Nouha Chaoued

Amel Borgi
email: amel.borgi@insat.rnu.tn

Anne Laurent

Camphor Odor Recognition within Unbalanced Multi-Sets

In fuzzy logic context, some works deal with the camphor odor perception. In this paper, we present a novel rule-based decision system for the camphor odor recognition within unbalanced multiset. Our first contribution consists in an adaptation of fuzzy knowledge representation and inference rules to the multi-valued logic context. The second contribution concerns the improvement of the knowledge base by changing facts representation and adding new rules. This proposition provides satisfactory results in term of precision, recall, F-measure and accuracy.

I. Introduction

Alexander Graham Bell said a century ago, "until you can measure their likenesses and differences you can have no science of odor" [START_REF] Bell | Discovery and Invention[END_REF]. Even so, scientists have tended to predict or identify odors based on their molecular structure.

Several factors influence odor perception as the external environment and the individual. The recognition of an odor by humans results from a stimulus. It is not objective as it is affected by the cognitive interpretation.

Basically, odor is quantified by three parameters: quality or identity, intensity or strength, hedonic character [START_REF] Kissi | Conception et élaboration d'un système expert à base de règles floues pour le traitement d'information chimique[END_REF]. Many methods have been used to quantify odor. But, these methods are affected by different problems as the detection cost, the long time took for analysis and detection, and exposing human to danger.

Artificial intelligence algorithms and techniques such as fuzzy logic, neural networks, and genetic algorithms had improved the process of decision-making in odor perception systems [3].

The authors in [START_REF] Medasani | An overview of membership function generation techniques for pattern recognition[END_REF] indicate that the performance of fuzzy logic approaches depends on the used membership function. The universe of discourse in the fuzzy context may be discrete or qualitative but more often it is numerical. Furthermore, the determination of fuzzy membership functions is not always trivial.

In this paper, we focus on multi-valued logic. It allows to symbolically represent imprecise knowledge using ordered linguistic symbols from natural language [START_REF] Glas | Knowledge representation in a fuzzy setting[END_REF]. Multi-valued logic allows the knowledge representation regardless of their reference domain type.

II. Preliminaries

Multi-valued logic is based on De Glas's theory [START_REF] Glas | Knowledge representation in a fuzzy setting[END_REF]. Linguistic variables are used to represent an opinion independent of a measuring system. In this approach, every linguistic variable is represented by an ordered and finite list of M linguistic symbols. They constitute a multi-set denoted by [5] [6]:

L M = {τ 0 , τ 1 , ..., τ M -1 }; M ∈ N * \{1} ( 1
)
τ i is the membership degree to the multi-set L M with i ∈ [0, M -1]. The membership relation in multi-valued logic is partial: x ∈ α A ⇔ x belongs to A at a degree α (2)
It should be noted that symbolic degrees are connected only by the total order relation ≤ defined by [START_REF] Akdag | A qualitative theory of uncertainty[END_REF]:

τ α ≤ τ β ⇔ α ≤ β; ∀ α and β ∈ [0, M -1]
(3)

A qualifier ϑ α is associated with each degree to express the imprecision of a predicate. A predicate is a feature of language used to attribute a property to a thing.

x is ϑ α A ⇐⇒ (x is ϑ α A) is true ⇐⇒ (x is A) is τ α true (4)
For example saying that the odor is little pleasant means that the odor satisfies the predicate pleasant with the degree τ 1 (Figure 1). Linguistic term sets are usually assumed to be uniformly distributed as the distance between every pair of successive terms is the same, i.e. 1 M -1 with M the number of terms in the set. Nevertheless, in some cases, a particular subdomain may be more informative than the remaining These linguistic term sets are said to be unbalanced, knowsn as unbalanced sets. Some authors [START_REF] Wang | A new version of 2-tuple fuzzy linguistic representation model for computing with words[END_REF]- [START_REF] Dong | Consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets and its use in the linguistic GDM with preference relation[END_REF] proposed examples from real life using these sets. Truck [START_REF] Truck | Comparison and links between two 2-tuple linguistic models for decision making[END_REF] indicates that such sets "are commonly used to describe erratic phenomenons and irregular progression in the patterns such as weather events or behavior under the influence of alcohol". Xia and Xu [START_REF] Xia | An approach to multiplicative linguistic group decision making based on possibility degrees[END_REF] give the example of diminishing marginal utility in economics. They indicate that the investment of a same amount is different if the company has a bad or a good performance. In this case, we need more precise bad information than good one. Hence, the gaps between the grades expressing bad information are smaller than the ones between the grades expressing good information.

In our work, we deal with unbalanced terms set in the context of multi-valued logic. This multi-set is defined as: L M = {τ 0 , τ 1 , ..., τ (M -1) }; M ∈ N * \{1}; with M its cardinality.

Each term τ i represents a possible value for a linguistic variable. It is defined by its position π(τ i ) (Figure 2-a):

π : L M → [0, 1]
This function represents the distribution of the terms τ i using real values between 0 and 1.

These terms are not necessarily symmetrically distributed or equidistant from each others, i.e. ∆(τ i , τ i+1 ) are not equal for all i ∈ [0, M -1] (Figure 2-b): ∆ :

L M × L M → [0, 1] ∆(τ i , τ j ) = π(τ j ) -π(τ i )
In this paper, we implement an inference engine for camphor odor perception. Factual knowledge and used rules are expressed within unbalanced multi-sets.

III. Inference Engine

An expert system consists of a knowledge base and an inference engine. It uses knowledge about its application domain, i.e. factual knowledge, and a reasoning process based on heuristic knowledge to solve problems that usually require human expertise.

Application areas include classification, diagnosis, monitoring, process control, design, scheduling and planning, and generation of options. In this work, we are interested in classification systems which allow to identify an object based on its characteristics.

An inference engine allows reasoning by applying logical rules to the knowledge base to deduct new knowledge. This process is iterated when a new fact in the knowledge base could trigger additional rules in the inference engine.

Approximate reasoning allows inference with approximate values and incomplete or ambiguous data. It lets us dealing with uncertainty in some expert systems and gives us more flexible handling of problems. It was defined by Zadeh in fuzzy logic context [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning[END_REF] as a set of methods that allow reasoning with imprecise inputs to obtain meaningful outputs. It is based on the Generalized Modus Ponens (GMP).

Reasoning uses production rules that consist of an IF part, i.e. a condition or premise, and a THEN part, i.e. an action or conclusion.

In the multi-valued logic context, the authors in [START_REF] Khoukhi | Approche logico-symbolique du traitement des connaissances incertaines et imprécises dans les systèmes à base de connaissances[END_REF]- [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multivalued framework[END_REF] have proposed different models with if-then rules. In the literature, rules are classified as: free or strong, simple or complex.

In this work, we use an adaptation of the GMP model proposed in [START_REF] Kacem | Generalized modus ponens based on linguistic modifiers in a symbolic multivalued framework[END_REF], [START_REF] Kacem | Extended symbolic approximate reasoning based on linguistic modifiers[END_REF] to infer within unbalanced multi-sets. This model deals with strong complex rules. The GMP model is as follow:

If (X 1 is ϑ α1 A 1 ) and ... and (X n is • n is the number of elements in the rule premise. The inference engine aims to deduct the inferred conclusion ϑ λ B. Our inference engine is based on forward chaining mode. The process starts with the known facts and deduce new facts.

ϑ αn A n ) Then (Y is ϑ β B) (X 1 is ϑ γ1 A 1 ) and ... and (X n is ϑ γn A n ) (Y is ϑ λ B) With: • A 1 , A 2 , ...,

IV. Odors classifier

The qualification of odors and the measurement of theirs intensities are based on human perception and are therefore subjective. The lack of numerical information due to the unavailability of measures for odors is the first problem when studying odors. Every odor has threedimensional different characteristics which are difficult to separate [START_REF] Kissi | Conception et élaboration d'un système expert à base de règles floues pour le traitement d'information chimique[END_REF]:

• Quality or identity: as floral, fruity, musk or camphor; • Intensity or strength: strong, medium, weak;

• Hedonic character: pleasant, unpleasant.

The semantic descriptors of odors are numerous, not precise, often referring to other odors, themselves described by a set of qualitative descriptors.

For this reason, we propose a tool for odor perception within multi-valued logic. We have implemented an inference engine based on the generic Knowledge Based System RAMOLI proposed in [START_REF]Ramoli: A generic knowledge-based systems shell for symbolic data[END_REF]. RAMOLI provides a reasoning process within uniform multi-sets. Our system aims to classify molecules according to the intensity of camphor odor within unbalanced multi-sets.

The olfactory properties of molecules have been described by Schnabel et al. [START_REF] Schnabel | Untersuchungen zur struktur-aktivitäts-beziehung bei geruchsstoffen 1. mitteilung: Wahrnehmungsschwellenwerte und geruchsqualitäten von gesättigten aliphatischen und alicyclischen verbindungen mit sauerstoff-funktion[END_REF]:

• R: the molecular length; The Schnabel score classifies molecules into three classes:

• non camphorated when the score is 0;

• slightly camphorated for ones with a score between 1 and 2; • very camphorated when the score is between 3 and 5.

The set of studied compounds includes 99 aliphatic alcohols: 29 primary, 52 secondary and 18 tertiary, with one or two carbon atoms.

To achieve the classification, the system uses facts within a multi-valued representation. To do so, we discretize the attribute values describing the molecules. We replace each numerical value with an equivalent linguistic term. Every molecule corresponds to a fact in the knowledge base.

Kissi, Ramdani et al. [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [24] proposed a representation of the membership functions for the linguistic variables spherical (Figure 3), surface (Figure 4) and camphorated (Figure 5) in the fuzzy logic context. We notice that these fuzzy sets are unbalanced. In a multi-valued logic context, we propose to represent these variables within the following unbalanced multi-sets:

• Spherical: L s 3 = {surely, almost, not} (Figure 6), with π(almost) = 0.3; • Surface: L 4 = {small, medium, large, very large}(Figure 7), with π(medium) = 0.62 and π(large) = 0.85; • Camphorated: L c 3 = {not, slightly, very} (Figure 8), with π(slightly) = 0.6. The variable density is added by the authors in [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [START_REF] Ramdani | Intelligent Sensory Evaluation: Methodologies and Applications[END_REF] to improve the accuracy of their system. Its corresponding multi-set is: L d 3 = {low, moderately, high} (Figure 9), with π(moderately) = 0.802. During the classification process, the facts are compared to the premise of existing rules. The rule triggered by the inference engine is that whose premise is the most similar to the fact.

V. Experimental Tests

To evaluate our inference engine, we proceed to three tests for camphor odor recognition. In the two first ones, we adapt to the multi-valued logic context the molecules representation and the fuzzy rules set proposed in [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [START_REF] Ramdani | Intelligent Sensory Evaluation: Methodologies and Applications[END_REF]. In the third test, we propose new molecules representation and new rules based on the set of studied compounds.

All these tests use unbalanced multi-sets.

A. First Test

The authors in [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [24] describe each molecule with the attributes: surface (small, medium, large) and spherical (yes, no). The camphor odor can be not or very.

The expert classification rules proposed in [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [24] are:

• If molecule is spherical and medium then its odor is very camphorated • If molecule is spherical and small then its odor is not camphorated • If molecule is spherical and large then its odor is not camphorated • If molecule is not spherical then its odor is not camphorated In this first test, in multi-valued logic context, we use the multi-sets L s 3 (Figure 6) and L 4 (Figure 7). For the variable camphorated, the multi-set includes only two terms: not and very (Figure 10). We adapt the expert fuzzy rules to the multi-valued logic context:

• R 1 : If molecule is surely spherical and surface is medium size then odor is very camphorated • R 2 : If molecule is surely spherical and surface is small size then odor is not camphorated • R 3 : If molecule is surely spherical and surface is large size then odor is not camphorated • R 4 : If molecule is not spherical then odor is not camphorated

The Table I gives a sample test of 4 molecules from the 99 molecules used. The classification rate of not camphorated molecule is good (93%) as all object in this class are homogeneous. However, the very camphorated class includes objects with a weak and strong odor of camphor, i.e. Schnabel score between 1 and 5. For this reason, we obtain a bad classification rate (37%). These results are similar to those obtained in [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [START_REF] Ramdani | Intelligent Sensory Evaluation: Methodologies and Applications[END_REF].

The accuracy of odor recognition is about 60% which is not a satisfactory rate.

B. Second Test

In [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [24], the authors proposed to consider the sphericity and the density of molecules to deduct the intensity of their camphor odors. The odor is described with two values: not and very. Camphorated perception is done using the following rules:

• if R/r>1.91 then not camphorated • if R/r ≤ 1.
7 and d ≤ 0.802 then very camphorated In a multi-valued logic context, the inference rules with the multi-sets L s 3 (Figure 6), L d 3 (Figure 9) and L 2 (Figure 10) are as follows:

• R 4 : If molecule is not spherical then odor is not camphorated • R 5 : If molecule is surely spherical and molecule is low dense then odor is very camphorated As in [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF] [24], we use in this test only rules R 4 and R 5 . A sample of 4 molecules from our test base is presented in Table II. The classification rate of non camphorated molecule is insufficient (66%). However, we obtain a good classification rate (95%) for the very camphorated class.

In [START_REF] Kissi | Determination of fuzzy logic membership functions using genetic algorithms: application to structure-odor modeling[END_REF], the accuracy of the non camphorated class is about 93%. While the accuracy of the very camphorated class is about 89%. We notice that we get a lower rate for the first class. While the rate for the second class is slightly better.

The accuracy of the classification is about 82% which is better than in the first test.

C. Third Test

In this test, we suggest to use the three attributes: surface, sphericity and density not only two of them as in the two first tests. We also propose to distinguish between molecules with a weak and strong camphor odor as in Schnabel et al. [START_REF] Schnabel | Untersuchungen zur struktur-aktivitäts-beziehung bei geruchsstoffen 1. mitteilung: Wahrnehmungsschwellenwerte und geruchsqualitäten von gesättigten aliphatischen und alicyclischen verbindungen mit sauerstoff-funktion[END_REF] work.

We introduce new rules obtained from the set of studied molecules using the new proposed knowledge representation.

The inference rules with the multi-sets L 4 (Figure 7), L s 3 (Figure 6), L d 3 (Figure 9) and L c 3 (Figure 8) are: The obtained classification rates are good for both non camphorated and camphorated molecules, respectively 83% and 97%. For the camphorated molecule, we obtain a rate of 70% for slightly camphorated class and 87% for very camphorated class.

Thus, the accuracy is about 90% for the classification in two classes non camphorated and camphorated; and 82% for the classification in three classes non camphorated, slightly camphorated and very camphorated.

D. Comparative results

In this paper, we implement a camphor odor classifier within unbalanced multi-sets. We test our tool with three different factual knowledge, i.e. facts representation, and distinct rules set.

We notice that the accuracy of the third test is the best. We also compare these tests using the metrics: precision, recall and F-measure. The table IV illustrates the values of theses metrics for all tests for the classification in two classes: Non camphorated and Camphorated.

The precision, the recall and the F-measure are improved as we use the attributes surface, sphericity and density in the molecule's description and in the new proposed rules. Moreover, the distribution of the camphorated components into two sub-classes, i.e. slightly and very, made the classes more homogeneous. It ameliorates the accuracy of our classification regarding to the first and second tests. The metrics for this test are presented in the Table V.

VI. Conclusion

The suitability of unbalanced multi-set theory in camphor odor recognition has been tested in this paper. To do so, an inference engine is implemented to deduct the odor of 99 aliphatic alcohols with one or two carbon atoms. The olfactory properties of molecules used are their measure of sphericity, their measure of surface and their density. In the same context of odors recognition, in a future study, we could test our inference engine to detect other odors as sandalwood odor.

In a further work, we will also test the suitability of unbalanced multi-sets in the diagnosis of childhood autism as described in the Childhood Autism Rating Scale (CARS). A tool was already implemented by Bel-Haj Kacem et al. in collaboration with the child psychiatry department of Razzi hospital in Tunisia but it uses uniform multi-sets [START_REF] Kacem | A diagnosis aid system of autism in a multi-valued framework[END_REF]. An adaptation of this proposal is required to deal with unbalanced multi-sets.

Figure 1 .

 1 Figure 1. Representation of a scale of 5 truth-degrees

Figure 2 .

 2 Figure 2. Terms Definitions

  • r: the molecular width; • R/r: the measure of sphericity of the molecule; • S: the measure of the surface of the molecule; • d: the density of the molecule (weight/volume).

Figure 3 .

 3 Figure 3. Membership functions for linguistic variable Spherical

Figure 4 .Figure 5 .

 45 Figure 4. Membership functions for linguistic variable Surface

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Representation of Spherical Multi-set

Figure 9 .

 9 Figure 9. Representation of Dense Multi-set

Figure 10 .

 10 Figure 10. Representation of Camphorated Multi-set

  A n are the rule predicates representing the premise and the observation, i.e. fact; • B is the predicate representing the rule conclusion;• A i and B are represented by the unbalanced multisets L M A i (i ∈ {1,..n}) and L M B respectively;

Table I

 I A sample First test results

	Inputs R/r S	Odor level Result
	1.90	8.96	very	TP
	1.55	11.85	not	FN
	2.46	11.63	not	TN
	2.17	12.97	not	FN

Table II A

 II sample Second test results

	Inputs R/r d	Odor level Result
	1.90	0.835	not	FN
	1.55	0.795	very	TP
	2.46	0.84	not	TN
	2.17	0.781	not	FN

  • R 6 : If surface is small size and molecule is low dense then odor is slightly camphorated • R 7 : If surface is small size and molecule is high dense then odor is not camphorated • R 8 : If surface is medium size and molecule is low dense then odor is very camphorated • R 9 : If surface is medium size and molecule is high dense then odor is slightly camphorated • R 10 : If molecule is surely spherical and surface is large size and molecule is low dense then odor is very camphorated • R 11 : If molecule is almost spherical and surface is large size and molecule is low dense then odor is slightly camphorated • R 12 : If molecule is not spherical and surface is large size and molecule is low dense then odor is not camphorated • R 13 : If surface is large size and molecule is high dense then odor is not camphorated In this test, we use the rules R 6 to R 13 . The Table III presents the obtained results for 4 molecules from our test base.

Table V

 V Precision Recall F-measure for the third test

		Not	Slightly	Very
		Camphorated	Camphorated	Camphorated
	Accuracy		0.82
	Precision	0.94	0.47	0.89
	Recall	0.83	0.69	0.87
	F-measure	0.88	0.56	0.88

  Three tests are done using the same training set. Results show that our approach can give similarly (first and second tests), if not better (third test), classification compared to Kissi, Ramdani et al. works.

				Table IV
		Precision Recall F-measure for the three tests
		Test 1	Test 2	Test 3
		Not	Camphorated Not	Camphorated Not	Camphorated
		Camphorated		Camphorated		Camphorated
	Accuracy	0.6		0.82	0.9
	Precision	0.49	0.875	0.9	0.8	0.94	0.89
	Recall	0.925	0.35	0.66	0.95	0.83	0.96
	F-measure	0.53	0.5	0.76	0.87	0.88	0.92

Acknowledgment

The authors would like to thank Mohamed Kissi and Mohammed Ramdani from the university of Hassan II Mohammedia -Morocco for their help and support in making this work possible by providing us the set of studied compounds.


Ngai Van

Huu Tron Huynh
email: nguyenhuutron@qnu.edu.vn

Nguyen

Vu Nguyen
email: nguyenvanvu@qnu.edu.vn

Michel Thera
email: michel.thera@unilim.fr

Huynh Van

Ngai • Nguyen
email: ngaivn@yahoo.com

Huu Tron

• Nguyen

Van Vu

Michel Théra

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

Keywords: subdifferential, Metric regularity, Directional metric regularity, Metric subregularity, Directional Hölder metric subregularity, Directional metric pseudo-subregularity, Coderivative, Slope Mathematics Subject Classification (2000) 49J52, 49J53, 90C30

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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; complementarity 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.

It is now well known that metric regularity is one of the pillars when studying these generalized equations. 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 claimed by Ioffe [START_REF] Ioffe | Metric regularity -a survey[END_REF][START_REF] Ioffe | Metric regularity -a survey[END_REF] ,"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] Ioffe | Convexity and variational analysis[END_REF][START_REF] Ioffe | Metric regularity -a survey[END_REF][START_REF] Ioffe | Metric regularity -a survey[END_REF][START_REF] Ioffe | Variational Analysis of Regular Mappings[END_REF][START_REF] Rockafellar | 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] Ioffe | Metric regularity -a survey[END_REF][START_REF] Ioffe | 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] Kruger | Nonlinear metric subregularity[END_REF][START_REF] Rockafellar | Variational Analysis[END_REF], etc.

As recently proposed by Arutyunov [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] Huynh | Directional metric regularity of multifunctions[END_REF], Ngai-Tron-Théra [START_REF] Huynh | Directional Hölder metric regularity[END_REF], Ngai-Tron-Tinh [START_REF] Ngai | Directional Hölder metric subregularity and application to tangent cones[END_REF], a new line of research is to build directional models for these objects. Characterizations of these concepts have been established and successfully applied to study optimality conditions for mathematical programs, for calculating tangent cones, etc. 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] 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 formally unifies all (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 and 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, Grundlehren der Mathematischen Wissenschaften[END_REF], Ioffe [START_REF] Ioffe | Variational Analysis of Regular Mappings[END_REF], Rockafellar & Wets [START_REF] Rockafellar | Variational Analysis[END_REF] and Penot [START_REF] Penot | 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 a quantity introduced by De Giovani, Marino & Tosques [START_REF] De 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)] + x -y otherwise.
where the notation a + means a if a ≥ 0 and 0 if a < 0. 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] Ioffe | 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. from X into the subsets of X * . Such an object, 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 a continuously differentiable function 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, 2, . . . , n such that | f i (x i ) -f i (x 0 )| < ε, 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 sub- differential ∂ , 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) for all y ∈ Y * ,
is called the coderivative of F associated with ∂ .

We assume further that ∂ satisfies a 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, we recall the definitions of metric sub-regularity and Hölder metric subregularity. We say that a set-valued mapping F : X ⇒ Y between metric spaces X,Y is metrically subregular at a point ( x, ȳ) ∈ gph F with constant τ > 0, if there exists a neighbourhood U of x such that

d(x, F -1 ( ȳ)) τd( 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 F is said to be Hölder metrically subregular at ( x, ȳ) ∈ gph F with modulus τ and order γ. For such a situation, (3.1) becomes

d(x, F -1 ( ȳ)) τd( 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 with closed graph 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 we consider γ := (γ 1 , ..., γ m ) ∈ R m such that γ i 1 for i = 1, ..., m. To begin with, let us first recall the definition of γ-metric pseudo subregularity (γ-MPSR, for short) w.r.t. a given direction u and order γ.

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, when m = 1, γ-metric pseudo subregularity in the direction u = 0 implies Hölder subregularity of order 1 γ . Note also that for m = 1, if we choose 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.
For 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;

• if one considers the case m = 1, h 1 (x) := d(x, T -1 ( ȳ)) one gets directional Hölder metric subregularity; • more generally, let be given m subsets S i , i = 1, ..., m, of T -1 i ( ȳ) containing x. In some situations, for suitable sets S i , when the distances d(x, S i ) can be computed explicitly, the pseudo-subregularity model with respect to h i (x) := d(x, S i ), i = 1, ..., m. seems to be more convenient than the Hölder metric subregularity one.

Especially, when γ i = 1 (i = 1, ..., m), one gets the usual metric subregularity.

As mentioned in the introduction, formally this pseudo-subregularity model covers all linear/nonlinear subregularity/error bound models considered in the literature. To see this, let us consider again a multifunction F : X ⇒ Y between two metric spaces X and Y, and for given ( x, ȳ) ∈ gph F, consider the inclusion

S := {x ∈ X : ȳ ∈ F(x)} . (3.7)
By means of a suitable residual function ϕ : X → [0, +∞), that is a function such that

x ∈ S ⇐⇒ ϕ(x) = 0,
an usual subregularity model for F at ( x, ȳ), called ϕ-subregularity, is defined as the following error bound property:

d(x, S) ≤ τϕ(x),
for all x near x, for some τ > 0. Obviously, ϕ-subregularity can be regarded as pseudo-subregularity by taking

h(x) := d( ȳ, F(x)) ϕ(x) , x ∈ X.
Let us give further an example for complementarity problems.

Example 3.1 Consider the complementarity system defined by

S := {x ∈ R n : q(x) = 0, f (x) ≥ 0, g(x) ≥ 0, f (x), g(x) = 0} , (3.8)
where,

q : R n → R d , f := ( f 1 , ..., f m ) : R n → R m ; g := (g 1 , ..., g m ) : R n → R m .
It is well known that a complementarity system like (3.8) covers, as a particular case, Karush-Kuhn-Tucker (KKT) systems associated to optimization problems, (see, e.g. [START_REF] Facchinei | An LP-Newton method: nonsmooth equations, KKT systems, and nonisolated solutions[END_REF][START_REF] Facchinei | Finite-dimensional variational inequalities and complementarity problems[END_REF][START_REF] Izmailov | Newton-type methods for optimization and variational problems[END_REF]). Given a solution x ∈ S, we say that the complementarity system (3.8) satisfies an error bound property at x, if there is τ > 0 such that for all x near x, one has

d(x, S) ≤ τ q(x) + m ∑ i=1 | min{ f i (x), g i (x)}| . (3.9)
This error bound plays a crucial role in the convergence analysis of some algorithms (e.g., the Newton-type methods) for solving system (3.8). It is easy to observe that the error bound property (3.9) is equivalent to the following pseudo-subregularity:

d(x, S) ≤ α q(x) + m ∑ i=1 (-f i (x)) + + m ∑ i=1 (-g i (x)) + m ∑ i=1 | f i (x)g i (x)| h i (x) , (3.10)
where, (f i (x)) + := max{-f i (x), 0}, and

h i (x) := | max{ f i (x), g i (x)}|, i = 1, ..., m.
In the error bound relation (3.9), the residual term for both the sign constraints and the complementarity constraint is represented by the function ∑ m i=1 | min{ f i (x), g i (x)}|, while the residual terms in relation (3.10) with respect to the sign constraints and to the complementarity constraint are separably represented. In some situations, this latter representation could be more convenient. In particular, regarding the residual term for the complementarity constraint, the functions min{ f i (x), g i (x)} (i = 1, ..., m) are generally less regular than the ones f i (x), g i (x), e.g., the functions min{ f i (x), g i (x)} being not necessarily smooth, even if f i , , g i are smooth.

Throughout the paper, it is convenient to keep in mind the notation used in [START_REF] Huynh | 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 of (γ, h)-metric pseudo subregularity (Definition 3.2). For such a purpose, the next proposition will be useful. First of all, we need the following lemma which gives a relation between the sets S and T -1 ( ȳ). Lemma 3.1 Let T : X ⇒ Y be a set-valued mapping. Then, one has S ⊂ T -1 ( ȳ). Furthermore, if T has closed graph then S = T -1 ( ȳ).

Proof The proof is straightforward. Proposition 3.1 Suppose that a closed mapping T is not (γ, h)-metrically pseudo subregular in the direction u at ( x, ȳ) ∈ gph T . Then, for each real sequence τ k ↓ 0, of non-negative numbers, there exists a sequence x k u -→ x which satisfies for large integers 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 ) √ τ k .
Consequently, one has

lim inf x u - →x,x ∈S, ψ T (x)/d(x,S)→0 |∇ψ T |(x) = 0. (3.11)
Proof Let us note from Lemma 3.1 that S = T -1 ( ȳ) since gph T is closed. 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 ). (3.12)
Hence, the lower semicontinuous (lsc for short) function ψ T inherits the following property:

0 < ψ T ( xk ) < τ k d xk , S , k = 1, 2, . . . (3.13)
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 ∈ X -→ ψ T (x) + ε k λ k
xxk attains its global minimum at xk . We are going to establish the following facts:

(i) xk ∈ S, k = 1, 2, . . . ; (ii) xk u - → x; (iii) ψ T ( xk ) τ k 1- √ τ k d( xk , S), for k sufficiently large; (iv) |∇ψ T |( xk ) √ τ k , for k sufficiently large.
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 (3.13).

To prove (ii), let us 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 xkxk λ k , we conclude that µ k → 1 as well as xkx → 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 .
(3.14)

Using (3.14) it yields xk -x xk -x -u u
→ 0, and therefore, (ii) is proved. For establishing (iii), we invoke (3.13) and obtain

ψ T ( xk ) d( xk , 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, the inequality

ψ T (x) + ε k λ k x -xk ψ T ( xk )
holds when x is nearby 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 when k is large enough

|∇ψ T |( xk ) ε k λ k = max ψ T ( xk ) √ τ k d xk , S , τ k max √ τ k , τ k = √ τ k . (3.15)
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 the condition

lim inf x u - →x,x ∈S, ψ T (x)/d(x,S)→0 |∇ψ T | (x) > 0 (3.16)
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 (3.16) 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, (3.17)
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 (3.17) implies the one in (3.16).

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, (3.18)
then T is γ-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 (3.19)
holds. 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 3.2 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). (3.20) 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 . (3.21)
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 . (3.22)
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 3.2, we obtain Lemma 3.3 used in the sequel for proving Proposition 3.4.

Lemma 3.3 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) < +∞ (3.23)
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 . (3.24)
Proof Applying Lemma 3.2 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 (3.25)
for each x ∈ S. According to [START_REF] Huynh | 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). (3.26)
But since h is locally Lipschitz, it holds that κ(x) := ∇h (x) is locally bounded around x. Let us rewrite (3.25) 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)] γ |∇ cl ρ T |(x) h(x) γ-1 -(γ -1)κ(x) ρ T (x) h(x) γ-1 d(x, S) • d(x, S) h(x) . (3.27)
Combining (3.23), (3.25) with (3.27) we infer 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) 1-γ ρ T (x) d(x,S) →0 |∇ cl ρ T |(x) h(x) γ-1 . (3.28) 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. (3.29)
By virtue of (3.23), we have lim sup

k→∞ d(x k , S) h(x k ) < +∞, which yields lim sup k→∞ 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. (3.30)
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
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 . (3.33)
This completes the proof of Lemma 3.3.

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 3.3. 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, (3.34)
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 3.3.

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 6 ): 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 6 ) is valid for all usual subdifferentials used in the literature of variational analysis. 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 sequence

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 k -x > 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 .
By (4.35), we have min j=1,...,m h j (x k )} > 0. 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 0 < 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 . (4.36)
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 .
(4.37)

Indeed, since 0 < r k min j=1,...,m h j (x k ) γ j η k , h j are continuous and

ψ T (x k ) = cl ϕ T (x k ), we can choose xk ∈ X such that xk -x k σ k r k and ϕ T ( xk ) ψ T (x k ) + σ k r k /2. (4.38)
Moreover, there exists ŷk ∈ T j ( xk ) such that

ŷk j -ȳ j d( ȳ j , T j ( xk )) + σ k r k 2 ∑ m j=1 h j ( xk ) -γ j +1 . It follows that m ∑ j=1 ȳ j -ŷk j h j ( xk ) γ j -1 m ∑ j=1 d( ȳ j , T j ( xk )) h j ( xk ) γ j -1 + σ k r k m ∑ j=1 1 h j ( xk ) γ j -1 2 m ∑ j=1 1 h j ( xk ) γ j -1 -1 = m ∑ j=1 d( ȳ j , T j ( xk )) h j ( xk ) γ j -1 + σ k r k /2 = ϕ T ( xk ) + σ k r k /2.

From these estimations and (4.38), one obtains that

m ∑ j=1 ȳ j -ŷk j h j ( xk ) γ j -1 ψ T (x k ) + σ k r k /2 + σ k r k /2 = ψ T (x k ) + σ k r k . Using (4.36), 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 . (4.39)
Define an extended real-valued function f k : X ×Y -→ R ∪ {+∞} by the formula

f k (x, y) := δ gph T (x, y) + m ∑ j=1 ȳ j -y j h j (x) γ j -1 + (τ k + σ k ) x -xk . (4.40)
Because gph T is closed, the continuity of h j implies that f k is lsc. 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 k (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 .
The last condition along with the properties (C 1 )-(C 6 ) of the subdifferential operator show that, there are some elements xi

k , x4 k j ∈ X, ỹi k ∈ Y , ỹ4 k j ∈ Y j and also xi k , x4 k j ∈ X * , ỹi k ∈ Y * , ỹ4
k j ∈ Y j which fulfill the following conditions:

(i) max xk -xi k , xk -x4 k j , ỹk -ỹi k , ỹk -ỹ4 k j ν k , where the parameter ν k is chosen such that 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 ( x4 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; (4.41a) 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; (4.41b) lim k→∞ 1 ψ T (x k ) m ∑ j=1 ȳ j -ŷk j h j ( xk ) γ j -1 = 1; (4.41c) 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; (4.41d) lim k→∞ 1 ψ T (x k ) m ∑ j=1 ȳ j -ỹ4 k j h j ( x4 k j ) γ j -1 = 1. (4.41e)
Proof of Fact 1 Firstly, we establish (4.41a). 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 (4.41a). For (4.41b), let us involve the Lipschitz property of 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 . (4.42)
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 . (4.43)
This allows us to write

h j ( xk ) h j (x k ) -1 L j r k h j (x k ) σ k . (4.44)
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 ). (4.45)
Since r k h j (x k ) γ η k , (4.41b) follows from (4.44) and (4.45).

In the next step, we prove (4.41c). 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 ( xk ). 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 jŷk j h j ( xk ) γ j -1 < ψ T (x k ) + σ k r k by the choice of ŷk j . Thus, the limit (4.41c) is a direct consequence of the fact that σ k = o(ψ T (x k )).

Based on (4.41b), we can derive that the proof of (4.41e) is similar to the one of (4.41d). Thus, it is sufficient to verify (4.41d) only. Indeed, fix an index i ∈ {1, 2, 3}. We infer from (4.41b) 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 ))
, we obtain (4.41d) by applying (4.41b).

Fact 2 We have ȳ ∈ T ( x1 k ). Let us define t k := x1 k -x, ỹ1 k -ȳ , (4.46a
)

u k := (t k ) -1 ( x1 k -x), (4.46b) v ki := (t k ) -1 ( ỹ1 ki -ȳi ), i = 1, . . . , m. (4.46c)
Then the following relations are valid as well

(u k , v k ) = 1, (4.47a) lim k→∞ u k = 1, (4.47b) lim k→∞ u u k -u = 0, (4.47c) lim k→∞ (t k ) 1-γ i v ki = 0. (4.47d)
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 (4.47a) is trivial by the definitions (4.46a)-(4.46c). To prove (4.47b), we note that x1 kx ∼ x kx = α k according to (4.41a). It follows from the choice of ŷk that (see (4.37)) ŷkiȳi 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 (4.47b) is obtained directly from (4.46a) and (4.46b).

With the aim of verifying (4.47c), let us define 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, (4.47c) follows from (4.47b) and (4.41a).

Finally, we establish (4.47d). Due to (4.46a), (4.46c) 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, (4.41a) and (4.41b)
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 ki -ȳi = o x k -x γ i . Thus, (4.47d) 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. (4.48)
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. (4.49)
By using (4.41b), 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 of the function ȳi -• -ỹ4 ki , • -ỹ4 ki . Thus,

ỹ4 ki = 1, ȳi -ỹ4 ki = -ỹ4 ki , ȳi -ỹ4 ki . (4.50)
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 (4.51)
Combining (4.41b), (4.49) with (4.51) we obtain (4.48).

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

u k := x1 k , (4.52a)
v ki := -h i ( x + t k u k ) γ i -1 ỹ1 ki , (4.52b)
ṽ ki := h i ( x + t k u k ) 1-γ i v ki . (4.52c)
For such elements, we have

lim k→∞ v ki = 1, (4.53a)
u k ,ṽ k1 , . . . ,ṽ km ∈ N gph T, (

x + t k u k , ȳ + t k v k ) , (4.53b)
lim k→∞ u k = 0, (4.53c)
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. (4.53d)
Proof of Fact 4 The relations (4.53a) is derived from (4.52b) and (4.48), while (4.53b) is a consequence of (ii). In order to obtain (4.53c), we invoke (iii), (iv), (vi), (vii), (4.41e) and (4.48). 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 , (4.54)
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 < +∞. (4.55)
Denoting γ * = max{γ 1 , . . . , γ m }, (4.54) 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 . (4.56)
Using (4.41b) 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, (4.53c) is obtained under the combination of (4.55), (4.56), (4.41e) and the assumption that ψ T (x k ) d(x k ,S) → 0. With the aim of establishing (4.53d), 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 . (4.57) 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 Taking into account the estimation

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
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 (4.58) 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 ), (4.61)
which implies

b ki - h i ( x4 ki ) h i ( x1 k ) γ i -1 a ki = o(b ki ). (4.62)
Taking the sum over the index i in (4.62), we reach the conclusion

b k - m ∑ i=1 h i ( x4 ki ) h i ( x1 k ) γ i -1 a ki = o(b k ). (4.63)
Observing that lim sup ii. the pair (x k , y k ) with x k := x + t k u k , y ki := ȳi

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, (4.64) 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 ).
+ (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 ; (4.67)
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 ) , (4.68a)
lim k→∞ u k = 0, lim k→∞ v ki = 1, (4.68b) lim k→∞ u k = 1, lim k→∞ u u k -u = 0, (4.68c) lim k→∞ (t k ) 1-γ 1 v k1 = • • • = lim k→∞ (t k ) 1-γ m v km = 0, (4.68d) 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. (4.68e)
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 , (4.69a) ûk := u k -1 u k , vki := (t k ) 1-γ i u k -γ i v ki , (4.69b)
û k := v k -1 u k , v k := v k -1 v k , (4.69c)
we obtain

x k = x + tk ûk , y ki = ȳi + (t k ) γ i vki , (4.70a) 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 ) , (4.70b
) ûk = 1, v k = 1, (4.70c) lim k→∞ vki = 0, lim k→∞ û k = 0. (4.70d)
On the other hand, combining (4.69c) with (4.68b) and (4.68e), 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 all assumptions of Proposition 4.2 but the pair (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 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) (4.71)
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 (4.71)

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 . (4.72)
Recall that x k ∈ T -1 (0). By virtue of (4.72), 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 . (4.73)
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

.74)

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 . (4.75)
Combining (4.73) with (4.75), 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 .
(4.76)

Passing to the limit w.r.t k in both sides of (4.76), 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. (4.77)
Since the left-hand side of (4.77) 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 T = f + F where f : X -→ Y = Y 1 is C 1 and F : X ⇒ Y 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. 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 Taking into account that u k → 0, and passing to the limit as k → ∞ in (4.88), we obtain that lim inf

h(x k ) γ-1 u k -∇ f (x k ) * v k ∈ D * F(x k , z k )(v k ).
h(x k ) γ-1 u k , xk -x k xk -x k + h(x k ) γ-1 τ + 1 k 2 -1 v k , y k y k 1 xk -x k v k , -f ( xk ) + f (x k ) + ∇ f (x k )( xk -x k ) .
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 Aknowledgements. We gratefully acknowledge the referees for their constructive comments and suggestions.

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

Proposition 4 . 1

 41 Let T : X ⇒ Y = Y 1 × • • • × Y mbe 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

1 .

 1 which is equivalent to (4.53d).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 *which fulfill 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. (4.66) 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 norm y Y * = y 1 , . . . , y m Y * = max y 1 Y * 1 , . . . , y m Y * m .

1 . 4 . 1

 141 .68c) implies lim k→∞ u ûku = 0. In summary, it follows that (0, 0) belongs to the set SCr γ,h T ( x, ȳ)(u), which contradicts the assumption of Theorem 4.Remark By letting h i (x) = xx , Theorem 4.1 subsumes to somewhat studied in[START_REF] Gfrerer | On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs[END_REF] 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].

Proposition 4 . 3

 43 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 ruleD * ( f + F)(x, f (x) + z) = ∇ f (x) * + D * F(x, z) (4.78) 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).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 estimationd x, T -1 (0) τh(x) 1-γ d 0, T (x) (4.79)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.

  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 (4.81)when (x, z) ∈ gph F. Taking (b) into account, we may apply (4.79) 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 . (4.82) 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 . (4.83) Substituting x = xk and z =f ( xk ) in (4.81), we get h(x k ) γ-1 u k -∇ f (x k ) * v k , xkx k +v k ,f ( xk )z k 0. (4.84)Consequently, after replacing z k by y kf (x k ), (4.84) 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 ) . (4.85) Combining inequalities (4.82), (4.83) and (4.85), yields

( 4 . 1 0[+ 1 k 2 - 1

 4121 86)Setting ûk := xkx k and applying the Taylor expansion to f gives:f ( xk ) = f (x k ) + ∇ f (x k )( ûk ) + ∇ f (x k + t ûk ) -∇ f (x k )]( ûk ) dt.From the Lipschitz continuity of the Jacobian ∇ f , we have lim supk→∞ 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 < +∞. (4.87)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 , (4.88) 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 ,

  .90) Reminding (4.82), 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 )). (4.91) Combining this relation with (4.88), we reach a contradiction as in the previous proof.

  Moreover, recall that ν k β k ȳiỹki (cf. (i)). Therefore, from the fact max{ ỹ1kiỹki , 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,	(4.60a)
		lim k→∞	ȳi -ỹ1 ki ȳi -ỹki	= lim k→∞	ȳi -ỹ4 ki ȳi -ỹki	= 1.	(4.60b)
	Combining (4.60a) with (4.60b), we obtain
	lim sup k→∞	| ỹ4 ki , ỹ1 ki -ỹ4 ki | ȳi -ỹ1 ki	lim sup k→∞	ỹ4 ki	ỹ1 ki -ỹ4 ki ki ȳi -ỹ1	= 0.
							.58)
	As in the proof of Fact 3 (see (4.50))
		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 .	(4.59)

Research of the first author was supported by VIASM. Research of the second author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2018.309. Research of the third author was supported by Minister of Education and Training, grant B2018-DQN-05. 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.