − y ¯ − g(¯ x)) k < ρ . For n sufficiently large, one has k x _n − x k < δ and

k (x n − x, ¯ y − g(x n ) − y) ¯ k = k x _n − x ¯ k + k y − y ¯ − g(x n ) k

≤ k x _n − x k + k x − x ¯ k + k y − y ¯ − g( x) ¯ k + k g( x) ¯ − g(x n ) k

≤ k x _n − x k + k (x − x, ¯ y − y ¯ − g( x)) ¯ k + λ _k x _n − x k

= ( λ + 1) k x _n − x k + 2 ρ

< ( λ + 3) ρ < δ (since ρ < δ λ +3 ).

Thus, one obtains (29). Next, we have

d (y − g(x n ), F(x n )) < ρ ( k (x n , y − g(x n )) − ( x, ¯ y) ¯ k + k g(x n ) − g( x) ¯ k )

≤ ρ ( k (x n , y − g(x n )) − ( x, ¯ y) ¯ k + λ _k x n − x ¯ k )

≤ ρ ( λ + 1) k (x n , y − g(x n )) − ( x, ¯ y) ¯ k

< 2 δ _k (x n ,y − g(x n )) − (¯ x, y) ¯ k (since ρ < _λ+2 ^δ ).

(32)

So, we receive (30). By relation (27), for n sufficiently large, say, n ≥ n ₀ , as (x n , y _n ) 6 = (¯ x, y ¯ +g( x)) ¯ we may find t _n > 0 and

(u n , w _n ) ∈ B((u, v+ Dg(¯ x)(u)), ρ ) such that

x _n = x ¯ +t n u _n , y = y ¯ + g( x) + ¯ t _n w _n . Set

y − g(x n ) = y ¯ + t _n v _n ; v _n = w _n − g(¯ x + t _n u _n ) − g( x) ¯

t _n .

If u = 0 then

k (u n , v _n ) − (0, v) k

= k (u n , (v n − w _n +Dg( x)) + (w ¯ n − v − Dg( x))) ¯ k

≤ k (u n w n − v − Dg( x)) ¯ k + k w n − v n − Dg(¯ x) k

≤ ρ + k ^{g( x+t ¯ n u} _{t n} ^{n ) − g( x) ¯} − Dg(¯ x) k

< ρ + ε < δ ; otherwise, u 6 = 0, since u _n ∈ B(u, ρ ), one has

t _n = k x n − x ¯ k k u _n k ≤ ρ

k u k − ρ ^< η , and therefore, by (24),

k (u n , v _n ) − (u,v) k ≤ k (u n ,w n ) − (u, v+ Dg(¯ x)(u)) k + k w _n − v _n − Dg( x)(u) ¯ k

≤ ρ +

g(¯ x+t n u n ) − g(¯ x)

t n − Dg( x)(u) ¯

≤ ρ + ε < δ .

Now, pick z n ∈ F ^{− 1} (y − g(x n )) such that

k x _n − z _n k ≤ (1 + 1/n)d(x n , F ^{− 1} (y − g(x n ))).

Hence, according to (23), (29), (30), (31), we obtain,

k x _n − z _n k ≤ (1 + 1/n) τ d(y − g(x n ), F(x n )). (33) Consequently,

k z _n − x ¯ k ≤ k x _n − z _n k + k x _n − x ¯ k

≤ (1 + 1/n) τρ _k (x n , y) − ( x, ¯ y ¯ + g(¯ x)) k + k x _n − x ¯ k

< (1 + 1/n) τρ ² + ρ < (1 + 1/n) τρ + ρ

< 2 τρ + ρ = ρ (1 + 2 τ )

< δ .

Using the local Lipschitz continuity around ¯ x of g gives, k g(z n ) − g(x n ) k ≤ λ _k z _n − x _n k . As y − g(x) ∈ / F (x) and lim n → ∞ x _n = x, then lim inf n → ∞ k x _n − z _n k > 0. Hence, by relation (33), one has

| Γ ϕ F+g ( · ,y) | (x) ≥ lim sup _{n → ∞} ^{ϕ F +g (x,y)} _{k x} ⁻ ₋ ^ϕ _z ^{F+g (z n ,y)}

n k

≥ lim sup _{n → ∞} ^{d(y − g(x n ),F(x} _k ⁿ _x ^{)) − d(y − g(z n ),F(z n ))}

n − z _n k

≥ lim sup _{n → ∞} ^{d(y − g(x n ),F(x n )) − d(y} _k ⁻ _x ^{g(x n ),F(z n )) −k g(x n ) − g(z n ) k}

n − z n k

≥ lim sup _{n → ∞} ^{d(y −} _k ^g(x _x ^{n ),F(x n ))}

n − z _n k − λ

≥ lim sup _{n → ∞} τ ^{− 1} _n+1 ⁿ ₋ λ = τ ^{− 1} ₋ λ . Therefore,

lim inf (x, y) −→

(u,v+Dg(¯ x)(u)) ( x, ¯ y ¯ + g( x)) ¯ x ∈ / (F + g) ^{− 1} (y), _{k (x,y)} ^{ϕ F+g} ₋ ^(x,y) _{( x,¯ ¯ y) k} → 0

| Γ ϕ F+g ( · , y) | (x) ≥ τ ^{− 1} ₋ λ .

Thanks to Theorem 3.1, the proof is complete.

5 Stability of Directional H¨older Metric Regularity under Mixed Coderivative-Tangency Conditions

For the usual metric regularity, sufficient conditions in terms of coderivatives have been given by various authors,

for instance, in [9, 21, 4, 24]. In this section, we establish a characterization of directional H¨older metric regularity

using the Fr´echet subdifferential in Asplund spaces, i.e., in Banach spaces in which every convex continuous function is

generically Fr´echet differentiable. There are many equivalent descriptions of Asplund spaces, see, e.g., Mordukhovich’s

book [4] and its bibliography. In particular, any reflexive space is Asplund, as well as each Banach space such that each

of its separable subspaces has a separable dual. We shall show that the proposed characterization also ensures the

stability of directional H¨older metric regularity under suitable differentiable perturbations. In this section, in order to formulate some coderivative characterizations of directional H¨older metric regularity, some additional definitions are required. Let X be a Banach space. Consider now an extended-real-valued function f : X → R ∪ { +∞ } . The Fr´echet subdifferential of f at ¯ x ∈ Dom f is given as

∂ f (¯ x) =

x ^∗ ∈ X ^∗ : lim inf

x → x, ¯ x 6 =¯ x

f (x) − f (¯ x) − h x ^∗ , x − x ¯ i k x − x ¯ k ≥ 0

.

For the convenience of the reader, we would like to mention that the terminology regular subdifferential instead of Fr´echet subdifferential is also popular due to its use in Rockafellar and Wets [36]. Every element of the Fr´echet subdif- ferential is termed as a Fr´echet (regular) subgradient. If ¯ x is a point where f ( x) = ¯ ∞, then we set ∂ f ( x) = ¯ /0. In fact one can show that an element x ^∗ is a Fr´echet subgradient of f at ¯ x iff

f (x) ≥ f (¯ x) + h x ^∗ ,x − x ¯ i + o( k x − x ¯ k ) where lim

x → x ¯

o( k x − x ¯ k ) k x − x ¯ k = 0.

It is well-known that the Fr´echet subdifferential satisfies a fuzzy sum rule in Asplund spaces (see [4], Theorem 2.33).

More precisely, if X is an Asplund space and f ₁ , f ₂ : X → R ∪ { ∞ } are such that f ₁ is Lipschitz continuous around x ∈ Dom f ₁ ∩ Dom f ₂ and f ₂ is lower semicontinuous around x, then for any γ > 0 one has

∂ ( f ₁ + f ₂ )(x) ⊂ ^[ { ∂ f ₁ (x 1 ) + ∂ f ₂ (x 2 ) | x _i ∈ x+ γ B _X , | f _i (x i ) − f _i (x) | ≤ γ , i = 1, 2 } + γ B _X ∗ . (34) For a nonempty closed set C ⊆ X, denote by δ C the indicator function associated with C (i.e. δ C (x) = 0, when x ∈ C and δ C (x) = ∞ otherwise). The Fr´echet normal cone to C at ¯ x is denoted by N(C, x). It is a closed and convex object in ¯ X ^∗ which is defined as ∂ δ C ( x). ¯ Equivalently a vector x ^∗ ∈ X ^∗ is a Fr´echet normal to C at ¯ x if

h x ^∗ ,x − x ¯ i ≤ o( k x − x ¯ k ), ∀ x ∈ C, where lim x → x ¯

o( k x − x ¯ k )

k x − x ¯ k = 0. Let F : X ⇉ Y be a set-valued map and (x, y) ∈ gph F. Then the Fr´echet coderivative at (x, y) is the set-valued map D ^∗ F (x, y) : Y ^∗ ⇉ X ^∗ given by

D ^∗ F(x, y)(y ^∗ ) :=

x ^∗ ∈ X ^∗ | (x ^∗ , − y ^∗ ) ∈ N(gph F, (x, y)) .

In the spirit of Gfrerer [45], see also Kruger [38], Ngai-Tinh [46] we introduce the following limit set for directional H¨older metric regularity of order γ ( γ _∈ (0, 1]).

Definition 5.1 Let F : X ⇉ Y be a closed multifunction and ( x, ¯ y) ¯ ∈ gph F be fixed. For a given (u,v) ∈ X × Y and each γ _∈ (0, 1], the critical limit set for metric regularity of order γ of F in the direction (u, v) at ( x, ¯ y) ¯ ∈ gph F is denoted by Cr ^γ F((¯ x, y), ¯ (u, v)) and is defined as the set of all (w, x ^∗ ) ∈ Y × X ^∗ such that there exist sequences { t _n } ↓ 0, { ε n } ↓ 0, (u n , v n ) ∈ cone B((u, v), ε n ) with k (u n , v n ) k = 1, w n ∈ Y , y ^∗ _n ∈ S _{Y ∗} (the unit sphere of Y ^∗ ),

x ^∗ _n ∈ D ^∗ F ( x+ ¯ t _n u _n , y+ ¯ t _n v _n +t n ^1/γ w _n )(y ^∗ _n ) with h y ^∗ _n , w _n i

k w n k → 1 and

w _n ,t n ^{(γ − 1)/γ} k w _n k ^{γ − 1} x ^∗ _n

→ (w, x ^∗ ).