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ESAIM: COCV 25 (2019) 55 ESAIM: Control, Optimisation and Calculus of Variationshttps://doi.org/10.1051/cocv/2018047 www.esaim-cocv.org

STABILITY OF ERROR BOUNDS FOR CONIC SUBSMOOTH

INEQUALITIES∗

Xi Yin Zheng1,∗∗ and Kung-Fu Ng2

Abstract. Under either linearity or convexity assumption, several authors have studied the stabilityof error bounds for inequality systems when the concerned data undergo small perturbations. In thispaper, we consider the corresponding issue for a more general conic inequality (most of the constraintsystems in optimization can be described by an inequality of this type). In terms of coderivatives forvector-valued functions, we study perturbation analysis of error bounds for conic inequalities in thesubsmooth setting. The main results of this paper are new even in the convex/smooth case.

Mathematics Subject Classification. 49J52, 49J53, 90C31, 90C25.

Received March 13, 2018. Accepted September 9, 2018.

1. Introduction

Let X be a Banach space and f : X → R∪{+∞} be a proper lower semicontinuous function with x ∈ dom(f),the domain of f . Recall that f has a local error bound at x if there exist τ, δ ∈ (0, +∞) such that

d(x, S(f, x)) ≤ τ [f(x)− f(x)]+ ∀x ∈ B(x, δ), (1.1)

where S(f, x) := {x ∈ X : f(x) ≤ f(x)} is the sublevel set of f at x and B(x, δ) is the open ball with center xand radius δ. Error bound theory has been recognized to be significant in sensitivity analysis and convergenceanalysis of some algorithms for solving optimization problems. Since Hoffman’s pioneering work [8], a great dealof works have been reported in mathematical programming literature discussing the error bound issues (fordetails see [2, 6, 9, 14, 15, 19, 21, 22, 24–26] and references therein).

In practical problems, all data obtained are not perfectly ideal, one is forced to use “test” data, and thegap between the ideal and “test” data is unavoidable. Therefore, from the points of view of theoretical interestas well as for applications, it is important to study the issue of stability when the system data undergo smallperturbations; error bound issue is of no exception. In 1994, Luo and Tseng [16] first studied the effect on errorbounds when linear inequalities (in Euclidean spaces) are perturbed. In 2005, Zheng and Ng [28] establishedthe stability results on error bounds for systems of conic linear inequalities in general Banach spaces. In 2010,

∗This work was supported by the National Natural Science Foundation of P.R. China (Grant No. 11371312) and the secondauthor was supported by Earmarked Grants (GRF) from the Research Grant Council of Hong Kong (Project nos. CUHK 14302516and 14304014) and CUHK direct grant no. 4053218.

Keywords and phrases: Conic inequality; error bound; subdifferential; quasi-subsmoothness.

1 Department of Mathematics, Yunnan University, Kunming 650091, P.R. China.2 Department of Mathematics (and IMS), The Chinese University of Hong Kong, Shatin, Hong Kong.

*∗ Corresponding author: [email protected]

Article published by EDP Sciences c© EDP Sciences, SMAI 2019

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https://www.esaim-cocv.org

mailto:[email protected]

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2 X.Y. ZHENG AND K.-F. NG

relaxing the linearity restriction and in terms of the subdifferential, Ngai et al. [20] established the stability oferror bounds for convex inequalities; this stability result was further extended in [31] to the subsmooth casefrom the convex case.

Let F be a proper vector-valued function from a Banach space X to another Banach space Y ordered by aclosed convex cone C and consider the following conic inequality

F (x) ≤C F (x), (CIE)

where ≤C is the pre-order induced by C. Many constraints in optimization can be described in terms of conicinequalities. In this paper, we study the stability of error bounds for (CIE). Recall that (CIE) is said to havean error bound at x if there exist τ, δ ∈ (0, +∞) such that

d(x, S(F, x, C)) ≤ τd(F (x)− F (x),−C) ∀x ∈ B(x, δ), (1.2)

where S(F, x, C) denotes the solution set of (CIE), that is,

S(F, x, C) := {x ∈ X : F (x) ≤C F (x)}.

We say that (CIE) has a strong error bound at x if there exist τ, δ ∈ (0, +∞) such that (1.2) holds and x is anisolated point of S(F, x, C), equivalently, there exist τ, δ ∈ (0, +∞) such that

‖x− x‖ ≤ τd(F (x)− F (x),−C) ∀x ∈ B(x, δ). (1.3)

In the case when Y = R and C = R+, (1.2) reduces to (1.1), while (1.3) reduces to

‖x− x‖ ≤ τ [F (x)− F (x)]+ ∀x ∈ B(x, δ).

It is known and easy to see that (CIE) has an error bound (resp. strong error bound) at x if and only if themultifunction FC is metrically subregular (resp. strongly metrically subregular) at (x, F (x)), where FC(x) :=F (x) + C. The notion of metric subregularity is useful in variational analysis and optimization and has beenwell studied (cf. [4, 9] and the references therein).

Let τ(F, x, C) denote the modulus of the error bound of conic inequality (CIE), that is,

τ(F, x, C) := inf{τ > 0 : (1.2) holds for some δ > 0}. (1.4)

Clearly, F has an error bound at x if and only if τ(F, x, C) < +∞. Our main concern of this paper is toconsider the stability of error bound for conic inequality (CIE) when the objective function F undergoes smallperturbations: (CIE) is said to have a stable error bound at x if there exist η, r ∈ (0, +∞) such that τ(G, x, C) ≤η for each function G : X → Y • := Y ∪ {∞Y } with

‖G− F‖x := lim supx→x

‖G(x)− F (x)− (G(x)− F (x))‖‖x− x‖

< r.

For convenience of presentation as well as printing, let ∂CF (x) denote D∗FC(x, F (x))(IC+), whereD∗FC(x, F (x)) is the coderivative of the multifunction FC at (x, F (x)) (see Sect. 2 for the detail) and

IC+ := {v∗ ∈ C+ : ‖v∗‖ = 1} and C+ := {v∗ ∈ Y ∗ : 〈v∗, y〉 ≥ 0 ∀y ∈ C}. (1.5)

In Section 3, as a natural extension of the convexity and smoothness, we adopt the notion of subsmoothvector-valued functions, and provide examples and results on such subsmooth functions. In Section 4, two types

STABILITY OF ERROR BOUNDS FOR CONIC SUBSMOOTH INEQUALITIES 3

of stability properties for error bounds of F at x ∈ dom(F ) are considered: one is that F has a stable strongerror bound at x, while the other is a weaker one that F has a stable error bound at x (the exact definitionsare given in Sect. 4). When F is assumed to be subsmooth at x, they are shown to be implied by the conditions0 ∈ int(∂CF (x)) and 0 6∈ bd(∂CF (x)), respectively. The converses are also established in Theorems 4.2, 4.7 and4.8 under the additional assumption that F is locally Lipschitz at x.

2. Preliminaries

Let X be a Banach space with topological dual X∗ and let BX denote the closed unit ball of X. We denoteby B(x, r) and B[x, r] the open and closed balls with center x and radius r, respectively. For a subset A of X,we define the boundary bd(A) of A as bd(A) := cl(A) \ int(A), where cl(A) and int(A) denote the closure andinterior of A, respectively. For a ∈ A, we use T (A, a) and N(A, a) to denote the Clarke tangent cone and theClarke normal cone of A to a, respectively, defined by

T (A, a) :={v ∈ X : ∀an

A→ a and ∀tn → 0+ ∃vn → v s.t. an + tnvn ∈ A ∀n ∈ N}

and

N(A, a) :={x∗ ∈ X∗ : 〈x∗, h〉 ≤ 0 for all h ∈ T (A, a)

}.

For ε ≥ 0, let Nε(A, a) denote the set of Frechet ε-normals of A to a, that is,

Nε(A, a) :=

{x∗ ∈ X∗ : lim sup

xA→a

〈x∗, x− a〉‖x− a‖

≤ ε

}.

When ε = 0, Nε(A, a) is a convex cone which is called the Frechet normal cone of A to a and is denoted byN(A, a). Let N(A, a) denote Mordukhovich’s limiting normal cone of A to a which is defined by

N(A, a) := lim supx

A→a,ε→0+

Nε(A, x).

That is, x∗ ∈ N(A, a) if and only if there exists a sequence {(xn, εn, x∗n)} in A×R+ ×X∗ such that (xn, εn)→(a, 0), x∗n

w∗

→ x∗ and x∗n ∈ Nεn(A, xn) for each n.Let f : X → R∪ {+∞} be a proper lower semicontinuous function. We use dom(f) and epi(f) to denote the

domain and the epigraph of f , respectively. The Clarke subdifferential ∂f(x) of f at x ∈ dom(f) is defined as

∂f(x) :={x∗ ∈ X∗ : (x∗,−1) ∈ N(epi(f), (x, f(x)))

}. (2.1)

If f is locally Lipschitz at x, then

∂f(x) :={x∗ ∈ X∗ : 〈x∗, h〉 ≤ f◦(x, h) := lim sup

z→x, t↓0

f(z + th)− f(z)

t∀h ∈ X

}.

The Frechet subdifferential ∂f(x) of f at x is defined by

∂f(x) :=

{x∗ ∈ X∗ : lim inf

u→x

f(u)− f(x)− 〈x∗, u− x〉‖u− x‖

≥ 0

},


or equivalently

∂f(x) ={x∗ ∈ X∗ : (x∗,−1) ∈ N(epi(f), (x, f(x)))

}.

Let ∂f(x) denote Mordukhovich’s limiting subdifferential of f at x, that is,

∂f(x) ={x∗ ∈ X∗ : (x∗,−1) ∈ N(epi(f), (x, f(x)))

}.

Recall that ∂f(x) ⊂ ∂f(x) ⊂ ∂f(x) and that if f is convex then

∂f(x) = ∂f(x) = ∂f(x) ={x∗ ∈ X∗ : 〈x∗, h〉 ≤ f(x+ h)− f(x) for all h ∈ X

}.

It is known that if X is a separable Banach space such that its dual X∗ is nonseparable (e.g., X = l1), then

there exists a Lipschitz function f on X such that ∂f(x) = ∂f(x) = ∅ for all x ∈ X. Thus, it cannot be expectedto develop Frechet/limiting subdifferential theory in the general Banach space framework. Recall that a Banachspace X is an Asplund space if every continuous convex function on X is Frechet differentiable at every point ofa dense subset of X. It is well known that X is an Asplund space if and only if every separable subspace of Xhas a separable dual space. With regard to these notions, the following two lemmas collect some useful resultson the Clarke and Frechet subdifferentials (and normal cones) (cf. [3, 17]).

Lemma 2.1. Let X be a Banach space, f : X → R ∪ {+∞} be a proper lower semicontinuous function, andlet g : X → R ∪ {+∞} be locally Lipschitz at x ∈ dom(f). Then ∂g(x) 6= ∅ and ∂(f + g)(x) ⊂ ∂f(x) + ∂g(x).

Lemma 2.2. Let X be an Asplund space, A be a closed subset of X, and let f, g : X → R ∪ {+∞} be properlower semicontinuous functions. Then the following statements hold:(i) N(A, a) = lim sup

xA→a

N(A, x) for all a ∈ A and N(A, a) is the weak∗ closed convex hull of N(A, a).

(ii) ∂f(x) = lim sup

zf→x

∂f(z) for all x ∈ dom(f).

(iii) If f is locally Lipschitz at x, then ∂f(x) is the weak∗ closed convex hull of ∂f(x), that is, ∂f(x) =cow

∗(∂f(x)).

(iv) If f is locally Lipschitz at x ∈ dom(g) and x∗ ∈ ∂(f + g)(x), then for any ε > 0 there exist x1, x2 ∈ B(x, ε)

with |g(x2)− g(x)| < ε such that x∗ ∈ ∂f(x1) + ∂g(x2) + εBX∗ .

Let X,Y be Banach spaces, and C ⊂ Y be a closed convex cone, which defines a pre-order ≤C in Y as follows:y1 ≤C y2 ⇔ y2 − y1 ∈ C. It is known that ≤C is a partial order in Y if and only if the closed convex cone Cis pointed (i.e., C ∩ −C = {0}). In the remainder of this paper, we always assume that C is a closed convexpointed cone in Y . Let C+ and IC+ be defined by (1.5). Note that, in our consideration, IC+ will play a roleof an “abstract one”; indeed, in the special case that Y = R and C = R+, one clearly has IC+ = {1}. Let ∞Y

denote an abstract infinity and Y • := Y ∪ {∞Y }. For a vector-valued function F : X → Y •, the epigraph of Fwith respect to the ordering cone C is defined by

epiC(F ) :={

(x, y) ∈ X × Y : F (x) ≤C y}.

Recall that F is C-convex if epiC(F ) is a convex subset of X × Y ; thus, F is C-convex if and only if

F (λx1 + (1− λ)x2) ≤C λF (x1) + (1− λ)F (x2) ∀x1, x2 ∈ dom(F ) and λ ∈ [0, 1],


where dom(F ) := {x ∈ X : F (x) 6=∞Y }. The following lemma regarding the epigraph epiC(F ) is useful for us.

Lemma 2.3. Let X,Y be Banach spaces and F : X → Y • be a function such that epiC(F ) is closed. Letx ∈ dom(F ) and c ∈ C. The following statements hold:(i) N(epiC(F ), (x, F (x) + c)) ⊂ X∗ ×−C+.(ii) If, in addition, F is continuous at x, then

T (epiC(F ), (x, F (x))) + {0} × C = T (epiC(F ), (x, F (x))) ⊂ T (epiC(F ), (x, F (x) + c)) (2.2)

and

N(epiC(F ), (x, F (x) + c)) ⊂ N(epiC(F ), (x, F (x))) ⊂ X∗ ×−C+. (2.3)

Proof. Let (x∗, y∗) ∈ N(epiC(F ), (x, F (x) + c)). Then, for any ε > 0 there exists δ > 0 such that⟨x∗, x′ − x

⟩+⟨y∗, y′ − F (x)− c

⟩≤ ε(‖x′ − x‖+ ‖y′ − F (x)− c‖

).

for all (x′, y′) ∈ epiC(F ) with ‖x′ − x‖+ ‖y′ − F (x)− c‖ < δ. Noting that {x} × (F (x) + c+ C) ⊂ epiC(F ), itfollows that 〈y∗, h〉 ≤ ε‖h‖ for all h ∈ C ∩BY (0, δ). Since C is a closed convex set and ε is an arbitrary positivenumber, 〈y∗, h〉 ≤ 0 for all h ∈ C. Hence y∗ ∈ −C+. This shows that (i) holds.

To prove (ii), suppose that F is continuous at x. Since (2.3) follows easily from (2.2) and the defini-tions concerned, we only need to prove (2.2). To do this, let (u, v) ∈ T (epiC(F ), (x, F (x))). Then, for any

(xn, yn)epiC(F )−→ (x, F (x)) and tn → 0+ there exists a sequence {(un, vn)} such that (un, vn)→ (u, v) and

(xn, yn) + tn(un, vn) ∈ epiC(F ) ∀n ∈ N,

and so

(xn, yn) + tn(un, vn + c) ∈ epiC(F ) ∀(n, c) ∈ N× C.

Therefore, (u, v + c) ∈ T (epiC(F ), (x, F (x))) for all c ∈ C and the equality in (2.2) is shown (noting that{0} × C contains the origin). To show the inclusion in (2.2), let (u, v) ∈ T (epiC(F ), (x, F (x))), and take any

(xn, yn)epiC(F )−→ (x, F (x) + c) and tn → 0+. Then, noting that (xn, F (xn)) → (x, F (x)) by the continuity of

F at x, there exists a sequence (un, vn) → (u, v) such that (xn, F (xn)) + tn(un, vn) ∈ epiC(F ) for all n ∈ N,and so (xn, yn) + tn(un, vn) ∈ epiC(F ) for all n ∈ N (because yn − F (xn) ∈ C). This implies that (u, v) ∈T (epiC(F ), (x, F (x) + c)), and the inclusion in (2.2) is shown. The proof is complete.

The following lemma, essentially known (cf. [17, 30]), is also useful for us.

Lemma 2.4. Let X,Y, Z be Banach spaces and let F : X → Z and G : Z → Y • be a continuously differentiablefunction and a C-convex function, respectively. Suppose that x ∈ F−1(dom(G)) satisfies the following Robinsonqualification condition:

F ′(x)(X) + R+(dom(G)− F (x)) = Z, (2.4)

where F ′(x) denotes the derivative of F at x. Then

N(epiC(G ◦ F ), (x, G(F (x)))) ={

(F ′(x)∗(z∗), y∗) : (z∗, y∗) ∈ N(epiC(G), (F (x), G(F (x))))}.


In the vector-valued setting, we adopt the following coderivative (cf. [17])

D∗eF (x)(y∗) :={x∗ ∈ X∗ : (x∗,−y∗) ∈ N(epiC(F ), (x, F (x)))

}∀y∗ ∈ Y ∗. (2.5)

Since the Clarke normal cone N(epiC(F ), (x, F (x))) is a weak∗-closed convex cone, D∗eF (x) is a weak∗-closedsublinear multifunction (and so a closed convex multifunction). Let FC(x) := F (x) + C for all x ∈ X. Thengph(FC) = epiC(F ) and so D∗eF (x) coincides with the coderivative D∗FC(x, F (x)) of the multifunction FC at(x, F (x)). For convenience, we will use the following subdifferential of F :

∂CF (x) : = D∗eF (x)(IC+)

={x∗ ∈ X∗ : (x∗,−y∗) ∈ N(epiC(F ), (x, F (x))) for some y∗ ∈ IC+

}(2.6)

(cf. [27]). It is easy to verify that ∂CF (x) =⋃

y∗∈IC+

∂(y∗ ◦ F )(x) if F is C-convex. In the special case when

Y = R and C = R+, it is clear from (2.1) that ∂CF (x) reduces to the Clarke subdifferential of F and hence isweak∗-closed. It is worth mentioning that ∂CF (x) is not necessarily closed when Y is a general Banach space.In the C-convex case, the following subdifferential is known:

∂F (x) :={T ∈ L(X,Y ) : T (x− x) ≤C F (x)− F (x) ∀x ∈ X

},

where L(X,Y ) denote the space of all continuous linear operators from X to Y . In fact, several kinds ofsubdifferentials for vector-valued functions have been introduced and studied (for the detail see [5, 7, 11] andthe references therein).

Recall that a function F : X → Y • is locally Lipschitz at x ∈ dom(F ) if ‖F (x1)− F (x2)‖ ≤ L‖x1 − x2‖ forall x1, x2 ∈ B(x, δ) and some L, δ ∈ (0, +∞). Next we adopt the following weaker notion: F is said to be locallyC-Lipschitz at x if there exist L, δ ∈ (0, +∞) such that F is continuous on B(x, δ) and

|〈y∗, F (x1)− F (x2)〉| ≤ L‖x1 − x2‖ ∀x1, x2 ∈ B(x, δ) and ∀y∗ ∈ IC+ . (2.7)

With notations given in (2.5) and (2.6), we present the following:

Lemma 2.5. Let X,Y be Banach spaces with Y ordered by a closed convex cone C and let F : X → Y • belocally C-Lipschitz at x ∈ dom(F ). Then

∂(y∗ ◦ F )(x) ⊂ D∗eF (x)(y∗) ∀y∗ ∈ IC+ . (2.8)

Consequently, ⋃y∗∈IC+

∂(y∗ ◦ F )(x) ⊂ ∂CF (x). (2.9)

Proof. Let y∗ ∈ IC+ and x∗ ∈ ∂(y∗ ◦ F )(x). We need to show that x∗ ∈ D∗eF (x)(y∗), namely (x∗,−y∗) ∈N(epiC(F ), (x, F (x))). To do this, let

(u, v) ∈ T (epiC(F ), (x, F (x))). (2.10)

We have to show that

〈x∗, u〉 − 〈y∗, v〉 ≤ 0. (2.11)


Since F is locally C-Lipschitz at x, there exist L, δ ∈ (0, +∞) such that F is continuous on B(x, δ) and (2.7)holds. Hence there exists a sequences {(xn, tn)} in X × (0, +∞) convergent to (x, 0) such that

(y∗ ◦ F )◦(x, u) = limn→∞

〈y∗, F (xn + tnu)− F (xn)〉tn

. (2.12)

Since (xn, F (xn)) → (x, F (x)), (2.10) implies that there exists a sequence {(un, vn)} in X × Y convergent to(u, v) such that (xn, F (xn)) + tn(un, vn) ∈ epiC(F ) for all n ∈ N. Then there exists a sequence {cn} in C suchthat

F (xn) + tnvn = F (xn + tnun) + cn ∀n ∈ N,

namely, vn = F (xn+tnun)−F (xn)+cntn

for all n ∈ N, and so

〈y∗, vn〉 ≥〈y∗, F (xn + tnun)− F (xn)〉

tn∀n ∈ N. (2.13)

On the other hand, by (2.7), one has

〈y∗, F (xn + tnun)− F (xn)〉tn

≥ 〈y∗, F (xn + tnu)− F (xn)〉

tn− L‖u− un‖

for all sufficiently large n ∈ N. Since (un, vn)→ (u, v), this and (2.13) imply that

〈y∗, v〉 = limn→∞

〈y∗, vn〉 ≥ limn→∞

〈y∗, F (xn + tnu)− F (xn)〉tn

.

It follows from (2.12) that 〈y∗, v〉 ≥ (y∗ ◦ F )◦(x, u). Since x∗ ∈ ∂(y∗ ◦ F )(x), this verifies (2.11). The proof iscomplete.

The main results in this paper require that ordering cone C of Y has a nonempty interior. For convenience,we provide some notations and properties on the ordering cone C under the assumption int(C) 6= ∅. Giveny ∈ C, let

C+y := {y∗ ∈ C+ : 〈y∗, y〉 = 1}, (2.14)

ry := sup{r > 0 : B(y, r) ⊂ C} and γC := sup{ry : y ∈ C and ‖y‖ = 1}. (2.15)

It is clear that γC > 0 if and only if C has a nonempty interior.The following lemma plays an important role in the proofs of the main results and might be of independent

interest.

Lemma 2.6. Let y ∈ Y with ‖y‖ = 1 and r > 0 be such that B(y, r) ⊂ C, and let F : X → Y • be a functionsuch that its epigraph epiC(F ) is closed. Then the following statements hold.(i) C+

y ⊂ [1, 1r ]IC+ and IC+ ⊂ [r, 1]C+

y ; consequently

D∗eF (x)(C+y ) ⊂

[1,

1

r

]∂CF (x) and ∂CF (x) ⊂ [r, 1]D∗eF (x)(C+

y ) ∀x ∈ dom(F ).


(ii) C+y is a weak∗-compact convex set.

(iii) D∗eF (x)(C+y ) is a weak∗-closed convex set for all x ∈ dom(F ).

(iv) d(v,−C) ≤ max{〈y∗, v〉 : y∗ ∈ C+y } ≤ 1

rd(v,−C) for all v ∈ Y \ −C.

(v) If {y∗n} ⊂ C+ satisfies ‖y∗n‖ → 1 and y∗nw∗

−→ y∗, then ‖y∗‖ ≥ γC .

Proof. By the assumption that B(y, r) ⊂ C, one has

〈y∗, y〉 − r‖y∗‖ = inf{〈y∗, z〉 : z ∈ B(y, r)} ≥ 0 ∀y∗ ∈ C+.

Hence, r‖y∗‖ ≤ 〈y∗, y〉 ≤ ‖y∗‖ for all y∗ ∈ C+ (thanks to ‖y‖ = 1). This implies that (i) holds. By (2.14), wehave that C+

y is a weak∗-closed convex set. Since every weak∗-closed bounded set is weak∗-compact, it followsfrom (i) that (ii) holds. Noting that

gph(D∗eF (x)) = {(y∗, x∗) : (x∗,−y∗) ∈ N(epiC(F ), (x, F (x)))}

is a weak∗-closed convex cone, it is easy from (ii) to verify that D∗eF (x)(C+y ) is a weak∗-closed convex set. This

shows that (iii) holds.To prove (iv), let v ∈ Y \−C. Then, d(v,−C) > 0 and B(v, d(v,−C))∩−C = ∅. By the separation theorem,

there exists y∗0 ∈ Y ∗ with ‖y∗0‖ = 1 such that infz∈B(v,d(v,−C))

〈y∗0 , z〉 = supz∈−C

〈y∗0 , z〉 = 0. This implies that y∗0 ∈ IC+

and 〈y∗0 , v〉 − d(v,−C) = 0. By (i), there exists t ∈ [1, 1r ] such that ty∗0 ∈ C+

y . Hence

d(v,−C) ≤ 〈ty∗0 , v〉 ≤ max{〈y∗, v〉 : y∗ ∈ C+y }. (2.16)

Let y∗ ∈ C+y . Then, by (i),

〈y∗, v〉 ≤ 〈y∗, v + c〉 ≤ ‖y∗‖‖v + c‖ ≤ 1

r‖v + c‖ ∀c ∈ C

and so 〈y∗, v〉 ≤ 1rd(v,−C). This and (2.16) imply that (iv) holds.

To prove (v), take an arbitrary γ in (0, γC). By the definition of γC , there exists y0 ∈ C such that ‖y0‖ = 1and B(y0, γ) ⊂ C. Hence

〈v∗, y0〉 − γ‖v∗‖ = inf{〈v∗, z〉 : z ∈ B(y0, γ)} ≥ 0 ∀v∗ ∈ C+. (2.17)

Let {y∗n} ⊂ C+ be such that ‖y∗n‖ → 1 and y∗nw∗

−→ y∗. Then, by (2.17),

‖y∗‖ ≥ 〈y∗, y0〉 = limn→∞

〈y∗n, y0〉 ≥ limn→∞

γ‖y∗n‖ = γ.

Hence ‖y∗‖ ≥ γC . This shows (v). The proof is complete.

3. Quasi-subsmooth vector-valued function

The notion of the subsmoothness of a closed set, a useful extension of convexity and smoothness, was intro-duced by Aussel et al. [1] and then was further studied by some authors (see [29, 31] and references therein).Let A be a closed subset of a Banach space X, and recall that A is said to be subsmooth at x ∈ A if for anyε > 0 there exists δ > 0 such that

〈x∗, u− x〉 ≤ ε‖u− x‖ ∀u, x ∈ A ∩B(x, δ) and ∀x∗ ∈ N(A, x) ∩BX∗ . (3.1)


Let f : X → R ∪ {+∞} be a proper lower semicontinuous function. Motivated by (3.1) and the primal-lower-nice property introduced by Poliquin (cf. [23]), Zheng and Wei [31] considered the following notion ofquasi-subsmoothness: f is said to be quasi-subsmooth at x if for any ε,M ∈ (0, +∞) there exists δ > 0 suchthat

〈x∗, u− x〉 ≤ f(u)− f(x) + ε‖u− x‖ ∀u, x ∈ B(x, δ) and x∗ ∈ ∂f(x) ∩MBX∗ . (3.2)

In extending to the vector-valued case, we have the corresponding notions in Definitions 3.1 and 3.2, where oneconsiders F : X → Y • with Banach spaces X and Y and C ⊂ Y being a closed convex cone.

Definition 3.1. We say that F is quasi-subsmooth at x ∈ dom(F ) with respect to C if for any ε,M ∈ (0, +∞)there exists δ > 0 such that

〈x∗, u− x〉 ≤ 〈y∗, F (u)− F (x)〉+ ε‖u− x‖ (3.3)

whenever u, x ∈ B(x, δ), y∗ ∈ IC+ and x∗ ∈ D∗eF (x)(y∗) ∩MBX∗ .

It is routine to check that F is quasi-subsmooth at x with respect to C if epiC(F ) is subsmooth at (x, F (x))and F is locally Lipschitz at x.

Setting x = x in (3.3), we introduce the following weaker notion.

Definition 3.2. We say that F is w-quasi-subsmooth at x ∈ dom(F ) with respect to C if for any ε,M ∈(0, +∞) there exists δ > 0 such that

〈x∗, u− x〉 ≤ 〈y∗, F (u)− F (x)〉+ ε‖u− x‖ (3.4)

whenever u ∈ B(x, δ), y∗ ∈ IC+ and x∗ ∈ D∗eF (x)(y∗) ∩MBX∗ .

For convenience, let Υ (X,Y ;C) denote the family of all functions F : X → Y • such that epiC(F ) is closed,and let

Γ (X,Y ;C) := {F ∈ Υ (X,Y ;C) : F is C− convex}.

In the case when Y = R and C = R+, the closedness of epiC(F ) means that F is lower semicontinuous. Forx ∈ dom(F ), let

QSx(X,Y ;C) : = {F ∈ Υ (X,Y ;C) : F is quasi-subsmooth at x},QSwx (X,Y ;C) : = {F ∈ Υ (X,Y ;C) : F is w-quasi-subsmooth at x}.

It is clear that

Γ (X,Y ;C) ⊂ QSx(X,Y ;C) ⊂ QSwx (X,Y ;C). (3.5)

Since D∗eF (x)(y∗) ⊃ ∂(y∗ ◦ F )(x) trivially holds for all y∗ ∈ C+, it is easy to verify from the sublinearity ofD∗eF (x) and Definition 3.2 that[

F ∈ QSwx (X,Y ;C)]

=⇒[D∗eF (x)(y∗) = ∂(y∗ ◦ F )(x) for all y∗ ∈ C+ \ {0}

]. (3.6)

The following proposition provides a useful subclass of vector-valued quasi-subsmooth functions.


Proposition 3.3. Let F be a continuously differentiable function between Banach spaces X and Z and letG ∈ Υ (Z, Y ;C) be C-convex. Let x ∈ F−1(dom(G)) be such that the Robinson qualification (2.4) is satisfied.Then the following statements hold.(i) G ◦ F ∈ QSwx (X,Y ;C).(ii) If, in addition, F (x) ∈ int(dom(G)), then G ◦ F ∈ QSx(X,Y ;C).

Proof. From (2.4) and Lemma 2.4, it is easy to verify that

∂(y∗ ◦G ◦ F )(x) = D∗e(G ◦ F )(x)(y∗) = F ′(x)∗(∂(y∗ ◦G)(F (x))) ∀y∗ ∈ C+. (3.7)

To prove (i), let Λ : X × Y × R ⇒ Z × Y be such that

Λ(x, y, t) :=

{(F ′(x)(x), y) + t(epiC(G)− (F (x), y)), if (x, y, t) ∈ X × Y × R+

∅, otherwise

where y := G(F (x)). Then, Λ is a closed convex multifunction and Λ(X × Y × R) = Z × Y (thanks to (2.4)).It follows from the Robinson–Ursecu theorem that there exists r > 0 such that

r(BZ ×BY ) ⊂ Λ(BX ×BY ×BR)

= F ′(x)(BX)×BY + [0, 1](epiC(G)− (F (x), y))

= F ′(x)(BX)×BY + epiC(G)− (F (x), y) (3.8)

(the last equality holds because epiC(G) is a convex set containing (F (x), y)). Let ε,M ∈ (0, +∞). Since F iscontinuously differentiable, take δ > 0 such that

‖F (x)− F (x′)− F ′(x′)(x− x′)‖ ≤ rε

M + 1‖x− x′‖ ∀x, x′ ∈ B(x, δ). (3.9)

Let y∗ ∈ IC+ and x∗ ∈ D∗e(G ◦ F )(x)(y∗)∩MBX∗ = ∂(y∗ ◦G ◦ F )(x)∩MBX∗ (thanks to (3.7)). To show thatG ◦ F is w-quasi-subsmooth at x, we only need to show that

〈x∗, x− x〉 ≤ 〈y∗, G(F (x))−G(F (x))〉+ ε‖x− x‖ ∀x ∈ B(x, δ). (3.10)

To do this, we use (3.7) to express x∗ as x∗ := F ′(x)∗(z∗) with some z∗ ∈ ∂(y∗ ◦ G)(F (x)). We claim that‖z∗‖ ≤ M+1

r . Granting this and setting x′ = x in (3.9), one has

〈x∗, x− x〉 = 〈z∗, F ′(x)(x− x)〉 ≤ 〈z∗, F (x)− F (x)〉+ ε‖x− x‖ ∀x ∈ B(x, δ);

this implies that (3.10) holds (thanks to z∗ ∈ ∂(y∗ ◦ G)(F (x)) and the convexity of y∗ ◦ G). To prove theinequality claimed, we consider an arbitrary z in BZ . Then, by (3.8), there exist (u, y′) ∈ BX ×BY , z′ ∈ Z andc′ ∈ C such that (rz, 0) = (F ′(x)(u) + z′ − F (x), y′ +G(z′) + c′ − y). Hence,

〈z∗, rz〉 = 〈z∗, F ′(x)(u) + z′ − F (x)〉≤ 〈z∗, F ′(x)(u)〉+ (y∗ ◦G)(z′)− (y∗ ◦G)(F (x))

≤ 〈F ′(x)(z∗), u〉+ 〈y∗, G(z′) + c′ − y〉= 〈x∗, u〉 − 〈y∗, y′〉≤ M + 1,

and so ‖z∗‖ ≤ M+1r is proved.


To prove (ii), suppose that F (x) ∈ int(dom(G)). Since F is continuously differentiable, there exist δ′ > 0and 1 > r > 0 such that rBZ ⊂ int(dom(G)) − F (x) for all x ∈ BX(x, δ′). Hence, as in the proof of part(i), the Robinson–Ursescu theorem enables us to strengthen (3.8) by replacing x and y = G(F (x)) with anyx ∈ BX(x, δ′) and y = G(F (x)), respectively. This, together with (3.9) and (3.7), implies that (ii) holds. Theproof is completed.

It is well known that if F : X → R is locally Lipschitz at x then ∂F is bounded on a neighborhood of x. Wedo not know whether or not the corresponding conclusion holds for vector-valued functions. However, under thequasi-subsmoothness assumption, we have the following result.

Proposition 3.4. Let F ∈ Υ (X,Y ;C) be locally Lipschitz at x. Then the following statements hold:(i) If F ∈ QSx(X,Y ;C), there exist δ,M ∈ (0, +∞) such that

sup{‖x∗‖ : x∗ ∈ D∗eF (x)(y∗)} ≤M‖y∗‖ ∀x ∈ B(x, δ) and y∗ ∈ C+. (3.11)

(ii) If F ∈ QSwx (X,Y ;C), there exists M ∈ (0, +∞) such that

sup{‖x∗‖ : x∗ ∈ D∗eF (x)(y∗)} ≤M‖y∗‖ ∀y∗ ∈ C+.

Proof. Since F is locally Lipschitz at x, there exist L, r ∈ (0, +∞) such that

‖F (x1)− F (x2)‖ ≤ L‖x1 − x2‖ ∀x1, x2 ∈ B(x, r). (3.12)

First suppose that F ∈ QSx(X,Y ;C). Then there exists δ ∈ (0, r) such that

〈x∗, u− x〉 ≤ 〈y∗, F (u)− F (x)〉+ ‖u− x‖ (3.13)

for all u, x ∈ B(x, δ), y∗ ∈ IC+ and x∗ ∈ D∗eF (x)(y∗) ∩ (L+ 2)BX∗ . Since D∗eF (x) is sublinear, to prove (i), itsuffices to show that

sup{‖x∗‖ : x∗ ∈ D∗eF (x)(y∗)} ≤ L+ 2 ∀(x, y∗) ∈ B(x, δ)× IC+ .

To do this, suppose to the contrary that there exist x0 ∈ B(x, δ) ⊂ B(x, r), y∗0 ∈ IC+ and x∗0 ∈ D∗eF (x0)(y∗0)such that L + 2 < ‖x∗0‖. On the other hand, by (3.12) and Lemmas 2.1 and 2.5, there exists x∗1 ∈ ∂(y∗0 ◦F )(x0) ⊂ D∗eF (x0)(y∗0) such that ‖x∗1‖ ≤ L. Since D∗eF (x0)(y∗0) is convex, there exists t ∈ (0, 1) such that‖tx∗1 + (1− t)x∗0‖ = L+ 2 and tx∗1 + (1− t)x∗0 ∈ D∗eF (y∗0). It follows from (3.12) and (3.13) that

〈tx∗1 + (1− t)x∗0, u− x0〉 ≤ 〈y∗0 , F (u)− F (x0)〉+ ‖u− x0‖ ≤ (L+ 1)‖u− x0‖ ∀u ∈ B(x, δ),

and so ‖tx∗1 + (1− t)x∗0‖ ≤ L+ 1, contradicting the choice of t. This shows that (i) holds. Similarly, (ii) can beproved.

Corollary 3.5. Let F ∈ QSwx (X,Y ;C) be locally Lipschitz at x. Then,

N(epiC(F ), (x, F (x))) = N(epiC(F ), (x, F (x))).

Proof. Let (x∗,−y∗) ∈ N(epiC(F ), (x, F (x))). We only need to show that

(x∗,−y∗) ⊂ N(epiC(F ), (x, F (x))). (3.14)


Note that y∗ ∈ C+ (by Lem. 2.3) and x∗ ∈ D∗eF (x)(y∗). It follows from (3.6) that if y∗ 6= 0 then x∗ ∈ ∂(y∗ ◦F )(x) ⊂ D∗eF (x)(y∗), which implies that (3.14) holds. If y∗ = 0, then x∗ = 0 (see Prop. 3.4(ii)); so (3.14) alsoholds in this case. The proof is complete.

Next, in the case when the ordering cone C has a nonempty interior, we prove that the quasi-subsmoothnessand weak qausi-subsmoothness admit better properties. To do this, we need the following lemma.

Lemma 3.6. Let F ∈ Υ (X,Y ;C) be locally Lipschitz at x ∈ dom(F ) and let y ∈ int(C). Then the followingstatements hold.(i) F ∈ QSx(X,Y ;C) if and only if for any ε > 0 there exists δ > 0 such that (3.3) holds whenever u, x ∈ B(x, δ),y∗ ∈ C+

y and x∗ ∈ D∗eF (x)(y∗), where C+y is defined by (2.14).

(ii) F ∈ QSwx (X,Y ;C) if and only if for any ε > 0 there exists δ > 0 such that (3.4) holds whenever u ∈ B(x, δ),y∗ ∈ C+

y and x∗ ∈ D∗eF (x)(y∗).

Proof. We only need to show (i) (because the proof of (ii) is similar). Since y ∈ int(C), there exists r > 0such that B(y, r) ⊂ C. Hence, by Lemma 2.6(i), IC+ ⊂ [r, 1]C+

y . Since D∗eF (x) is positively homogeneous, thesufficient part holds trivially. To prove the necessity part, suppose that F ∈ QSx(X,Y ;C). Since F is locallyLipschitz at x, by Proposition 3.4(i), there exist M, δ ∈ (0, +∞) such that D∗eF (x)(IC+) ⊂ MBX∗ for allx ∈ B(x, δ), and hence for any ε > 0 there exists δ0 ∈ (0, δ) such that (3.3) holds whenever u, x ∈ B(x, δ0),y∗ ∈ IC+ and x∗ ∈ D∗eF (x)(y∗). Noting (by Lem. 2.6(i)) that C+

y ⊂ [1, 1r ]IC+ , one sees that the necessity part

holds. The proof is complete.

Let T be a compact topological space and C(X×T,R) denote the space of all continuous functions on X×T .For ψ ∈ C(X × T,R), recall that the family {ψ(·, t)}t∈T is locally Lipschitz at x if there exist L, γ ∈ (0, +∞)such that

|ψ(x1, t)− ψ(x2, t)| ≤ L‖x1 − x2‖ ∀x1, x2 ∈ B(x, γ) and ∀t ∈ T. (3.15)

Further recall (cf. [29]) that the family {ψ(·, t)}t∈T is uniformly quasi-subsmooth at x ∈ X if for any ε,M ∈(0, +∞) there exists δ > 0 such that

〈x∗, u− x〉 ≤ ψ(u, t)− ψ(x, t) + ε‖u− x‖ (3.16)

whenever (u, t), (x, t) ∈ B(x, δ)× T and x∗ ∈ ∂ψ(·, t)(x) ∩MBX∗ .Note that (3.15) implies ∂ψ(·, t)(x) ⊂ LBX∗ for all (x, t) ∈ B(x, δ)× T . Thus, in the case when {ψ(·, t)}t∈T

is locally Lipschitz at x, {ψ(·, t)}t∈T is uniformly quasi-subsmooth at x ∈ X if and only if for any ε ∈ (0, +∞)there exists δ > 0 such that (3.16) holds whenever (u, t), (x, t) ∈ B(x, δ)× T and x∗ ∈ ∂ψ(·, t)(x). It is known(cf., Thm. 3.1 from [29]) that if {ψ(·, t)}t∈T is uniformly quasi-subsmooth and locally Lipschitz at x then

∂Φ(x) = cow∗

⋃t∈T (x)

∂ψ(·, t)(x)

, (3.17)

where Φ(x) := maxt∈T

ψ(x, t) and T (x) = {t ∈ T : Φ(x) = ψ(x, t)}.The following proposition will play important roles in our later analysis.

Proposition 3.7. Let X,Y be Banach spaces with the ordering cone C of Y having a nonempty interior. Letx ∈ X and F ∈ QSx(X,Y ;C) be such that F is locally Lipschitz at x. Let y ∈ int(C) with ‖y‖ = 1, and defineψ : X × C+

y → R ∪ {+∞} such that

ψ(x, y∗) := 〈y∗, F (x)〉 ∀(x, y∗) ∈ X × C+y ,


where C+y is defined by (2.14). Then the family {ψ(·, y∗)}y∗∈C+

yis uniformly quasi-smooth at x and

∂φ(x) = ∂φ(x) = D∗eF (x)(Λ(x)), (3.18)

where φ(x) := maxy∗∈C+

y

ψ(x, y∗) for all x ∈ X and Λ(x) :={v∗ ∈ C+

y : 〈v∗, F (x)〉 = maxy∗∈C+

y

〈y∗, F (x)〉}

.

Proof. Since F is locally Lipschitz at x, it is easy to verify from Lemma 2.6(i) and Lemma 2.5 that thereexist L, γ ∈ (0, +∞) such that (3.15) holds with T = C+

y and ∂(y∗ ◦ F )(x) ⊂ D∗eF (x)(y∗) for all (x, y∗) ∈B(x, γ)× C+. From Lemma 3.6(i) and the assumption that F ∈ QSx(X,Y ;C), it follows that {ψ(·, y∗)}y∗∈C+

y

is uniformly quasi-smooth at x. Thus, by the Lipschitz property of F at x and ([29], Thm. 3.1), one has

∂φ(x) = cow∗( ⋃

y∗∈Λ(x)

∂(y∗ ◦ F )(x))

= cow∗( ⋃

y∗∈Λ(x)

∂(y∗ ◦ F )(x)). This, together with (3.5) and (3.6), implies

that

∂φ(x) = cow∗(D∗eF (x)(Λ(x))). (3.19)

By Lemma 2.6(i) and the definition of Λ(x), Λ(x) is a bounded weak∗-closed convex set and so is a weak∗-compact convex set. Since gph(D∗eF (x)) is a weak∗-closed convex cone, D∗eF (x)(Λ(x)) is a weak∗-closed convex

set. This and (3.19) imply that ∂φ(x) = D∗eF (x)(Λ(x)). Since ∂φ(x) is trivially contained in ∂φ(x), it sufficesto show (for establishing (3.18)) that

D∗eF (x)(Λ(x)) ⊂ ∂φ(x). (3.20)

To prove this, let x∗ ∈ D∗eF (x)(Λ(x)). Then there exists y∗ ∈ Λ(x) ⊂ C+y such that x∗ ∈ D∗eF (x)(y∗). Thus, by

Lemma 3.6(i), for any ε > 0 there exists δ > 0 such that

〈x∗, x− x〉 ≤ 〈y∗, F (x)− F (x)〉+ ε‖x− x‖ = 〈y∗, F (x)〉 − φ(x) + ε‖x− x‖ ≤ φ(x)− φ(x) + ε‖x− x‖

for all x ∈ B(x, δ) (the above equality holds because y∗ ∈ Λ(x)). Hence, x∗ ∈ φ(x). This shows that (3.20)holds.

The following proposition provides the sum rule for subsmoothness and plays a key role in the proof of themain result in Section 4.

Proposition 3.8. Let X,Y be Asplund spaces and the ordering cone C of Y have a nonempty interior. LetF1, F2 ∈ Υ (X,Y ;C) be locally Lipschitz at x. Then the following statements hold:(i) If F1, F2 ∈ QSwx (X,Y ;C), then

F1 + F2 ∈ QSwx (X,Y ;C). (3.21)

(ii) If there exists a neighborhood V of x such that F1, F2 ∈ QSx(X,Y ;C) ∩ QSwx (X,Y ;C) for all x ∈ V , thenF1 + F2 ∈ QSx(X,Y ;C).

Proof. Suppose that F1, F2 ∈ QSwx (X,Y ;C). Then, by Corollary 3.5, the three kinds of normal cones of epiC(Fi)to (x, Fi(x)) coincide:

N(epiC(Fi), (x, Fi(x))) = N(epiC(Fi), (x, Fi(x))) = N(epiC(Fi), (x, Fi(x))), i = 1, 2 (3.22)

(and all are weak∗-closed convex cones). We claim that

D∗e(F1 + F2)(x)(y∗) ⊂ D∗eF1(x)(y∗) +D∗eF2(x)(y∗) ∀y∗ ∈ C+. (3.23)


To do this, let K be the convex cone defined by

K :={

(x∗1 + x∗2, v∗) : (x∗i , v

∗) ∈ N(epiC(Fi), (x, Fi(x))), i = 1, 2}.

Then, by ([17], Cor. 3.11) and the Lipschitz assumption on Fi at x, one has

N(epiC(F1 + F2), (x, F1(x) + F2(x))) ⊂ K. (3.24)

This and Lemma 2.2(i) imply that N(epiC(F1 + F2), (x, F1(x) + F2(x))) ⊂ cow∗(K) = K

w∗

. Thus, in order toshow (3.23), by (3.22) and the definition of D∗eF (see (2.5)), we only need to show that K is weak∗-closed.In turn, by the Krein–Smulian theorem, it suffices to show that K ∩ (BX∗ × BY ∗) is a weak∗-closed set. Todo this, let (x∗,−v∗) be a weak∗ cluster point of K ∩ (BX∗ × BY ∗). Since BX∗ × BY ∗ is weak∗-compact, it issufficient to prove that (x∗,−v∗) ∈ K. Take a net {(x∗α,−v∗α)} in K ∩ (BX∗ ×BY ∗) converging to (x∗,−v∗) withrespect to the weak∗ topology. Hence, for each α there exist u∗α ∈ D∗eF1(x)(v∗α) and w∗α ∈ D∗eF2(x)(v∗α) suchthat x∗α = u∗α + w∗α. It follows that v∗α ∈ C+ and max{‖u∗α‖, ‖w∗α‖} ≤M‖v∗α‖ ≤M for all α and some constantM (thanks to Prop. 3.4(ii)). Now consider any ε > 0. Since F1, F2 ∈ QSwx (X,Y ;C), there exists δ > 0 such that

〈u∗α, x− x〉 ≤ 〈v∗α, F1(x)− F1(x)〉+ ε‖x− x‖ and 〈w∗α, x− x〉 ≤ 〈v∗α, F2(x)− F2(x)〉+ ε‖x− x‖

for all x ∈ B(x, δ). Without loss of generality, we can assume that u∗αw∗

−→ u∗ and w∗αw∗

−→ z∗ (taking a subnet if

necessary). Then, x∗α = u∗α + w∗αw∗

−→ u∗ + z∗ = x∗,

〈u∗, x− x〉 ≤ 〈v∗, F1(x)− F1(x)〉+ ε‖x− x‖ and 〈z∗, x− x〉 ≤ 〈v∗, F2(x)− F2(x)〉+ ε‖x− x‖

for all x ∈ B(x, δ). Hence

(u∗,−v∗) ∈ N(epiC(F1), (x, F1(x))) and (z∗,−v∗) ∈ N(epiC(F2), (x, F2(x)))

and so (x∗,−v∗) = (u∗,−v∗) + (z∗,−v∗) ∈ K. This shows that (3.23) holds. To prove (3.21), let y∗ ∈ IC+ andx∗ ∈ D∗e(F1 +F2)(x)(y∗); then (3.23) implies that there exist x∗i ∈ D∗eFi(x)(y∗) (i = 1, 2) such that x∗ = x∗1 +x∗2.From Proposition 3.4(ii) and the assumption that F1, F2 ∈ QSwx (X,Y ;C) are locally Lipschitz at x, it followsthat for any ε > 0 there exists δ′ > 0 (independent on y∗ and x∗) such that 〈x∗i , x− x〉 ≤ 〈y∗, Fi(x)− Fi(x)〉+ε‖x− x‖ for all x ∈ B(x, δ′), and so

〈x∗, x− x〉 = 〈x∗1, x− x〉+ 〈x∗2, x− x〉 ≤ 〈y∗, F1(x) + F2(x)− F1(x)− F2(x)〉+ 2ε‖x− x‖

for all x ∈ B(x, δ′). This implies that (3.21) holds and so (i) is shown. Similar to the proof of (3.21), (ii) isimmediate from (3.23) (with x being replaced by each x in some neighborhood V of x) and the definition of thequasi-subsmoothness. The proof is complete.

Remark. Inclusion (3.24), as a consequence of ([17], Cor. 3.11), plays a key role in the above proof, while theproof of ([17], Cor. 3.11) relies on the assumption that X and Y are Asplund spaces. We do not know whether(3.24) holds with the Clarke normal cone replacing the limiting normal cone when X and Y are Banach spaces.

Let F,G ∈ Υ (X,Y ;C) and x ∈ dom(F ) ∩ dom(G). We introduce the following notation

‖G− F‖x := lim supx→x

‖G(x)− F (x)− (G(x)− F (x))‖‖x− x‖

.


Clearly, ‖G− F‖x < +∞ if and only if there exist L, δ ∈ (0, +∞) such that

‖(G− F )(x)− (G− F )(x)‖ ≤ L‖x− x‖ ∀x ∈ B(x, δ),

which is weaker than the local Lipschitz property of G− F at x. In the case, when Y = R, ‖G− F‖x was usedin [12, 13, 20, 31].

Proposition 3.9. Let F ∈ Υ (X,Y ;C) and G ∈ QSwx (X,Y ;C) be such that ‖G− F‖x < +∞. Then

D∗eG(x)(y∗) ⊂ D∗eF (x)(y∗) + ‖y∗‖‖G− F‖xBX∗ ∀y∗ ∈ IC+ (3.25)

Consequently, ∂CG(x) ⊂ ∂CF (x) + ‖G− F‖xBX∗ and

d(0, ∂CF (x)) ≤ ‖u∗‖+ ‖G− F‖x ∀u∗ ∈ ∂CG(x).

Proof. Let y∗ ∈ IC+ and x∗ ∈ D∗eG(x)(y∗) and take an arbitrary ε > 0. Then, by G ∈ QSwx (X,Y ;C), thereexists δ > 0 such that

〈x∗, x− x〉 ≤ 〈y∗, G(x)−G(x)〉+ ε‖x− x‖ ∀x ∈ B(x, δ). (3.26)

Since ‖G− F‖x < +∞, there exists δ′ ∈ (0, δ) such that

‖G(x)−G(x)− (F (x)− F (x))‖ ≤ (‖G− F‖x + ε)‖x− x‖ ∀x ∈ B(x, δ′).

Hence, for any x ∈ B(x, δ′),

〈y∗, G(x)−G(x)〉 ≤ 〈y∗, F (x)− F (x)〉+ ‖y∗‖‖G(x)−G(x)− (F (x)− F (x))‖≤ 〈y∗, F (x)− F (x)〉+ ‖y∗‖(‖G− F‖x + ε)‖x− x‖.

This and (3.26) imply that

〈x∗, x− x〉 ≤ 〈y∗, F (x)− F (x)〉+ ‖y∗‖(‖G− F‖x + 2ε)‖x− x‖ ∀x ∈ B(x, δ′). (3.27)

Noting that y∗ ∈ C+, it follows that

〈x∗, x− x〉 ≤ 〈y∗, F (x) + c− F (x)〉+ ‖y∗‖(‖G− F‖x + 2ε)‖x− x‖ ∀(x, c) ∈ B(x, δ′)× C,

that is,

〈x∗, x− x〉 − 〈y∗, y − F (x)〉 ≤ δepiC(F )(x, y) + ‖y∗‖(‖G− F‖x + 2ε)‖x− x‖

for all (x, y) ∈ B(x, δ′)× Y . Hence, by Lemma 2.1,

(x∗,−y∗) ∈ ∂δepiC(F )(x, F (x)) + ‖y∗‖(‖G− F‖x + ε)BX∗ × {0}= N(epiC(F ), (x, F (x))) + ‖y∗‖(‖G− F‖x + ε)BX∗ × {0}.

This means that x∗ ∈ D∗eF (x)(y∗) + ‖y∗‖(‖G− F‖x + ε)BX∗ . Letting ε→ 0 and noting that D∗eF (x)(y∗) andBX∗ are respectively weak∗-closed and weak∗-compact, we have x∗ ∈ D∗eF (x)(y∗) + ‖y∗‖‖G − F‖xBX∗ . Thisshows that (3.25) holds. The proof is completed.


4. Main results

Let F ∈ Υ (X,Y ;C), that is, F : X → Y • be a function such that its epigraph epiC(F ) is closed. For x ∈dom(F ), recall that F has an error bound at x if and only if τ(F, x, C) < +∞, where the error bound modulusτ(F, x, C) is defined by (1.4). We say that F has a stable error bound at x if there exist η, δ ∈ (0, +∞) suchthat

τ(G, x, C) ≤ η ∀G ∈ Υ (X,Y ;C) with ‖G− F‖x < δ. (4.1)

We say that F has a stable strong error bound at x if there exist η, δ ∈ (0, +∞) such that (4.1) holds and xis an isolated point of S(G, x, C) for all G ∈ Υ (X,Y ;C) with ‖G− F‖x < δ, that is, for any ε > 0 there existsr ∈ (0, +∞) such that

‖x− x‖ ≤ (η + ε)d(G(x)−G(x),−C) for all x ∈ B(x, r).

In this section, in terms of subdifferential ∂CF , we will consider the stability of error bounds for a quasi-subsmooth function F . Note that 0 6∈ bd(∂CF (x)) if and only if one of the following two properties holds:

(i) 0 ∈ int(cl(∂CF (x))) and (ii) 0 6∈ cl(∂CF (x)).

It is easy to verify that

(i) =⇒ d(0,bd(∂CF (x))) = sup{r > 0 : rBX∗ ⊂ cl(∂CF (x))} > 0

and

(ii) =⇒ d(0,bd(∂CF (x))) = d(0, ∂CF (x)) > 0.

For case (i), we have the following result.

Theorem 4.1. Let F ∈ QSwx (X,Y ;C) and 0 ∈ int(cl(∂CF (x))). Suppose that G ∈ Υ (X,Y ;C) is such that‖G− F‖x < d(0,bd(∂CF (x))). Then, for any ε ∈ (0, +∞) there exists δ > 0 such that

‖x− x‖ ≤ 1 + ε

d(0,bd(∂CF (x)))− ‖G− F‖xd(G(x)−G(x),−C) ∀x ∈ B(x, δ). (4.2)

Consequently, F has a stable strong error bound at x.

Proof. Let r := d(0,bd(∂CF (x))). From the assumption that 0 ∈ int(cl(∂CF (x))), it follows that

r = sup{t > 0 : tBX∗ ⊂ cl(∂CF (x))

}> 0 and rBX∗ ⊂ cl(∂CF (x)). (4.3)

Let G ∈ Υ (X,Y ;C) with ‖G− F‖x < r. Given an ε ∈ (0, ∞), take η ∈(

0, r−‖G−F‖x2

)and δ1 > 0 such that


r − ‖G− F‖x1 + ε

< r − ‖G− F‖x − 2η (4.4)

and

‖G(x)−G(x)− (F (x)− F (x))‖ ≤ (‖G− F‖x + η)‖x− x‖ ∀x ∈ B(x, δ1).

Noting that 〈y∗, F (x) − F (x)〉 ≤ d(F (x) − F (x),−C) for all y∗ ∈ IC+ , since F ∈ QSwx (X,Y ;C), there existsδ ∈ (0, δ1) such that

〈x∗, x− x〉 ≤ d(F (x)− F (x),−C) + η‖x− x‖≤ d(G(x)−G(x),−C) + (‖G− F‖x + 2η)‖x− x‖

for any x∗ ∈ rBX∗ ∩ ∂CF (x) and x ∈ B(x, δ). It follows from the inclusion in (4.3) that

〈x∗, x− x〉 ≤ d(G(x)−G(x),−C) + (‖G− F‖x + 2η)‖x− x‖ ∀(x∗, x) ∈ rBX∗ ×B(x, δ).

This means that

r‖x− x‖ ≤ d(G(x)−G(x),−C) + (‖G− F‖x + 2η)‖x− x‖ ∀x ∈ B(x, δ),

that is, (r − ‖G− F‖x − 2η)‖x− x‖ ≤ d(G(x)−G(x),−C) for all x ∈ B(x, δ). This and (4.4) imply that (4.2)holds. The proof is complete.

Under some mild assumptions, we will prove that 0 ∈ int(cl(∂CF (x))) is also a necessary condition for F tohave a stable strong error bound at x.

Theorem 4.2. Suppose that the ordering cone C has a nonempty interior and that F is quasi-subsmooth atx ∈ dom(F ). Further suppose that F : X → Y is locally Lipschitz at x ∈ X. Then F has a stable strong errorbound at x if and only if 0 ∈ int(cl(∂CF (x))).

Proof. The sufficiency part is immediate from Theorem 4.1 (and (3.5)). Next suppose that F has a stable strongerror bound at x. Then there exist κ, δ ∈ (0, +∞) such that

κ‖x− x‖ ≤ d(F (x)− F (x),−C) ∀x ∈ B(x, δ),

and so F (x)−F (x) 6∈ −C for all x ∈ B(x, δ)\{x}. Fix y ∈ int(C) with ‖y‖ = 1, and let φ(x) := maxy∗∈C+

y

〈y∗, F (x)−

F (x)〉 for all x ∈ X. Thus, by Lemma 2.6(iv), one has

κ‖x− x‖ ≤ d(F (x)− F (x),−C) ≤ φ(x) = φ(x)− φ(x) ∀x ∈ B(x, δ).

It follows that

κBX∗ ⊂ ∂φ(x). (4.5)

Noting that F − F (x) is locally Lipschitz and quasi-subsmooth at x, one has, by Proposition 3.7 (applied to Fbeing replaced by F − F (x)), that

∂φ(x) = D∗e(F − F (x))(x)(C+y ) = D∗eF (x)(C+

y ).


It follows from (4.5) and Lemma 2.6(i) that κrBX∗ ⊂ ∂CF (x), and so 0 ∈ int(cl(∂CF (x))). The proof is complete.

Next we consider the case when 0 6∈ cl(∂CF (x)). To do this, we need the following lemma.

Lemma 4.3. Let G ∈ Υ (X,Y ;C), x ∈ dom(G) and let β ∈ (0, τ(G, x, C)). Then there exists a sequence{(xn, cn, ηn)} ⊂ X × C × (0, +∞) such that

G(xn) + cn 6= G(x) ∀n ∈ N, (4.6)

(xn, G(xn) + cn, ηn)→ (x, G(x), 0) (4.7)

and

0 ∈ ∂(f + δepiC(G) +

1

β‖ · −(xn, G(xn) + cn)‖n

)(xn, G(xn) + cn) ∀n ∈ N, (4.8)

where

f(x, y) := ‖y −G(x)‖ and ‖(x, y)‖n := ‖x‖+ ηn‖y‖ ∀(x, y) ∈ X × Y. (4.9)

Proof. Since τ(G, x, C) > β, there exists a sequence {xn} in X \ S(G, x, C) converging to x such that

Ln := d(xn, S(G, x, C)) > βd(G(xn)−G(x),−C) ∀n ∈ N. (4.10)

Hence, 0 < Ln ≤ ‖xn − x‖ → 0, and for each n ∈ N there exists cn ∈ C such that

‖G(xn) + cn −G(x)‖ < Lnβ. (4.11)

This implies that f(xn, G(xn) + cn) < inf{f(x, y) + δepiC(G)(x, y) : (x, y) ∈ X × Y

}+ Ln

β , where f is defined

by (4.9). Noting that f + δepiC(G) is lower semicontinuous, it follows from the Ekeland variational principle thatthere exists (xn, cn) ∈ X × C such that

‖(xn, G(xn) + cn)− (xn, G(xn) + cn)‖n < Ln

and

f(xn, G(xn) + cn) ≤ f(x, y) + δepiC(G)(x, y) +1

β‖(x, y)− (xn, G(xn) + cn)‖n ∀(x, y) ∈ X × Y,

where ‖ · ‖n is defined by (4.9) with ηn :=√Ln → 0. Thus, (4.8) holds,

‖xn − xn‖ < Ln → 0 and ‖G(xn) + cn −G(xn)− cn‖ <√Ln → 0.

By (4.10), (4.11) and x ∈ S(G, x, C), we have

xn → x, G(xn) + cn → G(x), xn 6∈ S(G, x, C)

(so G(xn) + cn 6= G(x)). The proof is complete.


Theorem 4.4. Let X,Y be Banach spaces and suppose that the ordering cone C of Y has a nonempty interior.Let F,G ∈ Υ (X,Y ;C) and x ∈ dom(F ) be such that

0 6∈ cl(∂CF (x)) and ‖G− F‖x < d(0, ∂CF (x)).

Further suppose that one of the following three conditions is satisfied:(i) G is continuous on some neighborhood of x and G ∈ QSx(X,Y ;C).(ii) G ∈ Γ (X,Y ;C).(iii) X,Y are Asplund spaces and G ∈ QSwx (X,Y ;C).Then

τ(G, x, C) ≤ 1

γC(d(0, ∂CF (x))− ‖G− F‖x

) .Proof. Suppose to the contrary that τ(G, x, C) > 1

γC

(d(0,∂CF (x))−‖G−F‖x

) . Then there exists r ∈ (0, γC)

sufficiently close to γC such that

τ(G, x, C) > τr :=1

r(d(0, ∂CF (x))− ‖G− F‖x

) .By Lemma 4.3, there exists {(xn, cn, ηn)} ⊂ X ×C × (0, +∞) such that (4.6), (4.7) and (4.8) hold with β = τr.By (4.8) and Lemma 2.1, one has

(0, 0) ∈ ∂(f + δepiC(G) +

1

τr‖ · −(xn, G(xn) + cn)‖n

)(xn, G(xn) + cn)

⊂ ∂f(xn, G(xn) + cn) + ∂δepiC(G)(xn, G(xn) + cn) +1

τr∂‖ · −(xn, G(xn) + cn)‖n(xn, G(xn) + cn)

⊂ {0} × SY ∗ +N(epiC(G), (xn, G(xn) + cn)) +1

τr(BX∗ × ηnBY ∗),

where f and ‖ · ‖n are defined by (4.9). Hence, there exist y∗n ∈ SY ∗ , v∗n ∈ BY ∗ and x∗n ∈ BX∗ such that

(x∗nτr,−y∗n −

ηnv∗n

τr

)∈ N(epiC(G), (x, G(xn) + cn)). (4.12)

First suppose that (i) holds. Then G is continuous on some neighborhood of x. By (4.7), G is continuous ateach xn for all sufficiently large n. It follows from Lemma 2.3(ii) that

N(epiC(G), (xn, G(xn) + cn)) ⊂ N(epiC(G), (xn, G(xn))) ⊂ X∗ ×−C+

for all sufficiently large n. By (4.12), one has y∗n +ηnv

∗n

τr∈ C+ and

x∗n

τr∈ D∗eG(xn)(y∗n +

ηnv∗n

τr) for all sufficiently

large n. Since ‖y∗n +ηnv

∗n

τr‖ → 1 and G is quasi-subsmooth at x, it follows that for any ε > 0 there exists δ′ > 0

such that

⟨ x∗nτr, x− xn

⟩≤⟨y∗n +

ηnv∗n

τr, G(x)−G(xn)

⟩+ ε‖x− xn‖


for all x ∈ B(x, δ′) and all sufficiently large n. Noting that ‖xn − x‖ → 0 and ‖G(xn)−G(x)‖ → 0, it followsthat

lim infn→∞

⟨ x∗nτr, x− x

⟩≤ lim inf

n→∞

⟨y∗n +

ηnv∗n

τr, G(x)−G(x)

⟩+ ε‖x− x‖ ∀x ∈ B(x, δ′). (4.13)

Without loss of generality, we may assume that x∗nw∗

−→ x∗ ∈ BX∗ and y∗n +ηnv

∗n

τr

w∗

−→ y∗ ∈ BY ∗ ∩ C+ (passingto a subnet if necessary). Hence, by Lemma 2.6(v) and (4.13), one has γC ≤ ‖y∗‖ and

⟨x∗τr, x− x

⟩≤⟨y∗, G(x)−G(x)

⟩+ ε‖x− x‖ ∀x ∈ B(x, δ′).

It follows that x∗

τr∈ ∂(y∗ ◦G)(x) ⊂ D∗eG(x)(y∗), and so

x∗

τr‖y∗‖∈ D∗eG(x)

( y∗

‖y∗‖

)⊂ D∗eG(x)(IC+) = ∂CG(x). (4.14)

Therefore, by Proposition 3.9,

d(0, ∂CF (x)) ≤ ‖x∗‖τr‖y∗‖

+ ‖G− F‖x ≤1

τrγC+ ‖G− F‖x <

1

τrr+ ‖G− F‖x (4.15)

(thanks to the choice of r). This implies that τr <1

r(d(0,∂CF (x))−‖G−F‖x) , contradicting the definition of τr.

Next suppose that (ii) holds. The convexity of epiC(G), together with (4.12) and Lemma 2.3(i), implies that

y∗n +ηnv

∗n

τr∈ C+ and

⟨x∗nτr, x− xn

⟩−⟨y∗n +

ηnv∗n

τr, y −G(xn)− cn

⟩≤ 0 ∀(x, y) ∈ epiC(G).

Let (x∗

τr,−y∗) be a weak∗-cluster point of

{(x∗n

τr,−y∗n −

ηnv∗n

τr

)}. Then, by Lemma 2.6(v) and (4.7), one has

γC ≤ ‖y∗‖ and ⟨x∗

τr, x− x

⟩− 〈y∗, y −G(x)〉 ≤ 0 ∀(x, y) ∈ epiC(G).

It follows from the convexity of epiC(G) and Lemma 2.3(i) that (x∗

τr,−y∗) ∈ N(epiC(G), (x, G(x))) and y∗ ∈ C+.

Hence (4.14) and (4.15) hold, which implies τr <1


Finally suppose that (iii) holds. Since X and Y are Asplund space, Lemma 2.2(iv) together with (4.6)–(4.8)implies that there exist {(an, bn)}, {(dn, en}), {(un, vn)} ⊂ X × Y converging to (x, G(x)) such that bn 6= G(x)and

(0, 0) ∈ ∂f(an, bn) + ∂δepiC(G)(un, vn) +1

τr∂‖ · −(xn, G(xn) + cn)‖n(dn, en) +

1

n(BX∗ ×BY ∗).

It follows that

(0, 0) ∈ {0} × SY ∗ + N(epiC(G), (un, vn)

)+

1

τr(BX∗ × ηnBY ∗) +

1

n(BX∗ ×BY ∗).


Hence, there exist y∗n ∈ SY ∗ , v∗n ∈ BY ∗ and x∗n ∈ BX∗ such that((1

τr+

1

n

)x∗n,−y∗n −

(ηnτr

+1

n

)v∗n

)∈ N(epiC(G), (un, vn)). (4.16)

This and Lemma 2.3(i) imply that

‖y∗n +

(ηnτr

+1

n

)v∗n‖ → 1 and y∗n +

(ηnτr

+1

n

)v∗n ∈ C+.

Without loss of generality, we may assume that x∗nw∗

−→ x∗ ∈ BX∗ and y∗n + (ηnτr + 1n )v∗n

w∗

−→ y∗ ∈ C+ (passingto a subsequence if necessary). Thus, by Lemma 2.6(v) and (4.16), one has

r < γC ≤ ‖y∗‖,(x∗τr,−y∗

)∈ N(epiC(G), (x, G(x))) ⊂ N(epiC(G), (x, G(x))),

and so x∗

τr‖y∗‖ ∈ D∗eG(x)

(y∗

‖y∗‖

)⊂ ∂CG(x). This and Proposition 3.9 imply that

d(0, ∂CF (x)) ≤ ‖x∗‖τr‖y∗‖

+ ‖G− F‖x <1

τrr+ ‖G− F‖x,

namely τr <1


By Theorems 4.1 and 4.4, under the stated assumptions, we know that d(0, ∂CF (x)) > 0 is not only asufficient condition for F to have a stable error bound at x but also yields the following quantitative estimatefor error bound modulus

τ(G, x, C) ≤ 1

γC(d(0, ∂CF (x))− ‖G− F‖x

) .This leads to us to consider sufficient and necessary conditions for d(0, ∂CF (x)) > 0. For convenience, we usethe Clarke tangent derivative DCF (x) : X ⇒ Y defined by

DCF (x)(u) := {(v ∈ Y : (u, v) ∈ T (epiC(F ), (x, F (x))} ∀u ∈ X.

Proposition 4.5. Let X,Y be Banach spaces and suppose that the ordering cone C of Y has a nonemptyinterior. Given F ∈ Υ (X,Y ;C) and x ∈ dom(F ), the following statements hold:(i) If the Clarke tangent derivative DCF (x) is surjective, that is, DCF (x)(X) = Y , then d(0, ∂CF (x)) > 0.(ii) If F is continuous at x, then d(0, ∂CF (x)) > 0 if and only if DCF (x)(X) = Y .

Proof. To prove (i), suppose that DCF (x)(X) = Y . Since gph(DCF (x)) = T (epiC(F ), (x, F (x))) is a closedconvex cone, the Robinson–Ursescu theorem implies that there exists r > 0 such that rBY ⊂ DCF (x)(BX). Weclaim that

d(0, ∂CF (x)) ≥ r. (4.17)

Let x∗ be an arbitrary element in ∂CF (x) with y∗ ∈ IC+ such that x∗ ∈ D∗eF (x)(y∗). Then, for any ε ∈ (0, 1)there exist yε ∈ BY and uε ∈ BX such that

〈y∗,−yε〉 > ‖y∗‖ − ε = 1− ε and 〈x∗, uε〉 − 〈y∗, ryε〉 ≤ 0


(because ryε ∈ rBY ⊂ DCF (x)(BX)); consequently ‖x∗‖ ≥ r(1− ε) and so ‖x∗‖ ≥ r as ε is arbitrary in (0, 1).This shows that (4.17) holds.

To prove (ii), suppose that F is continuous at x. Since the sufficiency part is just (i), it suffices to show thenecessity part. To prove this, suppose that d(0, ∂CF (x)) > 0. We claim that

cl(DCF (x)(X)) = Y. (4.18)

Indeed, if this is not the case, cl(DCF (x)(X)) 6= Y . Since cl(DCF (x)(X)) is a closed convex cone in Y , it followsfrom the separation theorem that there exists v∗ ∈ Y ∗ with ‖v∗‖ = 1 such that sup{〈v∗, y〉 : y ∈ DCF (x)(X)} =0. Noting that DCF (x)(X) = DCF (x)(X) + C (thanks to Lemma 2.3(ii)), it follows that y∗ = −v∗ ∈ IC+ and(0,−y∗) ∈ N(epiC(F ), (x, F (x))) and so 0 ∈ D∗eF (x)(y∗) ⊂ ∂CF (x), contradicting d(0, ∂CF (x)) > 0. This showsthat (4.18) holds. Take an element c in int(C) and δ > 0 such that B(c, δ) ⊂ C. It follows from (4.18) that thereexists y ∈ DCF (x)(X) ∩ −B(c, δ2 ). Hence

B(

0,δ

2

)⊂ y +B(c, δ) ⊂ DCF (x)(X) + C = DCF (x)(X).

Since DCF (x)(X) is a cone, this implies that DCF (x)(X) = Y . The proof is complete.

From Theorem 4.4 and Proposition 4.5, we have the following corollary.

Corollary 4.6. Let X,Y be Banach spaces and suppose that the ordering cone C of Y has a nonempty interior.Let F ∈ Υ (X,Y ;C) and x ∈ dom(F ) be such that the sublinear conic inequality DCF (x)(u) <C 0 is solvablein the sense that there exists u0 ∈ X such that DCF (x)(u0) ∩ −int(C) 6= ∅. Further suppose that one of thefollowing three conditions is satisfied:(i) G is continuous on some neighborhood of x and G ∈ QSx(X,Y ;C).(ii) G ∈ Γ (X,Y ;C).(iii) X,Y are Asplund spaces and G ∈ QSwx (X,Y ;C).Then

τ(G, x;C) ≤ 1

γC[d(0, ∂CF (x))− ‖G− F‖x

]+

.

Remark. Let F : X → Y • be smooth around x ∈ dom(F ). It is easy to verify that DCF (x)(u) = F ′(x)(u) +Cfor all u ∈ X, where F ′(x) denotes the derivative of F at x, and so DCF (x)(u) <C 0 is solvable if and only ifF ′(x)(X) + C = Y . Thus, as a byproduct of Corollary 4.6, we have the following result: if F is smooth aroundx and F ′(x)(X) + C = Y then F has a stable error bound at x with respect to C. Clearly, F ′(x)(X) + C = Yis weaker than F ′(x)(X) = Y , the later condition being an important assumption for the famous Lyusternik–Graves theorem: if F is smooth around x and F ′(x)(X) = Y then F is metrically regular at (x, F (x)), namelythere exist τ, δ ∈ (0, +∞) such that

d(x, F−1(y)) ≤ τ‖y − F (x)‖ ∀(x, y) ∈ B(x, δ)×B(F (x), δ)).

Let L(X,Y ) denote the space of all bounded linear operators from X to Y . Recall that T ∈ L(X,Y ) is saidto satisfy the Slater condition with respect to C if

T (X) ∩ −int(C) 6= ∅ (equivalently, T (X) + C = Y ). (4.19)

By the Robinson–Ursescu theorem, the above Slater condition means

r(T,C) := sup{r ≥ 0 : rBY ⊂ T (BX) + C} > 0.


In the case when X = Rm, Y = Rn and C = Rn+, under the Slater condition, Luo and Tseng [16] consideredthe stability of error bounds for a conic linear inequality defined by a bounded linear operator T and C = Rn+.Under the Slater condition, with the help of r(T,C), we provide the following quantitative estimate of localerror bound moduli of conic linear inequalities:

τ(T , x, C) ≤ 1

γC(r(T,C)− ‖T − T‖)(4.20)

for all (T , x) ∈ L(X,Y ) × X with ‖T − T‖ < r(T,C). Indeed, noting that epiC(T ) = {(x, Tx + c) : (x, c) ∈X × C} is a closed convex cone, it is easy to verify that

T (epiC(T ), (x, Tx)) ⊃ epiC(T ) ∀x ∈ X.

Hence, by the definition of r(T,C),

rBY ⊂ T (BX) + C ⊂ DCT (x)(BX) ∀(x, r) ∈ X × (0, r(T,C)).

This and (4.17) imply that d(0, ∂CT (x)) ≥ r(T,C) for all x ∈ X. It follows from Corollary 4.6 that (4.20) holdsfor all (T , x) ∈ L(X,Y )×X with ‖T − T‖ < r(T,C).

Theorems 4.1 and 4.4 show that, in a sense, 0 6∈ bd(∂CF (x)) is a sufficient condition for (CIE) to have a stablelocal error bound at x. The following theorem shows that the converse also holds under some mild assumption.

Theorem 4.7. Let X,Y be Asplund spaces with the ordering cone C of Y having a nonempty interior. Supposethat F ∈ Υ (X,Y ;C) is quasi-subsmooth and locally Lipschitz on a neighborhood of x ∈ dom(F ). Then thefollowing statements are equivalent:

(i) 0 6∈ bd(∂CF (x)).(ii) There exist L, δ ∈ (0, +∞) such that τ(G, x, C) ≤ L whenever G ∈ QSwx (X,Y ;C) and ‖G− F‖x < δ.(iii) For any y ∈ int(C) with ‖y‖ = 1 there exist L, δ ∈ (0, +∞) such that

τ(F + ε(‖ · −x‖+ 〈x∗, ·〉)y, x, C

)≤ L ∀(ε, x∗) ∈ [0, δ)×BX∗ . (4.21)

(iv) There exist y ∈ int(C) with ‖y‖ = 1 and L, δ ∈ (0, +∞) such that (4.21) holds.

Proof. (i)⇒(ii) is immediate from Theorems 4.1 and 4.4. By Proposition 3.8, one has

F + ε(‖ · −x‖+ x∗)y ∈ QSwx (X,Y ;C) ∀(ε, y, x∗) ∈ (0, +∞)× int(C)×BX∗ .

Therefore, (ii)⇒(iii) holds. Since (iii)⇒(iv) is trivial, it remains to show (iv)⇒(i). To do this, suppose to thecontrary that 0 ∈ bd(∂CF (x)), that is,

0 ∈ cl(∂CF (x)) and 0 6∈ int(cl(∂CF (x))). (4.22)

Hence

εBX∗ 6⊂ ∂CF (x) ∀ε ∈ (0, +∞). (4.23)

By (iv), there exist y ∈ int(C) with ‖y‖ = 1 and L, δ, r ∈ (0, +∞) such that (4.21) holds and y + rBY ⊂ C. Itfollows from Lemma 2.6(i) that

1 ≤ ‖y∗‖ ≤ 1

r∀y∗ ∈ C+

y . (4.24)


We claim that

εBX∗ 6⊂ D∗eF (x)(C+y ) ∀ε ∈ (0, +∞). (4.25)

Indeed, if not, namely there exists ε0 > 0 such that

ε0BX∗ ⊂ D∗eF (x)(C+y ), (4.26)

then

rε0BX∗ ⊂ ∂CF (x) (4.27)

(contradicting (4.23)). To see how (4.27) follows from (4.26), let x∗ ∈ BX∗ . Then (4.26) implies that thereexist y∗0 , y

∗ ∈ C+y such that 0 ∈ D∗eF (x)(y∗0) and ε0x

∗ ∈ D∗eF (x)(y∗). Noting that, since D∗eF (x) is a convexmultifunction, it follows that

tε0x∗ = (1− t)0 + tε0x

∗ ⊂ D∗eF (x)((1− t)y∗0 + ty∗) ∀t ∈ [0, 1],

and so

tε0x∗

‖(1− t)y∗0 + ty∗‖∈ D∗eF (x)

((1− t)y∗0 + ty∗

‖(1− t)y∗0 + ty∗‖

)⊂ ∂CF (x) ∀t ∈ [0, 1]. (4.28)

Since the continuous function t 7→ t‖(1−t)y∗0+ty∗‖ takes value 1

‖y∗‖ at t = 1, there exists t0 ∈ (0, 1] such thatt0

‖(1−t0)y∗0+t0y∗‖ = r (thanks to (4.24)). This and (4.28) imply that rε0x∗ ∈ ∂CF (x), showing (4.27). Therefore

our claim (4.25) is valid. For each x ∈ dom(F ), let Φ(x) and Λ(x) be defined by

Φ(x) := maxy∗∈C+

y

〈y∗, F (x)− F (x)〉 and Λ(x) := {y∗ ∈ C+y : 〈y∗, F (x)− F (x)〉 = Φ(x)}. (4.29)

By assumption, we take δ′ ∈ (0, δ) such that F (and F − F (x)) are quasi-subsmooth and Lipschitz on B(x, δ′).From Proposition 3.7, it follows that

∂Φ(x) = ∂Φ(x) = D∗eF (x)(Λ(x)) ∀x ∈ B(x, δ′) (4.30)

and, in particular, cl(∂Φ(x)) = ∂Φ(x) = D∗eF (x)(C+y ) (because Λ(x) = C+

y ). Hence, by (4.22) and (4.25), one

has 0 ∈ ∂Φ(x) and εBX∗ 6⊂ ∂Φ(x) for all ε ∈ (0, +∞). Let ε ∈ (0, δ) be such that (2 + 1r )ε < 1

4L , and take

u∗ε ∈ εBX∗ \ ∂Φ(x); then there exist η ∈ (0, δ′) and a sequence {xn} in X converging to x such that

0 ≤ Φ(x)− Φ(x) + ε‖x− x‖ = Φ(x) + ε‖x− x‖ ∀x ∈ B[x, η], (4.31)

〈u∗ε, xn − x〉 > Φ(xn)− Φ(x), and so ε‖xn − x‖ > Φ(xn) for all n ∈ N. Hence

Φ(xn) + ε‖xn − x‖ < inf{Φ(x) + ε‖x− x‖ : x ∈ B[x, η]}+ 2ε‖xn − x‖ ∀n ∈ N.

By the Ekeland variational principle, for any n ∈ N there exists an ∈ B[x, η] such that ‖an − xn‖ < ‖xn − x‖and

Φ(an) + ε‖an − x‖ ≤ Φ(x) + ε‖x− x‖+ 2ε‖x− an‖ ∀x ∈ B[x, η]. (4.32)


Therefore, an 6= x for all n ∈ N and an → x (as xn → x). Without loss of generality we assume that an ∈ B(x, η)for all n ∈ N. Thus, by (4.32), one has

0 ∈ ∂(Φ + ε‖ · −x‖+ 2ε‖ · −an‖)(an) ⊂ ∂Φ(an) + 3εBX∗ ∀n ∈ N. (4.33)

By the uniform boundedness principle in functional analysis (cf., Thm. 1.6.9 from [18]), there exists x∗ ∈ BX∗

such that sup{∣∣∣⟨x∗, n(an−x)

‖an−x‖

⟩∣∣∣ : n ∈ N}

= +∞. Without loss of generality, we assume that 〈x∗, an − x〉 > 0

for all n ∈ N (taking a subsequence of {an} and replacing x∗ with −x∗ if necessary). It follows from (4.31) that

0 < Φ(an) + ε‖an − x‖+ ε〈x∗, an − x〉 ∀n ∈ N. (4.34)

Take y∗n ∈ Λ(an) (see (4.29)). Then 〈y∗n, y〉 = 1, Φ(an) = 〈y∗n, F (an)−F (x)〉, and hence we can rewrite (4.34) as

0 < 〈y∗n, F (an)− F (x) + ε(‖an − x‖+ 〈x∗, an − x〉)y〉 = 〈y∗n, G(an)−G(x)〉,

where G := F + ε(‖ · −x‖+ x∗)y. It follows that

G(an)−G(x) 6∈ −C. (4.35)

By (4.21), there exists δ > 0 such that

d(x, S(G, x, C)) ≤ 2Ld(G(x)−G(x),−C) ∀x ∈ B(x, δ). (4.36)

Take a sequence {un} in S(G, x, C) such that

‖an − un‖ ≤(

1 +1

n

)d(an, S(G, x, C)) ≤

(1 +

1

n

)‖an − x‖ → 0. (4.37)

Noting that G is quasi-subsmooth and Lipschitz at x (thanks to Prop. 3.8(ii)), it follows fromLemma 3.6(i) that we can assume without loss of generality that

〈a∗, un − an〉 ≤ 〈y∗, G(un)−G(an)〉+ ε‖un − an‖ ∀y∗ ∈ C+y and a∗ ∈ D∗eG(an)(y∗).

Since G(un) ≤C G(x), this implies that

〈a∗, un − an〉 ≤ 〈y∗, G(x)−G(an)〉+ ε‖un − an‖ ∀y∗ ∈ C+y and a∗ ∈ D∗eG(an)(y∗).

Hence

(‖a∗‖+ ε)‖un − an‖ ≥ 〈y∗, G(an)−G(x)〉 ∀y∗ ∈ C+y and a∗ ∈ D∗eG(an)(y∗). (4.38)

From (4.29) and the definition of G, it is easy to verify that

Λ(an) ={v∗ ∈ C+

y : 〈v∗, G(an)−G(x)〉 = maxy∗∈C+

y

〈y∗, G(an)−G(x)〉},

and so 〈y∗, G(an)−G(x)〉 ≥ d(G(an)−G(x),−C) for all y∗ ∈ Λ(an) (thanks to Lemma 2.6(iv) and (4.35)). Itfollows from (4.38) that

(‖a∗‖+ ε)‖un − an‖ ≥ d(G(an)−G(x),−C) ∀a∗ ∈ D∗eG(an)(Λ(an)).


Thus, by (4.36), one has

d(an, S(G, x, C)) ≤ 2L(‖a∗‖+ ε)‖un − an‖ ∀a∗ ∈ D∗eG(an)(Λ(an)).

This, together with (4.37) and (4.35), implies that n1+n ≤ 2L(‖a∗‖ + ε) for all a∗ ∈ D∗eG(an)(Λ(an)), namely

n2L(1+n) − ε ≤ d(0, D∗eG(Λ(an))). It follows from Proposition 3.9 and (4.24) that

n

2L(1 + n)− ε ≤ d(0, D∗eF (an)(Λ(an))) +

1

r‖G− F‖an ≤ d(0, D∗eF (an)(Λ(an))) +

2ε

r. (4.39)

By (4.33) and (4.30), one has 0 ∈ ∂Φ(an) + 3εBX∗ = D∗eF (an)(Λ(an)) + 3εBX∗ , and so d(0, D∗eF (an)(Λ(an))) ≤3ε. This and (4.39) imply that 1

2L ≤ (4 + 2r )ε, contradicting the choice of ε. The proof is complete.

It is worth mentioning that the perturbations of F appearing in (iii) and (iv) of Theorem 4.7 are all in thesame direction as y, which are different from the perturbations of F in (ii) of Theorem 4.7. In the special casewhen Y = R and C = R+, Theorems 4.1, 4.4 and 4.7 reduce to the corresponding results in [31].

In the convexity case, the Asplund space and Lipschitz assumptions of Theorem 4.7 can be relaxed; moreover,linear perturbation functions can be considered.

Theorem 4.8. Let X,Y be Banach spaces with the ordering cone C of Y having a nonempty interior. LetF ∈ Γ (X,Y ;C) = {H ∈ Υ (X,Y ;C) : epiC(H) is convex} and x ∈ dom(F ) be such that F is bounded abovewith respect to C at x, namely there exist γ,M ∈ (0, +∞) such that

F (B(x, γ)) ⊂MBY − C. (4.40)

Then the following statements are equivalent:(i) 0 6∈ bd(∂CF (x)).(ii) There exist L, δ ∈ (0, +∞) such that τ(G, x, C) ≤ L for all G ∈ Γ (X,Y ;C) with ‖G− F‖x < δ.(iii) There exist y ∈ int(C) with ‖y‖ = 1 and L, δ ∈ (0, +∞) such that

τ(F + ε〈x∗, ·〉y, x, C

)≤ L ∀(ε, x∗) ∈ [0, δ)×BX∗ . (4.41)

Proof. By the C-convexity of F , one has F ∈ QSwx (X,Y, ;C). Hence (i)⇒(ii) is immediate from Theorems 4.1and 4.4. Let y ∈ Y and x∗ ∈ X∗. From the C-convexity of F , it is easy to verify that F + ε〈x∗, ·〉y is C-convexand

∥∥F + ε〈x∗, ·〉y − F∥∥x≤ ε‖x∗‖‖y‖ for all ε > 0. Therefore, (ii)⇒(iii) is true. It remains to show (iii)⇒(i).

To do this, suppose to the contrary that (iii) holds but (i) does not hold. Then, (4.22) holds, and similar to thecorresponding part of the proof of Theorem 4.7, there exist y ∈ int(C) with ‖y‖ = 1 and L, δ, r ∈ (0, +∞) suchthat (4.41), (4.24) and (4.25) hold. Let Φ(x) and Λ(x) be as in (4.29). Then, Φ is convex. By (4.40) and (4.24),one has 〈y∗, F (x) − F (x)〉 ≤ 2M

r for all (y∗, x) ∈ C+y × B(x, γ). Since each function x 7→ 〈y∗, F (x) − F (x)〉 is

convex, there exists L′ ∈ (0, +∞) such that

|〈y∗, F (x1)− F (x)〉 − 〈y∗, F (x2)− F (x)〉| ≤ L′‖x1 − x2‖ ∀y∗ ∈ C+y and ∀x1, x2 ∈ B

(x,γ

2

).

Noting that C+y is weak∗-compact, Ioffe–Tikhomirov theorem (cf. Thm. 2 from [10]) implies that

∂Φ(x) = cow∗

⋃y∗∈C+

y

∂(y∗ ◦ F )(x)

= cow∗(D∗eF (x)(C+

y ))

= D∗eF (x)(C+y ),


where the last equality holds by Lemma 2.6(iii). Thus, by (4.22) and (4.25), one has

0 ∈ ∂Φ(x) and εBX∗ 6⊂ ∂Φ(x) ∀ε ∈ (0, +∞).

Hence

0 ≤ Φ(x)− Φ(x) = Φ(x) ∀x ∈ X. (4.42)

Let ε ∈ (0, δ) be such that 12L > 2ε+ ε

r , and take u∗ε ∈ εBX∗ \ ∂Φ(x). Then there exists a sequence {xn} in Xconverging to x such that 〈u∗ε, xn − x〉 > Φ(xn)− Φ(x) = Φ(xn) for all n ∈ N. Hence

Φ(xn) < infx∈X

Φ(x) + 〈u∗ε, xn − x〉 ≤ infx∈X

Φ(x) + ε‖xn − x‖ ∀n ∈ N.

By the Ekeland variational principle, for any n ∈ N there exists an ∈ X such that ‖an − xn‖ < ‖xn − x‖ andΦ(an) ≤ Φ(x) + ε‖x− an‖ for all x ∈ X. Therefore, an 6= x, an → x and

0 ∈ ∂(Φ + ε‖ · −an‖)(an) ⊂ ∂Φ(an) + εBX∗ ∀n ∈ N. (4.43)

Without loss of generality we assume that there exist x∗ ∈ BX∗ such that 〈x∗, an − x〉 > 0 for all n ∈ N (takinga subsequence of {an} if necessary). It follows from (4.42) that

0 < Φ(an) + ε〈x∗, an − x〉 ∀n ∈ N. (4.44)

On the other hand, by (4.41), there exists δ′ > 0 such that

d(x, S(G, x, C)) ≤ 2Ld(G(x)−G(x),−C) ∀x ∈ B(x, δ′), (4.45)

where G := F + ε〈x∗, ·〉y. Similar to the corresponding part of the proof of Theorem 4.7, we have

n

2L(1 + n)− ε ≤ d(0, D∗eF (an)(Λ(an))) +

1

r‖G− F‖an ≤ d(0, D∗eF (an)(Λ(an))) +

ε

r. (4.46)

Since the family {〈y∗, F − F (x)〉 : y∗ ∈ C+y } is convex and Lipschitz on B(x, γ2 ), one has ∂Φ(an) ⊂

D∗eF (an)(Λ(an)). It follows from (4.43) that 0 ∈ D∗eF (an)(Λ(an)) + εBX∗ . This and (4.46) imply thatn

2L(1+n) − ε ≤ ε+ εr , and so 1

2L ≤ 2ε+ εr , contradicting the choice of ε.

In the special case when Y = R, C = R+ and F is convex, the stability of the local error bound for F atx was characterized in terms of 0 6∈ ∂F (x) (as in Thm. 4.8) in [13, 20]. It is easy to verify that some resultsin [13, 20] can be recaptured by Theorems 4.1, 4.4 and 4.8. For example, in the case when Y = R, C = R+

and F is convex, Theorem 4.4 recaptures ([13], Thm. 8(i)), and Theorem 4.8 extends ([13], Thm. 8(iii)) to theinfinite-dimensional case from the finite-dimensional case.

Imitating the corresponding notion in a very recent paper [12] by Kruger et al., one can adopt the followingradii of error bound for a C-convex vector-valued functions F at x with respect to C:

Rc(F, x, C) := inf{ε > 0 : τPtbc(F, x, C, ε) = +∞} and Rl(F, x, C) := inf{ε > 0 : τPtbl

(F, x, C, ε) = +∞},

where

τPtbc(F, x, C, ε) := sup{τ(G, x, C) : G ∈ F + Γ (X,Y ;C) with ‖G− F‖x ≤ ε}


and

τPtbl(F, x, C, ε) := sup{τ(F + T, x, C) : T : X → Y is continuous linear operator with ‖T‖ ≤ ε}.

In the case when Y = R and C = R+, it was proved in [12] that

d(0,bd(∂CF (x))) = Rc(F, x, C) = Rl(F, x, C),

which implies the following stability

τ(G, x, C) < +∞ ∀G ∈ F + Γ (X,Y ;C) with ‖G− F‖x < Rc(F, x, C).

Replacing τ(G, x, C) < +∞, Theorems 4.1 and 4.4 provide a better estimate for error bound modulus τ(G, x, C):

τ(G, x, C) ≤ 1

γC(d(0,bd(∂CF (x)))− ‖G− F‖x

) ∀G ∈ QSwx (X,Y ;C) with ‖G− F‖x < d(0,bd(∂CF (x))).

From Theorems 4.1 and 4.4, it is clear that

d(0,bd(∂CF (x))) ≤ Rl(F, x, C) ≤ Rc(F, x, C).

Moreover, in the convex case, it is easy to verify from Theorem 4.8 that

γCRl(F, x, C) ≤ d(0,bd(∂CF (x))).

Therefore, in the special case when Y = Rn and C = Rn+, d(0,bd(∂CF (x))) = Rl(F, x, C) (because γC = 1).

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xi yin zheng and kung-fu ng4 x.y. zheng and k.-f. ng or equivalently @f^ (x) = n x 2x : (x; 1)...

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