Huynh Van

Ngai • Nguyen
email: ngaivn@yahoo.com

Huu Tron

• Nguyen

Van Vu

Michel Théra
email: michel.thera@unilim.fr

Directional Metric Pseudo Subregularity of Set-valued Mappings: a General Model

Keywords: subdifferential, Metric regularity, Directional metric regularity, Metric subregularity, directional Hölder metric subregularity, Coderivative Mathematics Subject Classification 49J52, 49J53, 90C30

In this paper, we introduce a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established.

Introduction

Over the past decades, mathematical ideas based on the use of advanced techniques of generalized differentiation have allowed to make significant advances in the study of generalized equations, that is inclusions governed by set-valued mappings. These inclusions/ generalized equations cover many important problems in various areas of mathematics and applied sciences such as physics, mechanics, and economics: equations; inequality systems; variational inequalities; complementary problems; optimal control, and various other topics. Also, due to their importance, as well as from their theoretical point of view and their broad applicability, variational systems have attracted the interest of many mathematicians.

One of the pillars of study of these generalized equations is the notion of metric regularity. This concept has emerged in the last decades, mainly after the contribution of Borwein [START_REF] Borwein | Stability and regular points of inequality systems[END_REF], even if according to Ioffe [START_REF]Metric regularity -a survey. Part I. Theory[END_REF][START_REF]Metric regularity -a survey[END_REF] in a recent survey, "the roots of this concept go back to a circle of fundamental regularity ideas of classical analysis embodied in such results as the implicit function theorem, the Banach open mapping theorem, theorems of Lyusternik and Graves, on the one hand, and the Sard theorem and the Thom-Smale transversality theory, on the other". Nowadays, this concept is commonly regarded as central for studying the existence and behavior of solutions of nonlinear equations under small perturbations of the data. The crucial role of metric regularity in Optimization and Variational Analysis as well as researcher's interest in this field can be found in many seminal works including for instance [START_REF] Borwein | Stability and regular points of inequality systems[END_REF][START_REF] Dontchev | Implicit Functions and Solution Mappings. A View from Variational Analysis[END_REF][START_REF] Gfrerer | On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs[END_REF][START_REF]Convexity and variational analysis[END_REF][START_REF]Metric regularity -a survey. Part I. Theory[END_REF][START_REF]Metric regularity -a survey[END_REF][START_REF]Variational Analysis of Regular Mappings[END_REF][START_REF] Wets | Variational Analysis[END_REF], and references therein. By the time and demand of use and applications, variants of this property have emerged suitable to practical problems. Weaker/stronger versions: calmness, (strong) (Hölder) metric sub/regularity, semiregularity or equivalent versions: pseudo Lipschitz, linear openness were studied and have proved to have an important role in various applications in Mathematics, especially in Variational Analysis and Optimization [START_REF] Cibulka | On semiregularity of mappings[END_REF][START_REF]Metric regularity -a survey. Part I. Theory[END_REF][START_REF]Metric regularity -a survey[END_REF][START_REF] Kruger | Error bounds and Hölder metric subregularity[END_REF][START_REF] Kruger | Error bounds and metric subregularity[END_REF][START_REF]Nonlinear metric subregularity[END_REF][START_REF] Wets | Variational Analysis[END_REF], ... Another direction in this line is to build directional models for these objects as recently proposed by Arutyunov-Avakov-Izmailov [START_REF] Arutyunov | Directional regularity and metric regularity[END_REF], Gfrerer [START_REF] Gfrerer | On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs[END_REF], Ngai-Théra [START_REF]Directional metric regularity of multifunctions[END_REF], Ngai-Tron-Théra [START_REF] Ngai | Directional Hölder metric regularity[END_REF], Ngai-Tron-Tinh [START_REF] Ngai | Directional Hölder metric subregularity and application to tangent cones[END_REF]. Characterizations of these concepts have been established and successfully applied to study optimality conditions for mathematical programs, for calculating tangent cones,...This notion of directional regularity is an extension of an earlier notion used by Bonnans and Shapiro [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF] to study sensitivity analysis. Later, Ioffe [START_REF] Ioffe | On regularity concepts in variational analysis[END_REF] has introduced and investigated an extension called relative metric regularity which covers many notions of metric regularity in the literature. In [START_REF] Penot | Metric regularity, openness and Lipschitzian behavior of multifunctions[END_REF] Penot studied this property to establish second-order optimality conditions.

The paper concerns a new type of directional metric regularity. In this article, motivated by the works mentioned above, especially those of (co)authors and by Gfrerer, we build a general new model which unifies almost (sub)regularity models in the literature; especially it covers both directional Hölder metric subregularity introduced by Ngai, Tron and Tinh and metric pseudo subregularity explored by Gfrerer. This property is given in Definition 3.2. Our aims are to give characterizations of the property. First, we establish a slope characterization, after that we move to a subdifferential/coderivative or a limit critical set characterization. Section 2 introduces the mathematical notation and basic definitions. In Section 3, we present the motivation of this paper and some slope characterizations of directional pseudo regularity. In the final section, we explore how the use of an abstract subdifferential may derive to characterizations of metric directional pseudo subregularity in terms of coderivatives.

Preliminaries and notations

For the convenience of the reader, we include in this section the material concerning set-valued analysis and variational analysis that will be extensively used throughout the sequel. We use the monographs of Mordukhovich [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation. I: Basic Theory[END_REF], Ioffe [START_REF]Variational Analysis of Regular Mappings[END_REF], Rockafellar & Wets [START_REF] Wets | Variational Analysis[END_REF] and Penot [START_REF]Calculus without Derivatives[END_REF] as our desk-copies.

For our purposes, we are going to work in the framework of real Banach spaces. If X is such a space, we denote by • the associated norm and by d(x, Ω ) the distance from x ∈ X to the subset Ω of X, that is, d(x, Ω ) := inf{ xy : y ∈ Ω }. Given X, we denote the topological dual (continuous dual) by X * , by • * the dual norm of • , by B X = {x ∈ X : x ≤ 1} the closed unit ball, by S X = {x ∈ X : x = 1} the unit sphere, by B(x, r) the closed ball with center x and radius r, respectively.

By a set-valued mapping (also named by some authors multifunction), we mean a mapping T from X into the subsets (possibly empty) of another Banach space Y and we use the notation T : X ⇒ Y. The graph of T denoted by gph T is the set of those points in X × Y such that y ∈ T (x), while T -1 : Y ⇒ X , the inverse of T (always defined), is given by (x, y) ∈ gph T ⇐⇒ (y, x) ∈ gph T -1 . We say that T is closed if its graph is closed with respect to the product topology on X ×Y . Given a set K ⊂ X, we use the notation cone K for the conic hull of K, that is for the set of all conic combinations ∑ i=n i=1 λ i x i of points of K where λ i ≥ 0 for each index i. Given an extended real-valued function f : X → R ∪ {+∞} we use the notation cl f to denote the lower semicontinuous envelope of f defined by cl f (x) = lim inf u→x f (u)), and Dom f will refer to the domain of f , that is, the set of those points x ∈ X such that f (x) is finite. We recall that the convex subdifferential of f at x ∈ Dom f is the set

∂ f (x) := {x ∈ X * : x , y -x ≤ f (y) -f (x) for all y ∈ X}, with the convention that ∂ f (x) = / 0, when f (x) = +∞.
For the purpose of this study, we use the concept of slope |∇ f |(x) of a function f at x ∈ Dom f . This is the quantity introduced by De Giovani, Marino & Tosques [START_REF] Giorgi | Evolution problerns in metric spaces and steepest descent curves[END_REF] and defined by

|∇ f |(x) :=    0 if x is a local minimum point of f lim sup y→x, y =x f (x) -f (y) d(x, y) otherwise. For x / ∈ Dom f , we set |∇ f |(x) = +∞.
When f is a convex function defined on a Banach space and x is not a minimum point, then according to Ioffe [START_REF]Convexity and variational analysis[END_REF]Poposition 3.8

] (see also Azé & Corvelec [2, Propo- sition 3.2]) |∇ f |(x) = sup y =x f (x) -f (y) x -y and |∇ f |(x) = d(0, ∂ f (x)).
In the following sections, we make use of the notion of abstract subdifferential operator.This operator denoted by ∂ satisfies the following conditions:

(C1) If f : X → R is a convex function which is continuous around x ∈ X and β : R → R is continuously differentiable at t = f (x), then ∂ (β • f )(x) ⊆ {β ( f (x))x ∈ X * : x , y -x ≤ f (y) -f (x) ∀y ∈ X}; (C2) ∂ f (x) = ∂ g(x) if f (y) = g(y)
for all y in a neighborhood of x;

(C3) Let f 1 : X → R ∪ {+∞} be a lower semicontinuous function and f 2 , ..., f n : X → R be Lipschitz functions. If f 1 + f 2 + ... + f n attains a local minimum at x 0 , then for any ε > 0, there exist

x i ∈ x 0 + εB X , x i ∈ ∂ f i (x i ), i ∈ 1, n, such that | f i (x i ) - f i (x 0 )| < ε, i ∈ 1, n, and x 1 + x 2 + ... + x n < ε.
We recall that the indicator function δ C of a closed set C in X is the function defined by δ C (x) = 0 when x ∈ C and δ C (x) = +∞, otherwise. Given an abstract subdifferential ∂ , the set N(C, x) := ∂ δ C (x) is called the normal cone to C at x associated with ∂ .

(C4) N(C, x) is assumed to be a cone for any closed subset C of X.

Let F : X ⇒ Y and ( x, ȳ) ∈ gph F. Given an abstract subdifferential ∂ and normal cone associated with ∂ , the set-valued mapping D * F( x, ȳ) : Y * ⇒ X * defined by

D * F( x, ȳ)(y ) = x ∈ X * : (x , -y ) ∈ N gph F, ( x, ȳ)
is called the coderivative of F associated with ∂ .

We assume further that ∂ satisfies the separable property in the following sense:

(C5) If f is a separable function defined on X × Y, that is, f (x, y) := f 1 (x) + f 2 (y), (x, y) ∈ X ×Y, where f 1 : X → R ∪ {+∞}, f 2 : Y → R ∪ {+∞}, then ∂ f (x, y) = ∂ f 1 (x) × ∂ f 2 (y), for all (x, y) ∈ X ×Y.
It is well known that the proximal subdifferential in Hilbert spaces, the Fréchet subdifferential in Asplund spaces, the viscosity subdifferentials in smooth spaces as well as the Ioffe and the Clarke-Rockafellar subdifferentials in the setting of general Banach spaces are subdifferentials verifying the conditions (C1)-(C5).

Slope characterizations of directional pseudo subregularity

For the understanding of the paper, it would be wise to recall the definitions of metric subregularity and of Hölder metric subregularity. We say that a set-valued mapping F : X ⇒ Y between metric spaces X,Y is called metrically subregular at a point ( x, ȳ) ∈ gph F with constant τ > 0, if there exists a neighbourhood U of x such that

τd(x, F -1 ( ȳ)) d( ȳ, F(x)) for all x ∈ U. (3.1)
If in relation (3.1) we replace d( ȳ, F(x)) by d( ȳ, F(x)) γ , with γ > 0, then we say that F is Hölder metrically subregular at ( x, ȳ) ∈ gph F with modulus τ and order γ and (3.1) can be equivalently rewritten as

τd(x, F -1 ( ȳ)) d( ȳ, F(x)) γ for all x ∈ U. (3.2)
Throughout the rest of the paper, we assume given Banach spaces X and Y i , (i = 1, • • • , m), as well as a finite family of set-valued mappings

T i : X ⇒ Y i , (i = 1, • • • , m). We note T := (T 1 , . . . , T m ) : X ⇒ Y 1 × • • • × Y m ,
the set-valued mapping defined by T (x) := T 1 (x) × . . . × T m (x) and consider γ := (γ 1 , ..., γ m ) ∈ R m such that γ 1 which means that γ i 1 for i = 1, ..., m. To begin with, let us first recall the definition of γmetric pseudo subregularity (γ-MPSR, for short) w.r.t. a given direction u and order γ 1. Definition 3.1 (directional pseudo subregularity, [START_REF] Gfrerer | On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs[END_REF]) A set-valued mapping T = (T 1 , . . . , T m ) : X ⇒ Y 1 × • • • ×Y m is said to be (γ 1 , . . . , γ m )-metrically pseudo subregular (γ i 1) in the direction u at ( x, ȳ) ∈ gph T with modulus τ > 0 iff there exist ε > 0 and δ > 0 such that

d x, T -1 ( ȳ) τ m ∑ i=1 x -x 1-γ i d ȳi , T i (x) (3.3) for x = x in B( x, ε) ∩ x + cone B(u, δ ) .
As observed by Gfrerer, γ i -metric pseudo subregularity (γ i 1) in the direction u = 0 implies Hölder subregularity of order 1 γ i . Note also that when m = 1, if we note T = T 1 and γ = γ 1 , then (3.3) becomes:

d x, T -1 ( ȳ) γ x -x γ-1 d x, T -1 ( ȳ) τd ȳ, T (x) . (3.4)
Hence, if a set-valued mapping T : X ⇒ Y is γ-MPSR at ( x, ȳ) in the direction u, then T is directionally Hölder metrically subregular of order γ at ( x, ȳ) in the direction u as mentioned in [START_REF] Ngai | Directional Hölder metric subregularity and application to tangent cones[END_REF].

The new idea in this contribution is to consider a general metric pseudo subregularity model called (γ, h)-pseudo subregularity associated with a given function h := (h 1 , . . . , h m ) : X -→ R m + and with γ := (γ 1 , ..., γ m ) ∈ R m with γ i 1 for i = 1, ..., m. To facilitate ease of reading, we shall introduce some useful real-valued functions corresponding to T = (T 1 , . . . , T m ) : X ⇒ Y 1 × • • • ×Y m and γ = (γ 1 , . . . , γ m ). For each j, we define ρ T j (•) = d ȳ j , T j (•) and set

ϕ T j (x) :=    ρ T j (x) h j (x) γ j -1 if h j (x) > 0, 0 otherwise.
Now let us denote by ϕ T the sum ϕ T = ∑ m j=1 ϕ T j , by ψ T = cl ϕ T the lower semicontinuous envelope of ϕ T and by S the sublevel set [ψ T 0]. Throughout the rest of the document, we make use of the following assumption:

(A ) h 1 , . . . , h m : X -→ R + are continuous functions and satisfy

h i (x) = 0 =⇒ ρ T i (x) = 0. (3.5) Definition 3.2 ((γ, h)-metric pseudo subregularity) Let T = (T 1 , . . . , T m ) : X ⇒ Y 1 × • •
• ×Y m be given and let ( x, ȳ) ∈ gph T . The mapping T is said to be (γ, h)-metrically pseudo subregular in the direction u at ( x, ȳ) ∈ gph T for h = (h 1 , . . . , h m ) if there exist τ > 0, ε > 0 and δ > 0 such that

d x, T -1 ( ȳ) τ ∑ 1 i m,h i (x)>0 d ȳi , T i (x) h i (x) γ i -1 (3.6) for x ∈ B( x, ε) ∩ x + cone B(u, δ ) with ∑ m i=1 h i (x) > 0.
When the direction u := 0, we will say briefly that T is (γ, h)-pseudo subregular at ( x, ȳ) (that is, the inequality (3.6) is satisfied for all x near x). Let us note that when considering some special mappings h, one recovers the concepts of directional metric subregularity mentioned above. For instance, if we choose the functions h i (x) = xx , one gets metric pseudo subregularity and if one considers the case m = 1, h 1 (x) := d(x, T -1 ( ȳ)) one gets directional Hölder metric subregularity. Especially, when γ i = 1 (i = 1, ..., m), one gets the usual metric subregularity.

Throughout the paper, it is convenient to keep in mind the notation used in [START_REF]Directional metric regularity of multifunctions[END_REF]:

x u - →
x is meant to be x → x if u = 0 and x → x and x-x xx → u u otherwise, as well. The rest of this section will be devoted to establish some characterizations for the (γ, h)-metric pseudo subregularity (Definition 3.2). For such a purpose, the next proposition will be useful. Proposition 3.1 Suppose that T is not (h, γ)-metrically pseudo subregular in the direction u at ( x, ȳ) ∈ gph T . Then, for each nonnegative real sequence (τ k ) ↓ 0, there exists a sequence x k u -→ x which satisfies for large integer k the following conditions:

i. d(x k , S) > 0; ii. ψ T (x k ) τ k 1- √ τ k d(x k , S); iii. ∇ψ T (x k ) max √ τ k , τ k . Consequently, one has lim inf x u - →x,x ∈S, ψ T (x)/d(x,S)→0 |∇ψ T |(x) = 0. (3.7)
Proof According to the definition of metric pseudo subregularity, we can find a sequence xk

u - → x such that xk ∈ S, τ k d xk , S > ∑ j=1,...,m h j ( xk )>0 d ȳ j , T j ( xk ) h j ( xk ) γ j -1 = m ∑ j=1 ϕ T j ( xk ) ψ T ( xk ). (3.8)
Hence, the lsc function ψ T inherits the following property:

0 < ψ T ( xk ) < τ k d xk , S , k = 1, 2, . . . (3.9) Let ε k := ψ T ( xk ) > 0 and λ k := min √ τ k d( xk , S), ε k τ k > 0.
By the Ekeland variational principle, there exists for each k an element xk such that

-xk -xk λ k ; -ψ T ( xk ) -ε k λ k xk -xk ψ T ( xk ); -the function f k : x -→ ψ T (x) + ε k λ k
xxk attains its minimum at xk . We are going to establish the following facts:

(i) xk ∈ S; (ii) xk u - → x; (iii) ψ T ( xk ) τ k 1- √ τ k d( xk , S); (iv) |∇ψ T |( xk ) max √ τ k , τ k .
Assuming that (i) does not hold, one has

τ k d xk , S τ k xk -xk τ k λ k ε k = ψ T ( xk ),
in contradiction with (3.9).

To prove (ii), let's note that λ k = o( xkx ) by virtue of the inequality

λ k √ τ k d xk , S √ τ k xk -x . Set µ k := xk -x xk -x .
After involving the triangle inequality, one deduces

1 - xk -xk xk -x xk -x xk -x 1 + xk -xk xk -x xk -x . Recalling that xk -xk λ k , we conclude that µ k → 1 as well as xk -x → 0. If u = 0, then a few straightforward calculations give us xk -x xk -x - u u = 1 µ k xk -x xk -x - u u + 1 µ k xk -xk xk -x + 1 µ k -1 u u .
(3.10)

Using (3.10) it yields xk -x xk -x -u u
→ 0, and therefore, (ii) is proved. For establishing (iii), we invoke (3.9) and obtain

ψ T ( xk ) d( x, S) ψ T ( xk ) d( xk , S) τ k d xk , S d( xk , S) . Since |d( xk , S) -d( xk , S)| xk -xk λ k √ τ k d( xk , S),
for large k we get

ψ T ( xk ) d( xk , S) τ k d xk , S d xk , S - √ τ k d xk , S = τ k 1 - √ τ k .
Hence, (iii) follows immediately.

In oder to verify (iv), remember that f k (•) attains a minimum at xk . As a result,

ψ T (x) + ε k λ k x -xk ψ T ( xk )
for x close to xk . Equivalently,

ψ T ( xk ) -ψ T (x) xk -x ε k λ k
for all x = xk belonging to a neighborhood of xk . In summary, we have

|∇ψ T |( xk ) ε k λ k = max ψ T ( xk ) √ τ k d xk , S , τ k max √ τ k , τ k . (3.11)
Letting x k = xk , the whole proof is established.

Based on Proposition 3.1, the next theorem offers a sufficient criterion for metric pseudo subregularity.

Theorem 3.1 Let T , γ, h and ψ T be defined as above. Suppose that lim inf

x u - →x,x ∈S, ψ T (x)/d(x,S)→0 |∇ψ T | (x) > 0 (3.12)
is fulfilled. Then, the set-valued mapping T is (γ, h)-metrically pseudo subregular w.r.t. the direction u at ( x, ȳ) ∈ gph T .

Proof When (3.12) is valid, the set-valued mapping T must be (γ, h)-metrically pseudo subregular as a direct consequence of Proposition 3.1.

Corollary 3.1 Under the same assumptions of Theorem 3.1, if now we have lim inf

x u - →x,x ∈S, ψ T (x)/ x-x →0 |∇ψ T | (x) > 0, (3.13)
then, the set-valued mapping T is (γ, h)-metrically pseudo subregular w.r.t. the direction u at ( x, ȳ) ∈ gph T .

Proof Relation (3.13) implies the one in (3.12).

Using Theorem 3.1 and taking h i (x) = xx , one obtains a slope characterization of directional pseudo subregularity of T .

Proposition 3.2 (slope characterization) If lim inf x u - →x,x ∈S x-x -γ ϕ T i (x)→0 ∇ cl m ∑ i=1 ρ T i (•) • -x γ i -1 (x) > 0 (3.14)
then T is (γ, h)-MPSR in the direction u at ( x, ȳ).

Similarly, considering the case m = 1, h 1 (x) := d x, T -1 ( ȳ) , we derive a characterization of directional Hölder metric subregularity of T .

Proposition 3.3 ( slope characterization) Suppose that the following condition

lim inf x u - →x,x ∈S, d( ȳ,T (x)) x-x d(x,T -1 ( ȳ)) γ-1 →0 ∇ cl d( ȳ, T (•)) d(•, T -1 ( ȳ)) γ-1 (x) > 0 (3.15)
does hold. Then T is directional Hölder γ-metric subregular in the direction u at ( x, ȳ).

In many applications, it is sufficient to focus on the case m = 1. For such a situation, when applying Theorem 3.1 we have to deal with the slope of the quotient of two functions. The next lemma will be useful to the computation of the slope of such a function.

Lemma 3.1 Let f , g : X -→ R ∪ {∞} be lsc extended real-valued functions and let x ∈ Dom f ∩ Dom g satisfying f (x) 0 and g(x) > 0. Suppose in addition that g is continuous at x. Under these assumptions, one has

∇ f g (x) |∇ f |(x) g(x) - f (x) g(x) 2 |∇g|(x). (3.16) Proof Let us denote Θ := f g . If x is an isolated point of Dom Θ , then x is a local minimum of both Θ and f , so |∇Θ |(x) = |∇ f |(x) = 0,
and the conclusion is trivial. On the contrary, we fix a sequence of nonnegative reals (ε k ) ↓ 0. Then, there is a sequence (δ k ) ↓ 0 for which the following property holds true:

z -x δ k =⇒ g(x) -g(z) |∇g|(x) + ε k x -z . (3.17)
For each k, we may select

z k ∈ B(x, δ k ) \ {x} such that f (x) -f (z k ) |∇ f |(x) -ε k x -z k . (3.18)
Due to the continuity of g, it is possible to assume g(z k ) > 0. We have

Θ (x) -Θ (z k ) x -z k = f (x) g(x) - f (z k ) g(z k ) x -z k = 1 g(z k ) • f (x) -f (z k ) x -z k - f (x) g(x)g(z k ) • g(x) -g(z k ) x -z k 1 g(z k ) |∇ f |(x) -ε k - f (x) g(x)g(z k ) |∇g|(x) + ε k .
From the last inequality we derive

lim sup k→∞ Θ (x) -Θ (z k ) x -z k lim sup k→∞ 1 g(z k ) |∇ f |(x) -ε k - f (x) g(x)g(z k ) |∇g|(x) + ε k = 1 g(x) |∇ f |(x) - f (x) g(x) 2 |∇g|(x). Since |∇Θ |(x) = lim sup z→x Θ (x) -Θ (z) x -z , we obtain |∇Θ |(x) lim sup k→∞ Θ (x) -Θ (z k ) x -z k 1 g(x) |∇ f |(x) - f (x) g(x) 2 |∇g|(x).
Invoking Lemma 3.1, we obtain Lemma 3.2 used in the sequel for proving Proposition 3.4.

Lemma 3.2 Let T : X ⇒ Y be a given set-valued mapping and let ( x, ȳ) ∈ gph T . Suppose that h : X -→ R + is locally Lipschitz around x and that the subsequent condition is valid as well lim sup

x u - →x,x ∈S d(x, S) h(x) < +∞ (3.19)
for some u ∈ X. Then,

lim inf x u - →x,x ∈S, h(x) 1-γ ρ T (x) d(x,S) →0 ∇ cl ρ T h γ-1 (x) lim inf x u - →x,x ∈S, h(x) -γ ρ T (x)→0 |∇ cl ρ T |(x) h(x) γ-1 . (3.20)
Proof Applying Lemma 3.1 with f = cl ρ T and g(•) = h(•) γ-1 we deduce

∇ cl ρ T h γ-1 (x) |∇ cl ρ T |(x) h(x) γ-1 - cl ρ T (x) h(x) γ-1 ∇ h γ-1 (x) h(x) γ-1 (3.21)
for each x ∈ S. According to [START_REF] Ngai | Error bounds in metric spaces and application to the perturbation stability of metric regularity[END_REF], we have

∇ h γ-1 (x) = (γ -1)h(x) γ-2 ∇h (x). (3.22)
But since h is locally Lipschitz, it holds that κ(x) := ∇h (x) is locally bounded around x. Let's re-write (3.21) as follows

∇ cl ρ T h γ-1 (x) |∇ cl ρ T |(x) h(x) γ-1 -(γ -1)κ(x) cl ρ T (x) h(x) |∇ cl ρ T |(x) h(x) γ-1 -(γ -1)κ(x)h(x) γ-1 ρ T (x) h(x) γ-1 d(x, S) • d(x, S) h(x) . ( 3
h(x) 1-γ ρ T (x) d(x,S) →0 ∇ cl ρ T h γ-1 (x) lim inf x u - →x,x ∈S, h(x) 1-γ ρ T (x) d(x,S) →0 |∇ cl ρ T |(x) h(x) γ-1 . (3.24)
Assume that (x k ) is a sequence in X such that

x k u - → x, x k ∈ S, h(x k ) 1-γ ρ T (x k ) d(x k , S) → 0. ( 3
h(x k ) -γ ρ T (x k ) = lim sup k→∞ h(x k ) 1-γ ρ T (x k ) d(x k , S) • d(x k , S) h(x k ) = 0. (3.26)
As a result, we get

lim inf k→∞ |∇ cl ρ T |(x k ) h(x k ) γ-1 lim inf x u - →x,x ∈S, h(x) -γ ρ T (x)→0 |∇ cl ρ T |(x) h(x) γ-1 . ( 3
h(x) 1-γ ρ T (x) d(x,S) →0 |∇ cl ρ T |(x) h(x) γ-1 lim inf x u - →x,x ∈S, h(x) -γ ρ T (x)→0 |∇ cl ρ T |(x) h(x) γ-1 . (3.28)
In summary, (3.24) and (3.28) give us lim inf

x u - →x,x ∈S, h(x) 1-γ ρ T (x) d(x,S) →0 ∇ cl ρ T h γ-1 (x) lim inf x u - →x,x ∈S, h(x) -γ ρ T (x)→0 |∇ cl ρ T |(x) h(x) γ-1 . (3.29)
This completes the proof of Lemma 3.2.

Based on the previous results, we present a robust version of Theorem 3.1 for the case m = 1 which might be more comfortable in practice.

Proposition 3.4 Let T , x, ȳ and h satisfy all assumptions of Lemma 3.2. Then, under the following condition

lim inf x u - →x,x ∈S, h(x) -γ ρ T (x)→0 |∇ cl ρ T |(x) h(x) γ-1 > 0, (3.30)
the set-valued mapping T is γ-metrically pseudo subregular at x for ȳ in the direction u.

Proof The proof follows by combining Theorem 3.1 with Lemma 3.2.

4 Coderivative characterization of directional metric pseudo subregularity Theorem 3.1 provides two sufficient conditions ensuring the validity of directional pseudo subregularity through the slopes corresponding to suitable functions. Gfrerer in his work [START_REF] Gfrerer | On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs[END_REF] dealt with such a property using the notion of coderivatives. In order to study the directional metric regularity property, the authors of [START_REF] Ngai | Metric subregularity of multifunctions: First and second order infinitesimal characterizations[END_REF], introduced the notion of limiting critical set around a reference point ( x, ȳ) ∈ gph T . Following a similar trend, we shall develop an infinitesimal criterion for directional metric pseudo subregularity using the coderivative associated with an abstract subdifferential ∂ . Hereinafter, we assume that the abstract subdifferential ∂ satisfies the following quotient fuzzy rule:

(C 5 ): Let f 1 , f 2 : X → R ∪ {+∞} be two locally Lipschitz functions around x 0 ∈ X with f 1 (x 0 ) ≥ 0, f 2 (x 0 ) > 0. For any ε > 0 one has ∂ f 1 f 2 (x 0 ) ⊆ x 1 ,x 2 ∈B(x 0 ,ε) f 2 (x 0 )∂ f 1 (x 1 ) -f 1 (x 0 )∂ f 2 (x 2 ) f 2 (x 0 ) 2 + εB X .
Note that (C 5 ) is valid for all usual subdifferentials in the literature of variational analysis.

Proposition 4.1 Let T : X ⇒ Y = Y 1 × • • • × Y m
be a closed set-valued mapping between two Banach spaces X and Y and ( x, ȳ) ∈ gph T . Suppose given functions h i : X -→ R + , (i = 1, • • • , m) locally Lipschitz around x. Let u ∈ X and γ 1 be given for which the following condition is fulfilled

lim sup x u - →x,x ∈S max i=1,...,m d(x, S) h i (x) < +∞. (4.1)
If T is not metrically pseudo subregular in the direction u at ( x, ȳ), then there exist some real sequence (t k ) ↓ 0 together with

(u k , v k ) ∈ S X×Y , u k ∈ X * and v k ∈ Y * such that (a). lim k→∞ u k = 1, lim k→∞ u u k -u = 0; (b). lim k→∞ max i=1,...,m (t k ) 1-γ i v ki = 0;
(c). the vector (u k ,ṽ k ) belongs to N gph T, ( x + t k u k , ȳ + t k v k ) , where ṽ k ∈ Y * is given by ṽ

ki = h i ( x + t k u k ) 1-γ i v ki ; (d). lim k→∞ ∑ m i=1 v ki , h i ( x + t k u k ) 1-γ i v ki h 1 ( x + t k u k ) 1-γ 1 v k1 , . . . , h m ( x + t k u k ) 1-γ m v km = 1; (e). lim k→∞ u k = 0, lim k→∞ v ki = 1.
Proof For convenience, we assume that the distance on Y is given by

y Y := y 1 Y 1 + • • • + y m Y m .
Further, with the aim of simplifying the notation, let us use • to indicate the norm in any Banach space. By Proposition 3.1, we can find some sequences

x k u - → x such that (i) x k ∈ S; (ii) lim k→∞ ψ T (x k ) d(x k , S) = 0; (iii) lim k→∞ ∇ψ T (x k ) = 0.
Denoting α k := x kx > 0 and assuming ψ T (x k ) = β k d(x k , S) where β k ↓ 0, for each k, choose some positive parameters σ k and η k satisfying σ k = o(ψ T (x k )) and η k < σ k . Making σ k and η k smaller if necessary, we may suppose

-2η k + σ k < ψ T (x k ); -ϕ T (z) > ψ T (x k ) -σ k whenever z -x k η k .
Let τ k := ∇ψ T (x k ), then according to the definition, x k is a local minimum to the function ψ

T (•)+(τ k +σ k ) •-x k .
Hence, there exists a radius r k min j=1,...,m

h j (x k ) γ j η k such that ψ T (x k ) = min x-x k 2r k ψ T (x) + (τ k + σ k ) x -x k . (4.2)
By the definition of a lsc envelope, we may select some ( xk , ŷk ) ∈ gph T which fulfills the two properties below:

xk -x k σ k r k and m ∑ j=1 ȳ j -ŷk j h j ( xk ) γ j -1 ψ T (x k ) + σ k r k . (4.3) Using (4.2), we deduce m ∑ j=1 ȳ j -ŷk j h j ( xk ) γ j -1 ψ T (x) + (τ k + σ k ) x -x k + σ k r k δ gph T (x, y) + m ∑ j=1 ȳ j -y j h j (x) γ j -1 + (τ k + σ k ) x -xk + (τ k + σ k ) x k -xk + σ k r k .
+ m ∑ j=1 ȳ j -y j h j (x) γ j -1 + (τ k + σ k ) x -xk . (4.5) Observe that f k ( xk , ŷk ) = ∑ m j=1 ȳ j -ŷk j h j ( xk ) γ j -1 and that f k ( xk , ŷk ) inf x-x k 2r k f (x, y) + ε k , where ε k := (τ k + σ k ) x k -xk + σ k r k > 0. Setting λ k := r k √ σ k > 0 and applying the Ekeland variational principle, take ( xk , ỹk ) ∈ X ×Y such that -( xk , ŷk ) -( xk , ỹk ) λ k ; -f k ( xk , ŷk ) -ε k λ k ( xk , ŷk ) -( xk , ỹk ) f k ( xk , ỹk ); -( xk , ỹk ) is a minimum to the function (x, y) ∈ X × Y -→ f k (x, y) + ε k λ k (x, y) - ( xk , ỹk ) subject to the constraint x -x k 2r k .
Observe that λ k = o(r k ), so xk = x, which allows for writing ỹk j = ȳ j . The last condition along with the properties (C 1 ) -(C 5 ) of the subdifferential operator show that, there are some elements xi k , xk j ∈ X, ỹi k ∈ Y , ỹk j ∈ Y j and also xi k , x k j ∈ X * , ỹi k ∈ Y * , ỹ k j ∈ Y j which fulfill the following conditions:

(i) max xk -xi k , xk -xk j , ỹk -ỹi k , ỹk -ỹk j ν k , where ν k min λ k , β k ȳ1 - ỹk1 , . . . , β k ȳm -ỹkm ; (ii) ( x1 k , ỹ1 k ) ∈ ∂ X×Y δ gph T ( x1 k , ỹ1 k ) = N gph T, ( x1 k , ỹ1 k ) ; (iii) x2 k ∈ ∂ X • -xk ( x2 k ); (iv) ( x3 k , ỹ3 k ) ∈ ∂ X×Y (•, •) -( xk , ỹk ) ( x3 k , ỹ3 k ); (v) ỹ4 k j ∈ ∂ Y j ȳ j -• ( ỹ4 k j ); (vi) x4 k j ∈ ∂ X h 1-γ j j ( x4 k j ) = (1 -γ j )h j ( x4 k j ) -γ j ∂ X h j ( x4 k j ); (vii) x1 k + (τ k + σ k ) x2 k + ε k λ k x3 k + ∑ m j=1 ȳ j -ỹ4 k j x4 k j σ k ; (viii) ỹ1 k j + ε k λ k ỹ3 k j + h j (x 4 k j ) 1-γ j ỹ4 k j σ k .
We shall establish the conclusion of Proposition 4.1 step-by-step through several auxiliary facts.

Fact 1 It holds that

lim k→∞ xk -x k x k -x = lim k→∞ xi k -x k x k -x = lim k→∞ x4 k j -x k x k -x = 0; (4.6a) lim k→∞ h j ( xk ) h j (x k ) = lim k→∞ h j ( xi k ) h j (x k ) = lim k→∞ h j ( x4 k j ) h j (x k ) = 1; (4.6b) lim k→∞ 1 ψ T (x k ) m ∑ j=1 ȳ j -ŷk j h j ( xk ) γ j -1 = 1; (4.6c) lim k→∞ 1 ψ T (x k ) m ∑ j=1 ȳ j -ỹi k j h j ( xi k ) γ j -1 = 1, i = 1, 2 , 3;
(4.6d)

lim k→∞ 1 ψ T (x k ) m ∑ j=1 ȳ j -ỹ4 k j h j ( x4 k j ) γ j -1 = 1.
(4.6e)

Proof of Fact 1 Firstly, we establish (4.6a). Equality lim

k→∞ xk -x k x k -x = 0 is trivial due to the choice of xk . Further, since max xi k -xk , x4 k j -xk λ k = o( x k -x ), whereas xk -xk λ k = o( x k -x ), one gets max xi k -x k , x4 k j -x k = o( x k -x ), which implies (4.6a).
For (4.6b), let us involve the Lipschitz property to each function h j . Indeed, for each j = 1, 2, . . . , m there is L j > 0 and r j > 0 for which one has

x -x r j , x -x r j x -x r j =⇒ |h j (x) -h j (x )| L j x -x . (4.7)
Thus, when k is large, it holds that

|h j ( xk ) -h j (x k )| L j xk -x k L j σ k r k . (4.8)
This allows us to write

h j ( xk ) h j (x k ) -1 L j r k h j (x k ) σ k . (4.9)
By interchanging in turn the role between xk with xk j , xi k and repeating the arguments above, we deduce

max h j ( x4 k j ) h j (x k ) -1 , h j ( xi k ) h j (x k ) -1 L j r k h j (x k ) (σ k + 2 √ σ k ). (4.10)
Since r k h j (x k ) γ η k , (4.6b) follows from (4.9) and (4.10).

In the next step, we prove (4.6c). According to the choice of xk , it is possible to derive

h j ( xk ) 1-γ j ȳ j -ŷk j h j ( xk ) 1-γ j d ȳ j , T j ( xk ) = ϕ T j (x k ).
As a result

m ∑ j=1 ȳ j -ŷk j h j ( xk ) γ j -1 ϕ T (x k ) > ψ T (x k ) -σ k . Further, we have ∑ m j=1 ȳ j -ŷk j h j ( xk ) γ j -1 < ψ T (x k ) + σ k r k
by the choice of ŷk j . Thus, the limit (4.6c) is a direct consequence of the fact that

σ k = o(ψ T (x k )).
Based on (4.6b), we can derive that the proof of (4.6e) is similar to the one of (4.6d). Thus, it is sufficient to verify (4.6d) only. Indeed, fix an index i ∈ {1, 2, 3}. We infer from (4.6b) that

lim k→∞ 1 ψ T (x k ) m ∑ j=1 ȳ j -ŷk j h j ( xi k ) γ j -1 = 1.
According to the choice of ỹi k j we get

ỹi k j -ŷk j ỹk j -ŷk j + ỹi k j -ŷk j 2λ k = 2r k √ σ k , which yields ỹi k j -ŷk j ψ T (x k )h j ( xi k ) γ j -1 2 √ σ k r k ψ T (x k )h j ( xi k ) γ j -1 2 √ σ k h(x k ) γ j η k ψ T (x k )h j ( xi k ) γ j -1 . Recalling that η k < σ k = o(ψ T (x k )),
u k := (t k ) -1 ( x1 k -x), (4.11b) v ki := (t k ) -1 ( ỹ1 ki -ȳi ), i = 1, . . . , m. (4.11c)
Then the following relations are valid as well

(u k , v k ) = 1, (4.12a) lim k→∞ u k = 1, (4.12b) lim k→∞ u u k -u = 0, (4.12c) lim k→∞ (t k ) 1-γ i v ki = 0. (4.12d)
Proof of Fact 2 Firstly, by the triangle inequality:

x k -x1 k x k -xk + xk -xk + xk -x1 k σ k r k + λ k + ν k = o(η k ).
Hence, for k large x k -x1 k < η k . Thus, we infer from the choice of η k that

ϕ T ( x1 k ) > ψ T ( x1 k ) -σ k > η k .
This shows that ȳ ∈ T ( x1 k ). Particularly, one has x1 k = x, which implies t k > 0. Hence, the elements u k , v ki are well-defined. The equality (4.12a) is trivial by the definitions (4.11a)-(4.11c). To prove (4.12b), we note that x1 kx ∼ x kx = α k according to (4.6a). It follows from the choice of ŷk that (see

(4.3)) ŷki -ȳi h i ( xk ) γ i -1 ψ T (x k ) + σ k r k = o(α k ),
which allows for writing

ỹ1 ki -ȳi ỹ1 ki -ỹki + ỹki -ŷki + ŷki -ȳi ỹ1 k -ỹk + ỹk -ŷk + ŷki -ȳi ν k + λ k + ŷki -ȳi = o(α k ).
Thus, the limit in (4.12b) is obtained directly from (4.11a) and (4.11b).

With the aim of verifying (4.12c), let us define the auxiliary element ûk := u x k -x (x kx)u. For such a notation, we have

u u k -u = (t k ) -1 u ( x1 k -ū) -u = u k ûk + u -u + u u k x1 k -x k x k -x = u k ûk + ( u k -1)u + u u k x1 k -x k x k -x .
Observe that x k u -→ x; we get ûk → 0, and hence, (4.12c) follows from (4.12b) and (4.6a).

Finally, we establish (4.12d). Due to (4.11a), (4.11c) we may write

(t k ) 1-γ i v ki = ỹ1 k -ȳ ( x1 k -x, ỹ1 k -ȳ) γ i ỹ1 k -ȳ x1 k -x γ i k→∞ ∼ ỹ1 k -ȳ x k -x γ i .
Recalling the estimation for ỹ1 kȳ as above, we find

ỹ1 ki -ȳi ν k + λ k + ŷki -ȳi ν k + λ k + h i ( xk ) γ i -1 ψ T (x k ) + σ k r k .
We know that lim sup

k→∞ h i (x k ) x k -x < +∞, lim k→∞ ψ T (x k ) d(x k , S) = 0.
Therefore, (4.6a) and (4.6b) imply lim sup

k→∞ h i ( xk ) γ i -1 ψ T (x k ) x k -x γ i = 0. Since r k h i (x k ) γ i η k and λ k = o(r k ), ν k = o(r k ),
we conclude that ỹ1 kiȳi = o x kx γ i . Thus, (4.12d) is thereby proved.

Fact 3 For each i = 1, 2, . . . , m it holds that

lim k→∞ [h i (x k )] γ i -1 ỹ1 ki = 1. (4.13)
Proof of Fact 3 Invoking (viii), there exists

w ki ∈ B Y * i such that σ k w ki = ỹ1 ki + ε k λ k ỹ3 ki + h(x 4 ki ) 1-γ i ỹ4 ki .
By virtue of (iv), the sequence ( ỹ3 ki ) is bounded in norm, so

lim k→∞ ỹ1 ki + h(x 4 ki ) 1-γ i ỹ4 ki = 0. (4.14)
By using (4.6b), it follows that h i ( x4 ki ) > 0 as k is sufficiently large (because of

h i ( x4 ki ) ∼ h i (x k ))
. This implies ỹ4 ki = ȳi unless a finite many of indexes k. Taking into account (v) and since the function ȳi -• is convex continuous on Y j , we can say that ỹ4

ki is a minimum to the function ȳi -• -ỹ4 ki , • -ỹ4 ki . Thus,

ỹ4 ki = 1, ȳi -ỹ4 ki = -ỹ4 ki , ȳi -ỹ4 ki . (4.15)
The latter permits us to obtain

h(x 4 ki ) 1-γ i -ỹ1 ki + h(x 4 ki ) 1-γ i ỹ4 ki ỹ1 ki h(x 4 ki ) 1-γ i + ỹ1 ki + h(x 4 ki ) 1-γ i ỹ4 ki (4.16)
Combining (4.6b), (4.14) with (4.16) we obtain (4.13).

Fact 4 Let us define with respect to each k the elements below:

u k := x1 k , (4.17a)
v ki := -h i ( x + t k u k ) γ i -1 ỹ1 ki , (4.17b)
ṽ ki := h i ( x + t k u k ) 1-γ i v ki . (4.17c)
For such elements, we have

lim k→∞ v ki = 1, (4.18a)
u k ,ṽ k1 , . . . ,ṽ km ∈ N gph T,

( x + t k u k , ȳ + t k v k ) , (4.18b)
lim k→∞ u k = 0, (4.18c)
lim k→∞ ∑ m i=1 v ki , h i ( x + t k u k ) 1-γ i v ki h 1 ( x + t k u k ) 1-γ 1 v k1 , . . . , h m ( x + t k u k ) 1-γ m v km = 1. (4.18d)
Proof of Fact 4 The relations (4.18a) is derived from (4.17b) and (4.13), while (4.18b) is a consequence of (ii). In order to obtain (4.18c), we invoke (iii), (iv), (vi), (vii), (4.6e) and (4.13). Indeed, thanks to (vii), there is û k ∈ B X * such that

σ k û k = x1 k + (τ k + σ k ) x2 k + ε k λ k x3 k + m ∑ i=1 (1 -γ i ) ȳi -ỹ4 ki h i ( x4 ki ) γ i ũ ki , (4.19)
where

ũ ki ∈ ∂ X h i ( x4 ki ) satisfies x4 ki = (1 -γ i )h i ( x4 ki ) -γ i ũ ki .
According to (iii), (iv) and by the Lipschitz property of each function h i , it is possible to check that lim sup

k→∞ max x2 k , x3 k , ũ k1 , . . . , ũ km < +∞. (4.20)
Denoting γ * = max{γ 1 , . . . , γ m }, (4.19) yields

x1 k σ k û k + (τ k + σ k ) x2 k + ε k λ k x3 k + (γ * -1) max i=1,...,m 1 h i ( x4 ki ) max i=1,...,m ũ ki m ∑ i=1 ȳi -ỹ4 ki h i ( x4 ki ) γ i -1 . (4.21)
Using (4.6b) and observing that lim sup

k→∞ d(x k , S) h i (x k ) < +∞, we deduce lim sup k→∞ d(x k , S) h i ( x4 ki ) < +∞.
As a result, (4.18c) is obtained under the combination of (4.20), (4.21), (4.6e) and the assumption that ψ T (x k ) d(x k ,S) → 0. With the aim of establishing (4.18d), we define some quantities

a ki := h i ( x + t k u k ) 1-γ i v ki , v ki , a k := ∑ m i=1 a ki , b ki := h i ( x + t k u k ) 1-γ i v ki , b k := ∑ m i=1 b ki .
Then, we have

-a ki = (t k ) -1 ỹ1 ki , ỹ1 ki -ȳi , b ki = (t k ) -1 h i (h i ( x1 k )) 1-γ i ỹ1 ki -ȳi . (4.22) Since ỹ1 ki + ε k λ k ỹ3 ki + h i ( x4 ki ) 1-γ i ỹ4 ki = σ k w ki (see in the proof of Fact 3), it follows that -t k h i ( x4 ki ) γ i -1 a ki = h i ( x4 ki ) γ i -1 ỹ1 ki , ỹ1 ki -ȳi = σ k h i ( x4 ki ) γ i -1 w ki , ỹ1 ki -ȳi - ε k λ k h i ( x4 ki ) γ i -1 ỹ3 ki , ỹ1 ki -ȳi -ỹ4 ki , ỹ1 ki -ȳi .
Consequently, we find

t k h i ( x + t k u k ) γ i -1 b ki - h i ( x4 ki ) h i ( x1 k ) γ i -1 a ki = ȳi -ỹ1 ki σ k h i ( x4 ki ) γ i -1 w ki , ỹ1 ki -ȳi ȳi -ỹ1 ki -ȳi -ỹ1 ki ε k λ k h i ( x4 ki ) γ i -1 ỹ3 ki , ỹ1 ki -ȳi ȳi -ỹ1 ki + ȳi -ỹ1 ki 1 - ỹ4 ki , ỹ1 ki -ȳi ȳi -ỹ1 ki . (4.23)
As in the proof of Fact 3 (see (4.15))

ỹ4 ki = 1, ȳi -ỹ4 ki + ỹ4 ki , ȳi -ỹ4 ki = 0,
and the latter implies

ỹ4 ki , ỹ1 ki -ȳi = ȳi -ỹ4 ki + ỹ4 ki , ỹ1 ki -ỹ4 ki . (4.24)
Moreover, recall that ν k β k ȳiỹki (cf. (i)). Therefore, from the fact max{ ỹ1 kiỹki , ỹ4 kiỹki } ν k (by virtue of (i)) one has

lim k→∞ ỹ1 ki -ỹki ȳi -ỹki = lim k→∞ ỹ4 ki -ỹki ȳi -ỹki = 0, (4.25a)
lim k→∞ ȳi -ỹ1 ki ȳi -ỹki = lim k→∞ ȳi -ỹ4 ki ȳi -ỹki = 1. (4.25b)
Combining (4.25a) with (4.25b), we obtain

lim sup k→∞ | ỹ4 ki , ỹ1 ki -ỹ4 ki | ȳi -ỹ1 ki lim sup k→∞ ỹ4 ki ỹ1 ki -ỹ4 ki ȳi -ỹ1 ki = 0.
Taking into account the estimation

1 - ỹ4 ki , ỹ1 ki -ȳi ȳi -ỹ1 ki 1 - ȳi -ỹ4 ki ȳi -ỹki + | ỹ4 ki , ỹ1 ki -ỹ4 ki | ȳi -ỹ1 ki , we find lim k→∞ 1 - ỹ4 ki , ỹ1 ki -ȳi ȳi -ỹ1 ki = 0.
Hence, we can infer from (4.23) that

t k h i ( x + t k u k ) γ i -1 b ki - h i ( x4 ki ) h i ( x1 k ) γ i -1 a ki = o( ȳi -ỹ1 ki ), (4.26)
which implies

b ki - h i ( x4 ki ) h i ( x1 k ) γ i -1 a ki = o(b ki ). (4.27)
Taking the sum over the index i in (4.27), we reach the conclusion ii. the pair (x k , y k ) with

b k - m ∑ i=1 h i ( x4 ki ) h i ( x1 k ) γ i -1 a ki = o(b k ). (4.28) Observe that lim sup k→∞ |a ki | b ki = lim sup k→∞ h i ( x1 k ) γ i -1 | ỹ1 ki , ȳi -ỹ1 ki | ȳi -ỹ1 ki lim sup k→∞ h i ( x1 k ) γ i -1 ỹ1 ki = 1, (4.29) we deduce b k -a k = b k - m ∑ i=1 h i ( x4 ki ) h i ( x1 k ) γ i -1 a ki + m ∑ i=1 h i ( x4 ki ) h i ( x1 k ) γ i -1 -1 a ki = o(b k ).
x k := x + t k u k , y ki := ȳi + (t k ) γ i v ki is in gph T but ȳ ∈ T (x k ); iii. u k , -h 1 (x k ) 1-γ 1 v k1 , . . . , -h m (x k ) 1-γ m v km ∈ N gph T, (x k , y k ) and one has lim k→∞ ∑ m i=1 v ki , h i (x k ) 1-γ i (y ki -ȳi ) h 1 (x k ) 1-γ 1 (y k1 -ȳ1 ), . . . , h m (x k ) 1-γ m (y km -ȳm ) = 1.
Using this new notion, we are now ready to present the infinitesimal characterization for the property of (γ, h)-metric pseudo subregularity. The next theorem is in this sense. Suppose on the contrary that T is not (γ, h)-metrically pseudo subregular in the direction u at ( x, ȳ). Let t k > 0, (u k , v k ) ∈ S X×Y , and (u k , v k ) ∈ X * ×Y * be the sequences in the conclusion of Proposition 4.1. Let us define

x k := x + t k u k , y k := ȳ + t k v k ; (4.32)
then it is clear that

u k , -h 1 (x k ) 1-γ 1 v k1 , . . . , -h m (x k ) 1-γ m v km ∈ N gph T, (x k , y k ) , (4.33a)
lim k→∞ u k = 0, lim k→∞ v ki = 1, (4.33b)
lim k→∞ u k = 1, lim k→∞ u u k -u = 0, (4.33c)
lim k→∞ (t k ) 1-γ 1 v k1 = • • • = lim k→∞ (t k ) 1-γ m v km = 0, (4.33d) lim k→∞ ∑ m i=1 v ki , h i (x k ) 1-γ i v ki h 1 (x k ) 1-γ 1 v k1 , . . . , h m (x k ) 1-γ m v km = 1. (4.33e)
Further, as proved in Fact 2, we also have ȳ

∈ T ( x + t k u k ) = T (x k ). Setting tk := u k t k , (4.34a)
ûk := u k -1 u k , vki := (t k ) 1-γ i u k -γ i v ki , (4.34b)
û k := v k -1 u k , v k := v k -1 v k , (4.34c)
we obtain

x k = x + tk ûk , y ki = ȳi + (t k ) γ i vki , (4.35a) û k , -h 1 (x k ) 1-γ i v k1 , . . . , -h m (x k ) 1-γ m v km ∈ N gph T, (x k , y k ) , (4.35b) ûk = 1, v k = 1, (4.35c) lim k→∞ vki = 0, lim k→∞ û k = 0. (4.35d)
On the other hand, combining (4.34c) with (4.33b) and (4.33e), we find

lim k→∞ ∑ m i=1 v ki , h i (x k ) 1-γ i (y ki -ȳi ) h 1 (x k ) 1-γ 1 (y k1 -ȳ1 ), . . . , h m (x k ) 1-γ m (y km -ȳm ) = lim k→∞ v k -1 ∑ m i=1 v ki , h i (x k ) 1-γ i v ki h 1 (x k ) 1-γ 1 v k1 , . . . , h m (x k ) 1-γ m v km = lim k→∞ max i=1,...,m v ki -1 ∑ m i=1 v ki , h i (x k ) 1-γ i v ki h 1 (x k ) 1-γ 1 v k1 , . . . , h m (x k ) 1-γ m v km = 1.
In addition, taking into account the representation Note that certain applications concern set-valued mapping having a convex (and closed) graph. For such situations, the counterpart of Theorem 4.1 might be also fulfilled. The next result is in this sense. Proposition 4.2 (Set-valued mapping with convex graph) Suppose that the setvalued mapping T : X ⇒ Y has closed convex graph. Fix some given ( x, ȳ) ∈ gph T and u ∈ X. Let γ and h be the same as in Theorem 4.1. If T is (γ, h)-metrically pseudo subregular in the direction u at ( x, ȳ), then one has (0, 0) ∈ SCr γ,h T ( x, ȳ)(u).

u ûk -u = u k -1 ( u u k -u) + u k -1 -1 u, ( 4
Proof Replacing T by T (•) := T (•+ x)-ȳ if necessary, we may assume x = 0 and ȳ = 0. Let T satisfy the assumptions of Proposition 4.2 but (0, 0) be in the set SCr γ,h T (0, 0)(u). The definition of SCr γ,h T (0, 0)(u) shows that, there exist a real sequence (s k ) ↓ 0 together with some sequences

(u k , v k ) ∈ S X × S Y * and (v k , u k ) Y ×X * ---→ (0, 0) which fulfill the conditions below: -lim k→∞ u u k -u = 0; -we have y k ∈ T (x k ) but 0 = ȳ ∈ T (x k ), in which x k = s k u k ∈ X and y ki = (s k ) γ i v ki ; -u k , -h 1 (x k ) 1-γ 1 v k1 , . . . , -h m (x k ) 1-γ m v km is an element of the normal cone N gph T, (x k , y k ) ; -lim k→∞ ∑ m i=1 v ki , h i (x k ) 1-γ i y ki h 1 (x k ) 1-γ 1 y k1 , . . . , h m (x k ) 1-γ m y km = 1.
Let τ, δ and r be positive real parameters such that

d x, T -1 (0) τ ∑ i:h i (x)>0 h i (x) 1-γ i d 0, T i (x) (4.36)
whenever x ∈ cone B(u, δ ) with 0 < x < r. Since lim k→∞ u u ku = 0, x k will be in cone B(u, δ ) after skipping a few first indexes k. Hence, it is possible to apply (4.36) at

x = x k d x k , T -1 (0) τ m ∑ i=1 h i (x k ) 1-γ i d 0, T i (x k ) τ m ∑ i=1 h i (x k ) 1-γ i y ki .
(4.37)

Recall that x k ∈ T -1 (0). By virtue of (4.37), for some 0 < σ k < 1 there exists z k ∈ T -1 (0) which fulfills the inequalities

0 < x k -z k τ(1 + σ k ) m ∑ i=1 h i (x k ) 1-γ i y ki . (4.38)
Because the set gph T is convex, we may derive from the choice of u k and v k that

u k , z k -x k + m ∑ i=1 -h i (x k ) 1-γ i v ki , -y ki 0. ( 4

.39)

As a result, we obtain

∑ m i=1 h i (x k ) 1-γ i y ki z k -x k ∑ m i=1 v ki , h i (x k ) 1-γ i y ki h 1 (x k ) 1-γ 1 y k1 , . . . , h m (x k ) 1-γ m y km u k , x k -z k z k -x k . (4.40)
Combining (4.38) with (4.40), we deduce

1 τ(1 + σ k ) ∑ m i=1 v ki , h i (x k ) 1-γ i y ki h 1 (x k ) 1-γ 1 y k1 , . . . , h m (x k ) 1-γ m y km u k , x k -z k z k -x k u k .
(4.41)

Passing to the limit w.r.t k in both sides of (4.41), we obtain the following estimate lim inf

k→∞ 1 τ(1 + σ k ) ∑ m i=1 v ki , h i (x k ) 1-γ i y ki h 1 (x k ) 1-γ 1 y k1 , . . . , h m (x k ) 1-γ m y km 0. ( 4

.42)

Since the left-hand side of (4.42) takes a positive value, we reach a contradiction. This completes our proof.

As arising from many applications, for instance, in generalized equations, we restrict our consideration to the case m = 1 and for every (x, z) ∈ gph F near ( x, z) ∈ gph F. Then, T = f + F is (γ, h)-metrically pseudo subregular in the direction u at ( x, ȳ) ∈ gph T if and only if (0, 0) ∈ SCr γ,h T ( x, ȳ)(u).

T = f + F where f : X -→ Y = Y 1 is C 1 and F : X ⇒ Y
Proof It is sufficient to prove only the necessary part. Without any loss of generality, we may assume x = 0, f ( x) = ȳ = 0. Suppose T is (γ, h)-metrically pseudo subregular in the direction u at (0, 0). Let κ, δ , ε be positive real numbers under which the following estimation In view of (c), one has

h(x k ) γ-1 u k ∈ D * T (x k , y k )(v k ).
Applying the sum rule formula to T = f + F at (x k , z k ), the latter inclusion yields

f ( xk ) = f (x k ) + ∇ f (x k )( ûk ) + 1 0 [∇ f (x k + t ûk ) -∇ f (x k )]( ûk ) dt.
From the Lipschitz continuity of the Jacobian ∇ f , we have Taking into account that u k → 0, and passing to the limit as k → ∞ in (4.53), we obtain that lim inf

k→∞ τ + 1 k 2 -1 v k , y k y k 0.
However, the latter relation obviously contradicts (d). Thus, the proof is complete.

Remark 4.2

The conclusion of the preceding proposition is still valid for γ = 2 if we choose h(x) = xx . Indeed, following the proof above, we have

y k = (s k ) γ v k = x k γ v k = h(x k ) γ v k . ( 4

( 4 . 4 )

 44 Define a lsc function f k : X ×Y -→ R ∪ {+∞} by the formula f k (x, y) := δ gph T (x, y)

1 .

 1 which is equivalent to (4.18d).Combining Facts 1, 2, 3, 4 we obtain a full proof for Proposition 4.Definition 4.1 ((γ, h)-limiting critical set) Let T , γ and h as similar as in Definition 3.2 and let ( x, ȳ) ∈ gph T . For some fixed element u ∈ X, we define the limiting critical set SCr γ,h T ( x, ȳ)(u) with respect to T , γ, h and u at the reference point ( x, ȳ) as follows. A pair(v, u ) ∈ Y × X * lies in SCr γ,h T ( x, ȳ)(u) if it is possible to find some sequences (t k ) ↓ 0, (v k , u k ) Y ×X * ---→ (v, u ) and (u k , v k ) ∈ S X × S Y * whichfulfill simultaneously the following conditions: i. lim k→∞ u u ku = 0;

Theorem 4 . 1

 41 Let T , γ, h and ( x, ȳ) as in Definition 4.1. Suppose that each function h i is locally Lipschitz around x and that lim supx→ x,x ∈S d(x, S) h i (x) < +∞, i = 1, . . . , m. (4.31)If (0, 0) ∈ SCr γ,h T ( x, ȳ)(u), then the set-valued mapping T is (γ, h)-metrically pseudo subregular in the direction u ∈ X at ( x, ȳ).ProofThe proof is almost based on Proposition 4.1. Assume that the norm in Y * = Y * 1 × • • • ×Y * m coincides with the maximum y Y * = y 1 , . . . , y m Y * = max y 1 Y * 1 , . . . , y m Y * m .

  .33c) implies lim k→∞ u ûk -u = 0. In summary, it follows that (0, 0) ∈ SCr γ,h T ( x, ȳ)(u), which contradicts the assumption of Theorem 4.1.Remark 4.1 By letting h i (x) = xx , Theorem 4.1 subsumes to somewhat studied in [9,Theorem 1]. Taking m = 1, h 1 = d x, T -1 ( ȳ) , then Theorem 4.1 recovers the results presented in the works[START_REF] Ngai | Metric subregularity of multifunctions: First and second order infinitesimal characterizations[END_REF][START_REF] Ngai | Directional Hölder metric subregularity and application to tangent cones[END_REF].

  has a closed and convex graph. Under some robust condition imposed on the given data f and the abstract subdifferential ∂ , the next proposition has the advantage of offering a necessary and sufficient condition for (γ, h)-metric pseudo subregularity.Proposition 4.3 () Let γ = γ 1 ∈ [1,2) and h = h 1 : X -→ R be the same as in statement of Theorem 4.1. Suppose in addition that the Jacobian map ∇ f is Lipschitz around x while the coderivative associated with ∂ obeys the sum rule D * ( f + F)(x, f (x) + z) = ∇ f (x) * + D * F(x, z) (4.43)

d x, T - 1

 1 (0) τh(x) 1-γ d 0, T (x)(4.44) holds whenever x = 0 ∈ cone B(u, δ )∩εB. If (0, 0) ∈ SCr γ,h T (0, 0)(u), we may select for each k, some elementss k > 0, u k ∈ S X , v k ∈ Y , u k ∈ X * and v k ∈ S Y * such that (a). the pair (x k , y k ) is in gph T with x k = s k u k and y k = f (x k ) + z k = (s k ) γ v k ; (b). lim k→∞ u u ku = lim k→∞ v k = lim k→∞ u k = 0; (c). u k , -h(x k ) 1-γ v k ∈ N(gph T, (x k , y k )); (d). lim k→∞ v k , h(x k ) 1-γ y k h(x k ) 1-γ y k = 1.

h 2 - 1

 21 (x k ) γ-1 u k -∇ f (x k ) * v k ∈ D * F(x k , z k )(v k ). (4.45)According to the definition of a coderivative, and noticing that gph F is closed and convex, we deduceh(x k ) γ-1 u k -∇ f (x k ) * v k , xx k +v k , zz k 0 (4.46)when (x, z) ∈ gph F. Taking (b) into account, we may apply (4.44) with x = x k and getd x k , T -1 (0) τh(x k ) 1-γ d 0, T (x k ) τh(x k ) 1-γ y k . (4.47) Recall that S = T -1 (0). Let σ k > 0 be such that lim sup k→∞ σ k d(x k , S) = 0 and let xk ∈ S \ {x k } satisfy x kxk d(x k , S) + min σ k , 1 k 2 h(x k ) 1-γ y k . (4.48) Substituting x = xk and z =f ( xk ) in (4.46), we get h(x k ) γ-1 u k -∇ f (x k ) * v k , xkx k +v k ,f ( xk )z k 0. (4.49)Consequently, after replacing z k by y kf (x k ), (4.49) readsh(x k ) γ-1 u k , xkx k + v k , y k v k ,f ( xk ) + f (x k ) + ∇ f (x k )( xkx k ) . (4.50)Combining inequalities (4.47), (4.48) and (4.50), yieldsh(x k ) γ-1 u k , xkx k xkx k + h(x k ) γ-1 τ + 1 k v k , y k y k 1 xkx k v k ,f ( xk ) + f (x k ) + ∇ f (x k )( xkx k ) .

( 4 .

 4 51)Setting ûk := xkx k and applying the Taylor expansion to f gives:

+ 1 k 2 - 1

 21 lim sup k→∞ sup t∈[0,1] ∇ f (x k + t ûk ) -∇ f (x k ) ûk < +∞,and therefore, we obtainlim sup k→∞ f ( xk )f (x k ) -∇ f (x k )( ûk ) ûk 2 < +∞.(4.52)This leads to the estimationu k , xkx k xkx k + τ v k , y k y k θ k h(x k ) 1-γ xkx k , (4.53) in which the real sequence (θ k ) is bounded. Due to the choice of xk , it is possible to write d x k , S xkx k d x k , S + σ k , which permits to deduce that lim sup k→∞ xkx k h(x k ) = lim sup k→∞ xkx k d(x k , S) • d(x k , S) h(x k ) < +∞. (4.54)

  we obtain (4.6d) by applying (4.6b).

	Fact 2 We have ȳ ∈ T ( x1 k ).
	Let us define
	t k := x1 k -x, ỹ1 k -ȳ ,	(4.11a)

  .55) Reminding (4.47), this yields d(x k , S) = o(h(x k )), and hence, from the choice of xk , this implies xkx k = o(h(x k )). (4.56) Combining this relation with (4.53), we reach a contradiction as in the previous proof.

Research of the last author was supported by the Australian Research Council (ARC) grant DP160100854 and benefited from the support of the FMJH Program PGMO and from the support of EDF.