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

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

van Ngai Huynh, Huu Tron Nguyen, van Vu Nguyen, Michel Thera

To cite this version:

van Ngai Huynh, Huu Tron Nguyen, van Vu Nguyen, Michel Thera. Directional Metric Pseudo

Subregularity of Set-valued Mappings: a General Model. 2019. �hal-02307887�

(will be inserted by the editor)

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

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

Tribute to Professor Alexander Kruger on his sixty-fifth birthday. With recognition for research achievement and friendship

Received: date / Accepted: date

Abstract This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the liter- ature. Some slope and abstract coderivative characterizations are established.

Keywords Abstract subdifferential·Metric regularity·Directional metric regu- larity·Metric subregularity·Directional H¨older metric subregularity·Directional metric pseudo-subregularity·Coderivative·Slope

Mathematics Subject Classification (2000) 49J52·49J53·90C30

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.

First author

Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet- nam E-mail: [email protected]

Second author

Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet- nam

E-mail: [email protected] Third author

Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet- nam

E-mail: [email protected] Fourth author

XLIM UMR-CNRS 7252, Universit´e de Limoges, France and Centre for Informatics and Applied Optimi- sation (CIAO), Federation University Australia

E-mail: [email protected]

1 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 [4], even if claimed by Ioffe [17,18] ,“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 map- ping 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 cru- cial 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 [4,7,11,16–19,29], and references therein. By the time and demand of use and applications, variants of this property have emerged suitable to practical prob- lems. Weaker/stronger versions: calmness, (strong) (H¨older) metric sub/regularity, semiregularity or equivalent versions: pseudo Lipschitz, linear openness were stud- ied and have proved to have an important role in various applications in Mathematics, especially in Variational Analysis and Optimization [5,17,18,21–23,29],etc.

As recently proposed by Arutyunov [1], Gfrerer [10], Ngai-Th´era [14], Ngai- Tron-Th´era [12], Ngai-Tron-Tinh [26], 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 [3] to study sensitivity analysis.

Later, Ioffe [15] introduced and investigated an extension called relative metric regu- larity which covers many notions of metric regularity in the literature. In [27] Penot studied this property to establish second-order optimality conditions.

The paper concerns a new type of directional metric regularity. In this article, mo- tivated 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 H¨older metric subregularity intro- duced by Ngai, Tron and Tinh and metric pseudo subregularity explored by Gfrerer.

This property is given in Definition3.2. Our aims are to give characterizations of the property. First, we establish a slope characterization and after that we move to a subd- ifferential/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.

2 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 [24], Ioffe [19], Rockafellar &

Wets [29] and Penot [28] as our desk-copies.

For our purposes, we are going to work in the framework of real Banach spaces. If Xis such a space, we denote byk · kthe associated norm and byd(x,Ω)the distance fromx∈X to the subsetΩ ofX, that is,d(x,Ω):= inf{kx−yk:y∈Ω}. GivenX, we denote the topological dual (continuous dual) byX^∗, byk · k^∗the dual norm of k · k, byBX={x∈X:kxk ≤1}the closed unit ball, byS_X={x∈X:kxk=1}the unit sphere, byB(x,r)the closed ball with centerxand radiusr, respectively.

By a set-valued mapping (also named by some authors multifunction), we mean a mappingT fromX into the subsets (possibly empty) of another Banach spaceY and we use the notationT :X ⇒Y.The graph ofT denoted by gphT is the set of those points inX×Y such thaty∈T(x), whileT⁻¹:Y ⇒X , the inverse of T (always defined), is given by(x,y)∈gphT ⇐⇒ (y,x)∈gphT⁻¹.We say thatT is closed if its graph is closed with respect to the product topology onX×Y. Given a setK⊂X, we use the notation coneK for the conic hull of K, that is for the set of all conic combinations∑ⁱ⁼ⁿ_i=1λixiof points ofKwhereλi≥0 for each indexi.

Given an extended real-valued function f :X →R∪ {+∞}, we use the nota- tion clf to denote the lower semicontinuous envelope of f defined by clf(x) = lim infu→xf(u)), and Domf will refer to the domain of f, that is, the set of those pointsx∈Xsuch that f(x)is finite. We recall that the convex subdifferential of f at x∈Domf is the set

∂f(x):= {x^?∈X^∗:hx^?,y−xi ≤f(y)−f(x) for all y∈X}, with the convention that∂f(x) =/0, whenf(x) = +∞.

For the purpose of this study, we use the concept of slope|∇f|(x)of a function f atx∈Domf. This is a quantity introduced by De Giovani, Marino & Tosques [6]

and defined by

|∇f|(x):=







0 ifxis a local minimum point off

lim sup

y→x,y6=x

[f(x)−f(y)]₊

kx−yk otherwise.

where the notation a₊ means a if a≥0 and 0 if a<0. For x∈/ Domf, we set

|∇f|(x) = +∞.

When f is a convex function defined on a Banach space andxis not a minimum point, then according to Ioffe [16, Poposition 3.8] (see also Az´e & Corvelec [2, Propo- sition 3.2])

|∇f|(x) =sup

y6=x

f(x)−f(y)

kx−yk and |∇f|(x) =d(0,∂f(x)).

In the following sections, we make use of the notion of abstract subdifferential operator. fromX into the subsets ofX^∗.Such an object, denoted by∂, satisfies the following conditions:

(C1) If f :X →Ris a convex function which is continuous around ¯x∈X andβ : R→Ris a continuously differentiable function att= f(x), then

∂(β◦f)(x)⊆ {β⁰(f(x))x^?∈X^∗:hx^?,y−xi ≤f(y)−f(x) ∀y∈X};

(C2) ∂f(x) =∂g(x)if f(y) =g(y)for allyin a neighborhood ofx;

(C3) Letf₁:X→R∪ {+∞}be a lower semicontinuous function and f₂, ...,f_n:X→ Rbe Lipschitz functions. If f₁+f₂+...+f_nattains a local minimum atx₀,then for any ε>0, there exist x_i∈x₀+εBX,x^?_i ∈∂f_i(x_i), i=1,2, . . . ,n such that

|f_i(x_i)−f_i(x₀)|<ε, andkx^?₁+x^?₂+...+x^?_nk<ε.

We recall that the indicator function δC of a closed setC inX is the function defined byδC(x) =0 whenx∈CandδC(x) = +∞, otherwise. Given an abstract sub- differential∂, the setN(C,x):=∂ δC(x)is called thenormal conetoCatxassociated with∂.

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

LetF:X⇒Y and(x,¯ y)¯ ∈gphF. Given an abstract subdifferential∂ and normal cone associated with∂, the set-valued mappingD^∗F(x,¯ y)¯ :Y^∗⇒X^∗defined by

D^∗F(x,¯ y)(y¯ ^?) =

x^?∈X^∗:(x^?,−y^?)∈N gphF,(x,¯ y)¯ for ally^?∈Y^∗, is called the coderivative ofFassociated with∂.

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

(C5) If f is a separable function defined onX×Y, that is, f(x,y):= f₁(x) +f₂(y), (x,y)∈X×Y,wheref₁:X→R∪ {+∞}, f₂:Y →R∪ {+∞},then

∂f(x,y) =∂f₁(x)×∂f₂(y), for all (x,y)∈X×Y.

It is well known that the proximal subdifferential in Hilbert spaces, the Fr´echet subd- ifferential in Asplund spaces, the viscosity subdifferentials in smooth spaces as well as the Ioffe and the Clarke-Rockafellar subdifferentials in the setting of general Ba- nach spaces are subdifferentials verifying the conditions (C1)-(C5).

3 Slope characterizations of directional pseudo subregularity

For the understanding of the paper, we recall the definitions of metric sub-regularity and H¨older metric subregularity. We say that a set-valued mappingF:X⇒Ybetween metric spacesX,Y ismetrically subregular at a point (¯x,y)¯ ∈gphF with constant τ>0, if there exists a neighbourhoodUof ¯xsuch that

d(x,F⁻¹(y))¯ 6τd(y,¯ F(x))for allx∈U. (3.1) If in relation (3.1) we replaced(y,¯ F(x))byd(y,¯F(x))^γ, withγ>0, thenF is said to beH¨older metrically subregularat(¯x,y)¯ ∈gphFwith modulusτand orderγ. For such a situation, (3.1) becomes

d(x,F⁻¹(y))¯ 6τd(y,¯F(x))^γ for all x∈U. (3.2) Throughout the rest of the paper, we assume given Banach spacesX andY_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 noteT:= (T₁, . . . ,T_m):X⇒Y₁× · · · ×Y_m, the set-valued mapping defined byT(x):=T₁(x)×. . .×T_m(x)and we considerγ:= (γ₁, ...,γm)∈ R^msuch thatγ_i>1 fori=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 directionuand order γ.

Definition 3.1 (directional pseudo subregularity, [11])A set-valued mappingT= (T₁, . . . ,Tm):X⇒Y1× · · · ×Y_mis said to be(γ₁, . . . ,γm)-metrically pseudo subregular (γ_i>1) in the directionuat(x,¯y)¯ ∈gphT with modulusτ>0 iff there existε>0 andδ >0 such that

d x,T⁻¹(y)¯ 6τ

m

∑

i=1

kx−xk¯ ^1−γid y¯_i,T_i(x)

(3.3) forx6=x¯inB(x,¯ε)∩ x+¯ coneB(u,δ)

.

As observed by Gfrerer, whenm=1,γ-metric pseudo subregularity in the direction u=0 implies H¨older subregularity of order¹_γ. Note also that form=1, if we choose T =T₁andγ=γ₁, then (3.3) becomes:

d x,T⁻¹(¯y)γ

6kx−xk¯ ^γ−1d x,T⁻¹(y)¯

6τd y,¯ T(x)

. (3.4)

Hence, if a set-valued mappingT :X⇒Y isγ-MPSR at(x,¯y)¯ in the directionu, then T is directionally H¨older metrically subregular of orderγat(x,¯ y)¯ in the directionu as mentioned in [26].

The new idea in this contribution is to consider a general metric pseudo sub- regularity model called(γ,h)−pseudo subregularity associated with a given func- tion h:= (h₁, . . . ,h_m):X −→R^m+ and withγ := (γ₁, ...,γm)∈R^m withγi >1 for i=1, ...,m. To facilitate ease of reading, we shall introduce some useful real-valued

functions corresponding toT= (T₁, . . . ,T_m):X⇒Y₁× · · · ×Y_mandγ= (γ₁, . . . ,γm).

For each j,we defineρTj(·) =d y¯_j,T_j(·) and set

ϕ_Tj(x):=







ρ_Tj(x)

h_j(x)^γj−1 ifh_j(x)>0,

0 otherwise.

Now let us denote byϕT the sumϕT=∑^m_j=1ϕT_j, byψT=clϕT the lower semicon- tinuous envelope ofϕT and bySthe sublevel set[ψ_T60]. Throughout the rest of the document, we make use of the following assumption:

(A) h₁, . . . ,h_m:X−→R+are continuous functions and satisfy

h_i(x) =0=⇒ρTi(x) =0. (3.5) Definition 3.2 ((γ,h)-metric pseudo subregularity)LetT= (T₁, . . . ,Tm):X⇒Y1×

· · · ×Y_mbe given and let(x,¯y)¯ ∈gphT. The mappingT is said to be(γ,h)-metrically pseudo subregular in the directionuat(x,¯y)¯ ∈gphT forh= (h₁, . . . ,h_m)if there exist τ>0,ε>0 andδ>0 such that

d x,T⁻¹(y)¯

6τ

∑

16i6m,hi(x)>0

d y¯_i,T_i(x)

h_i(x)^γi−1 (3.6)

forx∈B(x,¯ ε)∩ x¯+coneB(u,δ)

with∑^m_i=1h_i(x)>0.

For the direction u:=0,we will say briefly thatT is(γ,h)−pseudo subregular at (x,¯y)¯ (that is, the inequality (3.6) is satisfied for allxnear ¯x). Let us note that when considering some special mappingsh, one recovers the concepts of directional metric subregularity mentioned above. For instance,

◦ if we choose the functionsh_i(x) =kx−xk, one gets metric pseudo subregularity;¯

◦ if one considers the casem=1,h₁(x):=d(x,T⁻¹(y))¯ one gets directional H¨older metric subregularity;

◦ more generally, let be givenmsubsetsS_i, i=1, ...,m,ofT_i⁻¹(y)¯ containing ¯x.In some situations, for suitable setsS_i, when the distancesd(x,S_i)can be computed explicitly, the pseudo-subregularity model with respect toh_i(x):=d(x,S_i),i= 1, ...,m.seems to be more convenient than the H¨older metric subregularity one.

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

As mentioned in the introduction, formally this pseudo-subregularity model cov- ers 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 spacesXandY,and for given(x,¯y)¯ ∈gphF,consider the inclusion

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

x∈S ⇐⇒ ϕ(x) =0,

an usual subregularity model forFat(x,¯ y), called¯ ϕ−subregularity, is defined as the following error bound property:

d(x,S)≤τ ϕ(x),

for all x near ¯x, for someτ >0. Obviously, ϕ−subregularity can be regarded as pseudo-subregularity by taking

h(x):=d(y,¯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∈Rⁿ: q(x) =0, f(x)≥0, g(x)≥0, hf(x),g(x)i=0}, (3.8) where,q:Rⁿ→R^d, f:= (f₁, ...,f_m):Rⁿ→R^m;g:= (g₁, ...,g_m):Rⁿ→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. [8,9,20]). Given a solution ¯x∈S,we say that the complementarity system (3.8) satisfies an error bound property at ¯x, if there isτ>0 such that for allxnear ¯x,one has

d(x,S)≤τ kq(x)k+

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 kq(x)k+

m i=1

∑

(−f_i(x))₊+

m i=1

∑

(−g_i(x))₊

m i=1

∑

|f_i(x)g_i(x)|

hi(x)

!

, (3.10) where,(−fi(x))₊:=max{−fi(x),0},andhi(x):=|max{fi(x),gi(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 lat- ter 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_iare smooth.

Throughout the paper, it is convenient to keep in mind the notation used in [14]:

x−→^u x¯is meant to bex→x¯ifu=0 andx→x¯and_kx−¯^x−¯x_xk→_kuk^u otherwise, as well.

The rest of this section will be devoted to establish some characterizations of (γ,h)-metric pseudo subregularity (Definition3.2). For such a purpose, the next propo- sition will be useful. First of all, we need the following lemma which gives a relation between the setsSandT⁻¹(y).¯

Lemma 3.1 Let T :X ⇒Y be a set-valued mapping. Then, one has S⊂T⁻¹(y).¯ Furthermore, if T has closed graph then S=T⁻¹(y).¯

Proof The proof is straightforward. ut

Proposition 3.1 Suppose that a closed mapping T is not(γ,h)-metrically pseudo subregular in the direction u at(x,¯y)¯ ∈gphT . Then, for each real sequenceτ_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. ψT(x_k)61−^τ√^k

τ_kd(x_k,S);

iii.

∇ψT

(x_k)6√ τk. Consequently, one has

lim inf

x−→^u x,x6∈S,¯ ψ_T(x)/d(x,S)→0

n|∇ψ_T|(x)o

=0. (3.11)

Proof Let us note from Lemma3.1thatS=T⁻¹(y)¯ since gphTis closed. According to the definition of metric pseudo subregularity, we can find a sequence ˜x_k−→^u x¯such that

˜

x_k6∈S, τkd x˜_k,S

>

∑

j=1,...,m hj(x˜_k)>0

d y¯_j,T_j(x˜_k) h_j(x˜_k)^γj−1 =

m

∑

j=1

ϕTj(x˜_k)>ψT(x˜_k). (3.12)

Hence, the lower semicontinuous (lsc for short) functionψ_T inherits the following property:

0<ψ_T(x˜_k)<τ_kd x˜_k,S

,k=1,2, . . . (3.13) Letε_k:=ψ_T(x˜_k)>0 andλ_k:=minn√

τ_kd(˜x_k,S),^εk

τ_k

o>0. By the Ekeland varia- tional principle, there exists for eachkan element ˆx_ksuch that

◦ kxˆ_k−x˜_kk6λ_k;

◦ ψ_T(x˜_k)−^εk

λ_kkxˆ_k−x˜_kk>ψ_T(xˆ_k);

◦ the functionf_k:x∈X7−→ψ_T(x) +^εk

λ_kkx−xˆ_kkattains its global minimum at ˆx_k. We are going to establish the following facts:

(i) ˆx_k6∈S,k=1,2, . . .;

(ii) ˆx_k−→^u x;¯ (iii) ψT(xˆ_k)61−^τ√^k

τ_kd(xˆ_k,S), forksufficiently large;

(iv) |∇ψ_T|(ˆx_k)6√

τ_k, forksufficiently large.

Assuming that (i) does not hold, one has τ_kd x˜_k,S

6τ_kkx˜_k−xˆ_kk6τ_kλ_k6ε_k=ψ_T(x˜_k), in contradiction with (3.13).

To prove (ii), let us note thatλk=o(kx˜_k−xk)¯ by virtue of the inequality λ_k6√

τ_kd x˜_k,S 6√

τ_kkx˜_k−xk.¯ Setµ_k:=^kˆ_k˜^xk−¯xk

xk−¯xk. After involving the triangle inequality, one deduces

1−kxˆ_k−x˜_kk kx˜_k−xk¯

kx˜_k−xk¯ 6kxˆ_k−xk¯ 6

1+kxˆ_k−x˜_kk kx˜_k−xk¯

kx˜_k−xk.¯

Recalling thatkxˆ_k−x˜_kk6λ_k, we conclude thatµ_k→1 as well askxˆ_k−xk →¯ 0. If u6=0, then a few straightforward calculations give us

ˆ x_k−x¯ kxˆ_k−xk¯ − u

kuk = 1

µk

˜ x_k−x¯ kx˜_k−xk¯ − u

kuk

+ 1

µ_k

xˆ_k−x˜_k kx˜_k−xk¯ +

1 µ_k−1

u kuk.

(3.14)

Using (3.14) it yields

ˆ xk−¯x

kˆx_k−¯xk−_kuk^u

→0, and therefore, (ii) is proved.

For establishing (iii), we invoke (3.13) and obtain ψ_T(xˆ_k)

d(xˆ_k,S)6 ψ_T(x˜_k)

d(ˆx_k,S)6τ_kd x˜_k,S d(xˆ_k,S) . Since

|d(ˆx_k,S)−d(x˜_k,S)|6kxˆ_k−x˜_kk6λ_k6√

τ_kd(x˜_k,S), for largekwe get

ψ_T(ˆx_k)

d(xˆ_k,S)6 τ_kd x˜_k,S d x˜_k,S

−√

τkd x˜_k,S= τ_k 1−√

τk

. Hence, (iii) follows immediately.

In oder to verify (iv), remember that f_k(·)attains a minimum at ˆx_k. As a result, the inequality

ψ_T(x) +ε_k λk

kx−xˆ_kk>ψ_T(xˆ_k) holds whenxis nearby ˆx_k. Equivalently,

ψT(ˆx_k)−ψ_T(x) kxˆ_k−xk 6 εk

λk

for allx6=xˆ_kbelonging to a neighborhood of ˆx_k. In summary, we have whenkis large enough

|∇ψ_T|(xˆ_k)6ε_k λk

=max (

ψ_T(x˜_k)

√

τkd x˜_k,S,τk

)

6max√

τk,τk =√

τk. (3.15) Lettingx_k=xˆ_k, the whole proof is established. ut

Based on Proposition3.1, the next theorem offers a sufficient criterion for metric pseudo subregularity.

Theorem 3.1 Let T ,γ, h andψ_T be defined as above. Suppose that the condition lim inf

x−→^u x,x6∈S,¯ ψ_T(x)/d(x,S)→0

n|∇ψ_T|(x)o

>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,¯y)¯ ∈gphT .

Proof When (3.16) is valid, the set-valued mappingTmust be(γ,h)-metrically pseudo subregular as a direct consequence of Proposition3.1. ut Corollary 3.1 Under the same assumptions of Theorem3.1, if now we have

lim inf

x−→^u x,x6∈S,¯ ψ_T(x)/kx−¯xk→0

n|∇ψ_T|(x)o

>0, (3.17)

then, the set-valued mapping T is(γ,h)-metrically pseudo subregular w.r.t. the direc- tion u at(¯x,y)¯ ∈gphT .

Proof Relation (3.17) implies the one in (3.16). ut Using Theorem3.1and takinghi(x) =kx−xk, one obtains a slope characterization¯ of directional pseudo subregularity ofT.

Proposition 3.2 (slope characterization)If lim inf

x−→^u x,x6∈S¯ kx−¯xk^−γϕ_Ti(x)→0

∇cl ^m

∑

i=1

ρ_Ti(·) k· −xk¯ ^γi−1

(x)>0, (3.18) then T isγ-MPSR in the direction u at(x,¯ y).¯

Similarly, considering the casem=1,h₁(x):=d x,T⁻¹(y)¯

, we derive a characteri- zation of directional H¨older metric subregularity ofT.

Proposition 3.3 ( slope characterization)Suppose that the following condition lim inf

x−→^u x,x6∈S,¯

d(¯y,T(x)) kx−¯xkd(x,T−1(y))¯ γ−1→0

∇cl d(y,¯T(·)) d(·,T⁻¹(y))¯ ^γ−1

(x)>0 (3.19)

holds. Then T is directional H¨olderγ-metric subregular in the direction u at(x,¯ y).¯ In many applications, it is sufficient to focus on the casem=1. For such a situation, when applying Theorem3.1we 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∈Domf∩Domg satisfying f(x)>0and 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 denoteΘ := _g^f. Ifxis an isolated point of DomΘ, thenxis a local minimum of bothΘ 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:

kz−xk6δ_k=⇒g(x)−g(z)6 |∇g|(x) +ε_k

kx−zk. (3.21)

For eachk, we may selectz_k∈B(x,δk)\ {x}such that f(x)−f(z_k)> |∇f|(x)−ε_k

kx−z_kk. (3.22)

Due to the continuity ofg, it is possible to assumeg(z_k)>0. We have

Θ(x)−Θ(z_k) kx−z_kk =

f(x)

g(x)− f(z_k) g(z_k) kx−z_kk

= 1

g(z_k)· f(x)−f(z_k)

kx−z_kk − f(x)

g(x)g(z_k)·g(x)−g(z_k) kx−z_kk

> 1

g(z_k) |∇f|(x)−εk

− f(x)

g(x)g(z_k) |∇g|(x) +εk

.

From the last inequality we derive lim sup

k→∞

Θ(x)−Θ(z_k) kx−z_kk

>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)

kx−zk , we obtain

|∇Θ|(x)>lim sup

k→∞

Θ(x)−Θ(z_k) kx−z_kk > 1

g(x)|∇f|(x)− f(x)

g(x)²|∇g|(x).

u t Invoking Lemma3.2, we obtain Lemma3.3used in the sequel for proving Proposi- tion3.4.

Lemma 3.3 Let T :X ⇒Y be a given set-valued mapping and let (x,¯y)¯ ∈gphT . Suppose that h:X −→R+ is locally Lipschitz around x and that the subsequent¯ condition is valid as well

lim sup

x−→^u x,x6∈S¯

d(x,S)

h(x) <+∞ (3.23)

for some u∈X . Then, lim inf

x−→^u x,x6∈S,¯

h(x)1−γ ρT(x)

d(x,S) →0

∇

clρ_T h^γ−1

(x)> lim inf

x−→^u x,x6∈S,¯ h(x)^−γρ_T(x)→0

|∇clρ_T|(x) h(x)^γ−1

. (3.24)

Proof Applying Lemma3.2withf =clρ_T andg=h^γ−1we deduce

∇

clρ_T h^γ−1

(x)>|∇clρ_T|(x)

h(x)^γ−1 −clρ_T(x) h(x)^γ−1

∇ h^γ−1 (x)

h(x)^γ−1 (3.25)

for eachx6∈S. According to [13], we have ∇ h^γ−1

(x) = (γ−1)h(x)^γ−2

∇h

(x). (3.26)

But since h is locally Lipschitz, it holds thatκ(x):=

∇h

(x) is locally bounded around ¯x. Let us rewrite (3.25) as follows

∇

clρ_T h^γ−1

(x)>|∇clρT|(x)

h(x)^γ−1 −(γ−1)κ(x)clρ_T(x) [h(x)]^γ

>|∇clρ_T|(x)

h(x)^γ−1 −(γ−1)κ(x) ρT(x)

h(x)^γ−1d(x,S)·d(x,S) h(x) .

(3.27)

Combining (3.23), (3.25) with (3.27) we infer lim inf

x−→^u x,x6∈S,¯

h(x)1−γ ρT(x)

d(x,S) →0

∇

clρ_T h^γ−1

(x)> lim inf

x−→^u x,x6∈S,¯

h(x)1−γ ρT(x)

d(x,S) →0

|∇clρ_T|(x)

h(x)^γ−1 . (3.28)

Assume that(x_k)is a sequence inXsuch that

x_k−→^u x,¯ x_k6∈S, h(x_k)^1−γρ_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→∞

n

h(x_k)^−γρT(x_k)o

=lim sup

k→∞

h(x_k)^1−γρT(x_k)

d(x_k,S) ·d(x_k,S) h(x_k)

=0. (3.30)

As a result, we get lim inf

k→∞

|∇clρ_T|(x_k) h(x_k)^γ−1

> lim inf

x−→^u x,x6∈S,¯ h(x)^−γρ_T(x)→0

|∇clρ_T|(x) h(x)^γ−1

. (3.31)

Combining (3.29) with (3.31), it holds that lim inf

x−→^u x,x6∈S,¯

h(x)1−γ ρT(x)

d(x,S) →0

|∇clρT|(x)

h(x)^γ−1 > lim inf

x−→^u x,x6∈S,¯ h(x)^−γρ_T(x)→0

|∇clρT|(x) h(x)^γ−1

. (3.32)

In summary, (3.28) and (3.32) give us lim inf

x−→^u x,x6∈S,¯

h(x)1−γ ρT(x)

d(x,S) →0

∇

clρT

h^γ−1

(x)> lim inf

x−→^u x,x6∈S,¯ h(x)^−γρ_T(x)→0

|∇clρ_T|(x) h(x)^γ−1

. (3.33)

This completes the proof of Lemma3.3. ut

Based on the previous results, we present a robust version of Theorem3.1for the casem=1 which might be more comfortable in practice.

Proposition 3.4 Let T ,x,¯ y and h satisfy all assumptions of Lemma¯ 3.3. Then, under the following condition

lim inf

x−→^u x,x6∈S,¯ h(x)^−γρ_T(x)→0

|∇clρ_T|(x)

h(x)^γ−1 >0, (3.34)

the set-valued mapping T isγ-metrically pseudo subregular atx for¯ y in the direction¯ u.

Proof The proof follows by combining Theorem3.1with Lemma3.3. ut

4 Coderivative characterization of directional metric pseudo subregularity Theorem3.1provides two sufficient conditions ensuring the validity of directional pseudo subregularity through the slopes corresponding to suitable functions. Gfrerer in his work [11] dealt with such a property using the notion of coderivatives. In or- der to study the directional metric regularity property, the authors of [25], introduced the notion of limiting critical set around a reference point(x,¯ y)¯ ∈gphT. Follow- ing 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:

(C₆): Let f₁,f₂:X→R∪ {+∞}be two locally Lipschitz functions aroundx₀∈X with f₁(x₀)≥0, f₂(x₀)>0. For anyε>0 one has

∂ f1

f₂

(x₀)⊆ ^[

x₁,x₂∈B(x₀,ε)

f2(x₀)∂f1(x₁)−f1(x₀)∂f2(x₂) f₂(x₀)² +εB_X^?

.

Note that(C₆)is valid for all usual subdifferentials used in the literature of vari- ational analysis.

Proposition 4.1 Let T :X ⇒Y =Y₁× · · · ×Y_m be a closed set-valued mapping between two Banach spaces X and Y and(x,¯ y)¯ ∈gphT . Suppose given functions h_i :X −→R+,(i=1,· · ·,m), locally Lipschitz aroundx. Let u¯ ∈X and γ>1 be given for which the following condition is fulfilled

lim sup

x−→^u x,x6∈S¯ i=1,...,mmax

d(x,S)

h_i(x) <+∞. (4.35)

If T is not metrically pseudo subregular in the direction u at(x,¯y), then there exist¯ some real sequence(t_k)↓0together with(u_k,v_k)∈SX×Y, u^?_k∈X^∗and v^?_k∈Y^∗such that

(a). lim

k→∞

u_k

=1,lim

k→∞

kuku_k−u =0;

(b). lim

k→∞max_i=1,...,m

(t_k)^1−γikv_kik =0;

(c). the vector(u^?_k,−˜v^?_k)belongs to N gphT,(x¯+t_ku_k,y¯+t_kv_k)

, wherev˜^?_k∈Y^∗is given byv˜^?_ki=h_i(x¯+t_ku_k)^1−γiv^?_ki;

(d). lim

k→∞

(

∑^m_i=1hv^?_ki,hi(x¯+t_ku_k)^1−γiv_kii

h₁(x+¯ t_ku_k)^1−γ1v_k1, . . . ,h_m(¯x+t_ku_k)^1−γmv_km

)

=1;

(e). lim

k→∞ku^?_kk=0,lim

k→∞kv^?_kik=1.

Proof For convenience, we assume that the distance onY is given by kyk_Y :=ky₁k_Y1+· · ·+ky_mk_Ym.

Further, with the aim of simplifying the notation, let us usek·kto indicate the norm in any Banach space. By Proposition3.1, we can find some sequence x_k−→^u x¯such that

(i) x_k6∈S;

(ii) lim

k→∞

ψT(x_k) d(x_k,S)=0;

(iii) lim

k→∞

n

∇ψ_T (x_k)o

=0.

Denotingαk:=kx_k−xk¯ >0 and assumingψT(x_k) =βkd(x_k,S)where(β_k)↓0, for eachk, choose some positive parametersσk andηksatisfyingσk=o(ψ_T(x_k))and ηk<σk. Makingσkandηksmaller if necessary, we may suppose

◦ 2η_k+σk<ψT(x_k);

◦ ϕT(z)>ψT(x_k)−σ_kwheneverkz−x_kk6ηk. By (4.35), we have min_j=1,...,m

h_j(x_k)}>0.Letτ_k:=

∇ψ_T

(x_k), then according to the definition,x_kis a local minimum to the functionψ_T(·) + (τ_k+σ_k)k· −x_kk. Hence, there exists 0<r_k6 min

j=1,...,m

h_j(x_k)^γjη_k such that

ψ_T(x_k) = min

kx−x_kk62r_k

n

ψ_T(x) + (τ_k+σ_k)kx−x_kko

. (4.36)

By the definition of a lsc envelope, we may select some(ˆx_k,yˆ_k)∈gphTwhich fulfills the two properties below:

kxˆ_k−x_kk6σ_kr_k and

m

∑

j=1

ky¯_j−yˆ_{k j}k

h_j(xˆ_k)^γj−1 6ψ_T(x_k) +σ_kr_k. (4.37) Indeed,since 0<r_k6 min

j=1,...,m

h_j(x_k)^γjηk ,h_jare continuous and ψT(x_k) =clϕ_T(x_k), we can choose ˆx_k∈Xsuch thatkxˆ_k−x_kk6σ_kr_kand

ϕT(xˆ_k)6ψT(x_k) +σkr_k/2. (4.38) Moreover, there exists ˆy_k∈T_j(ˆx_k)such that

kyˆ_{k j}−y¯_jk6d(y¯_j,T_j(xˆ_k)) + σ_kr_k 2∑^m_j=1h_j(xˆ_k)^−γj+1. It follows that

m

∑

ky¯_j−yˆ_{k j}k h_j(xˆ_k)^γj−16

m

∑

d(y¯_j,T_j(xˆ_k)) h_j(xˆ_k)^γj−1 +σ_kr_k

m j=1

∑

1 h_j(ˆx_k)^γj−1 2

m

∑

1 h_j(xˆ_k)^γj−1

!−1

=

m

∑

j=1

d(y¯j,Tj(xˆ_k))

h_j(xˆ_k)^γj−1 +σ_kr_k/2

=ϕ_T(ˆx_k) +σ_kr_k/2.

From these estimations and (4.38), one obtains that

m

∑

ky¯_j−yˆ_{k j}k

h_j(xˆ_k)^γj−1 6ψT(x_k) +σ_kr_k/2+σ_kr_k/2=ψT(x_k) +σ_kr_k. Using (4.36), we deduce

m

∑

j=1

ky¯_j−yˆ_{k j}k

h_j(xˆ_k)^γj−16ψ_T(x) + (τ_k+σ_k)kx−x_kk+σ_kr_k 6δgphT(x,y) +

m

∑

ky¯_j−y_jk

h_j(x)^γj−1+ (τ_k+σk)kx−xˆ_kk + (τ_k+σ_k)kx_k−xˆ_kk+σ_kr_k.

(4.39)

Define an extended real-valued function f_k:X×Y−→R∪ {+∞}by the formula

f_k(x,y):=δ_gphT(x,y) +

m

∑

ky¯_j−y_jk

h_j(x)^γj−1+ (τ_k+σ_k)kx−xˆ_kk. (4.40)

Because gphT is closed, the continuity of hj implies that f_k is lsc. Observe that f_k(ˆx_k,yˆ_k) =∑^m_j=1 ^{k¯yj−ˆyk jk}

h_j(ˆx_k)^γj−1 and that

f_k(xˆ_k,yˆ_k)6 inf

kx−x_kk62rk

f_k(x,y) +εk,

whereε_k:= (τ_k+σ_k)kx_k−xˆ_kk+σ_kr_k>0. Settingλ_k:=r_k√

σ_k>0 and applying the Ekeland variational principle, take(x˜_k,y˜_k)∈X×Y such that

◦ k(xˆ_k,yˆ_k)−(x˜_k,y˜_k)k6λ_k;

◦ f_k(ˆx_k,yˆ_k)−^εk

λ_kk(ˆx_k,yˆ_k)−(x˜_k,y˜_k)k>f_k(x˜_k,y˜_k);

◦ (x˜_k,y˜_k)is a minimum to the function(x,y)∈X×Y 7−→ f_k(x,y) +^εk

λ_kk(x,y)− (x˜_k,y˜_k)ksubject to the constraintkx−x_kk62r_k.

The last condition along with the properties(C₁)−(C₆)of the subdifferential operator show that, there are some elements ˜xⁱ_k,x˜⁴_{k j}∈X, ˜yⁱ_k∈Y, ˜y⁴_{k j}∈Y_jand also ˜x^i?_k,x˜^4?_{k j}∈X^∗,

˜

y^i?_k ∈Y^∗, ˜y^4?_{k j} ∈Y_j^?which fulfill the following conditions:

(i) max

kx˜_k−x˜ⁱ_kk,kx˜_k−x˜⁴_{k j}k,ky˜_k−y˜ⁱ_kk,ky˜_k−y˜⁴_{k j}k 6ν_k, where the parameterν_k is chosen such thatν_k6min

λ_k,β_kky¯1−y˜_k1k, . . . ,β_kky¯m−y˜_kmk ; (ii) (x˜^1?_k ,y˜^1?_k )∈∂X×YδgphT(x˜¹_k,y˜¹_k) =N gphT,(x˜¹_k,y˜¹_k)

; (iii) ˜x^2?_k ∈∂_Xk· −xˆ_kk(˜x²_k);

(iv) (x˜^3?_k ,y˜^3?_k )∈∂X×Yk(·,·)−(˜x_k,y˜_k)k(x˜³_k,y˜³_k);

(v) ˜y^4?_{k j}∈∂Y_jky¯j− ·k(y˜⁴_{k j});

(vi) ˜x^4?_{k j}∈∂X h^1−γ_j ^j

(˜x⁴_{k j}) = (1−γj)h_j(x˜⁴_{k j})^−γj ∂Xh_j (˜x⁴_{k j});

(vii)

x˜^1?_k + (τ_k+σ_k)˜x^2?_k +^εk

λ_kx˜^3?_k +∑^m_j=1ky¯j−y˜⁴_{k j}kx˜^4?_{k j} 6σ_k; (viii)

y˜^1?_{k j}+^εk

λ_ky˜^3?_{k j}+h_j(˜x⁴_{k j})^1−γjy˜^4?_{k j} 6σ_k.

We shall establish the conclusion of Proposition4.1step-by-step through several aux- iliary facts.

Fact 1 It holds that

k→∞lim

kxˆ_k−x_kk kx_k−xk¯ =lim

k→∞

kx˜ⁱ_k−x_kk kx_k−xk¯ =lim

k→∞

kx˜⁴_{k j}−x_kk

kx_k−xk¯ =0; (4.41a)

k→∞lim h_j(ˆx_k) h_j(x_k)=lim

k→∞

h_j(˜xⁱ_k) h_j(x_k)=lim

k→∞

h_j(x˜⁴_{k j})

h_j(x_k) =1; (4.41b)

k→∞lim ( 1

ψ_T(x_k)

m

∑

ky¯_j−yˆ_{k j}k h_j(xˆ_k)^γj−1

)

=1; (4.41c)

k→∞lim ( 1

ψT(x_k)

m

∑

j=1

ky¯_j−y˜ⁱ_{k j}k h_j(x˜ⁱ_k)^γj−1

)

=1,i=1,2,3; (4.41d)

k→∞lim ( 1

ψT(x_k)

m

∑

ky¯j−y˜⁴_{k j}k hj(x˜⁴_{k j})^γj−1

)

=1. (4.41e)

Proof of Fact 1Firstly, we establish (4.41a). Equality lim

k→∞

kˆx_k−x_kk

kx_k−¯xk =0 is trivial due to the choice of ˆx_k. Further, since

max

kx˜_kⁱ−x˜_kk,kx˜⁴_{k j}−x˜_kk 6λk=o(kx_k−xk),¯ whereaskx˜_k−xˆ_kk6λ_k=o(kx_k−xk), one gets¯

max

kx˜ⁱ_k−x_kk,kx˜⁴_{k j}−x_kk =o(kx_k−xk),¯ which implies (4.41a).

For (4.41b), let us involve the Lipschitz property of each functionh_j. Indeed, for each j=1,2, . . . ,mthere isL_j>0 andr_j>0 for which one has

kx−xk¯ 6r_j,kx⁰−xk¯ 6r_j kx−x⁰k6r_j

=⇒ |h_j(x)−h_j(x⁰)|6L_jkx−x⁰k. (4.42) Thus, whenkis large, it holds that

|h_j(xˆ_k)−h_j(x_k)|6L_jkxˆ_k−x_kk6L_jσ_kr_k. (4.43) This allows us to write

h_j(xˆ_k) h_j(x_k)−1

6Lj

r_k

h_j(x_k)σ_k. (4.44)

By interchanging in turn the role between ˆx_kwith ˜x_{k j}, ˜xⁱ_kand repeating the arguments above, we deduce

maxn

h_j(x˜⁴_{k j}) h_j(x_k) −1

,

h_j(x˜ⁱ_k) h_j(x_k)−1

o

6L_j r_k

h_j(x_k)(σ_k+2√

σ_k). (4.45) Sincer_k6h_j(x_k)^γηk, (4.41b) follows from (4.44) and (4.45).

In the next step, we prove (4.41c). According to the choice of ˆx_k, it is possible to derive

h_j(xˆ_k)^1−γjky¯_j−yˆ_{k j}k>h_j(xˆ_k)^1−γjd y¯_j,T_j(xˆ_k)

=ϕTj(xˆ_k).