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Error bound characterizations of Guignard’s constraint qualification in convex programming
A Barbara, A Jourani
To cite this version:
A Barbara, A Jourani. Error bound characterizations of Guignard’s constraint qualification in convex programming. 2021. �hal-03225245�
Error bound characterizations of Guignard’s constraint qualification in convex programming
*A. Barbara, A. Jourani
Institut de Math´ematiques de Bourgogne UMR 5584 CNRS
Universit´e Bourgogne Franche-Comt´e F-21000 Dijon, France
Abstract. This paper deals with error bound characterizations of Guignard’s qualification condition for a convex inequality system in a Banach space X. We establish necessary and sufficient conditions for a closed convex setS defined by a convex function g to have Guignard’s condition. These conditions are expressed in terms of the notion of error bound. Our results show that these characterizations hlod in the following special cases:
1. g is the maximum of a finite number of differentiable convex functions.
2. S is closed convex and polyhedral.
3. The dimension of the subspace lin(S) is less than 2 andg is positively homogeneous.
We construct technical examples showing that these characterizations are limited to the three situations above.
We introduce a new condition in terms of the gauge function which allows us to give an error bound character- ization of convex nondifferentiable systems and to obtain as a direct consequence different characterizations of the concept of strong conical hull intersection property (CHIP) for a finite collection of convex sets.
Keywords. Subdifferential, Normal cone, Guignard’s condition, SCHIP property.
MSC. 49J52, 49J53, 90C31, 90C25
1 Introduction
Necessary optimality conditions are known to be very important in optimization in the computation of (possible) local or global minima. To be more concrete, consider the following optimization problem
ß minf(x)
g(x)≤0 (1)
Here f, g : X 7→R∪ {+∞} are exented lower semicontinous convex functions and X is a Banach space. We know that for a feasible point for problem (14), that isg(¯x)≤0, then the following assertions are equivalent :
1. ¯xis a solution of (14), 2. 0∈∂f(¯x) +N(S,¯x)
provided thatf is Lipschitz continous around ¯x. Where
S:={x∈X : g(x)≤0}
∂f(¯x) is the Fenchel subdifferential off at ¯xandN(S,x) is the normal cone in the sense of convex analysis to¯ S at ¯x. The problem is
How to computeN(S,x) in terms of the data¯ g?
Without additional constraint qualification, there is no way to obtain this computation (take g(x) =x2). So we are looking for conditions which allow us to get this computation. One of them is known to be Abadie’s
*This work was partially supported by the EIPHI Graduate School (contract ANR-17-EURE-0002)
constraint qualification expressed as follows :
T(S,x) =¯ {h∈X: g0(¯x, h)≤0} (2)
where T(S,x) denotes the tangent cone of¯ S at ¯x, that is, the negative polar of the normal cone N(S,x), and¯ whereg0(¯x, h) is the directional derivative ofgat ¯xin the directionh, that is,g0(¯x, h) = lim
s→0+
g(¯x+th)−g(¯x)
t .
Using a separation theorem, we may easily obtain the equivalence between the two following assertions:
1. Abadie’s constraint qualification holds at ¯x, withg(¯x) = 0, 2. The following property holds at ¯x, withg(¯x) = 0,
N(S,x) = cl¯ ∗[R+∂g(¯x)]. (3)
So that one of these conditions, guarantees the following characterization for a feasible point ¯xfor (14):
1. ¯xis a solution of (14), 2. 0∈∂f(¯x) + cl∗[R+∂g(¯x)].
Unfortunately, these two conditions are not sufficient to get the existence of Karush-Kuhn-Tucker (KKT) multipliers, that there existsλ≥0, such that
0∈∂f(¯x) +λ∂g(¯x)
provided that f is Lipschitz continuous around ¯x(see Example 1.1). The real number λis a KKT-lagrange multiplier for problem (14).
Example 1.1 [10] EndowR2with the usual scalar producth·,·iand the associated eulidean normk·k. Consider the convex functionsf andg defined on R2 by
f(x, y) =x, g(x, y) =k(x, y)k −y.
Then we have
T(S,(0,0)) ={0} ×R+ andN(S,(0,0)) =R×R− while
∂g(0,0) =B((0,−1),1)andR+∂g(0,0) =R×]− ∞,0[∪{(0,0)}.
The feasible pointx¯= (0,0)is a solution of (14) but
0∈/ ∂f(¯x) +R+∂g(¯x).
The situation is quite different when g is the maximum of finite number of convex differentiable functions.
Indeed, the Abadie’s constraint qualification can be expressed at ¯xas
T(S,x) =¯ {h∈X: h∇gi(¯x), hi ≤0i∈I(¯x)} (4) whereg1,· · ·, gmare convex real-valued functions onX which are differentiable at ¯x∈S,I(¯x) ={i: gi(¯x) = 0}
andg(x) = max
i=1,···,mgi(x). Using Farkas lemma, we obtain that condition (4) is equivalent to
N(S,x) =¯ R+co{∇gi(¯x) : i∈I(¯x)} (5) where “co” stands for the convex hull. As∂g(¯x) = co{∇gi(¯x) : i∈I(¯x)}, then condition (5) can be expressed as
N(S,x) =¯ R+∂g(¯x). (6)
This last one is calledGuignard constraint qualification and ensures the existence of KKT-Lagrange multiplier.
Example 1.1 is very instructive because it allows to say that Guignard’s and Abadie’s constraint qualifications are clearly distinct, with only the affirmation that Guignard’s condition leads to that of Abadie.
Using the subdifferential calculus∂g+(¯x) = co{0, ∂g(¯x)}, where a+= max(0, a), we easily show that condition (6) is equivalent to
N(S,x) =¯ R+∂g+(¯x). (7)
Note that all classical constraint qualifications (Slater condition, Mangasarian-Fromovitz condition, ...) imply the Guignard’s constraint qualification (6). One of them is the so-called error bound.
Definition 1.1 (Local error bound) We say that the following system
g(x)≤0 (8)
satisfies error bound atx, with¯ g(¯x) = 0, if there exist two real numbers α >0 andr >0 such that
d(x, S)≤αg+(x) ∀x∈B(¯x, r). (9)
Where
d(x, S) = inf
u∈Sku−xk
is the distance function ofStox. If error bound holds at everyx, with¯ g(¯x) = 0, we say that the system satisfies error bound.
This concept is equivalent to say that the set-valued mappingM :R⇒X defined by M(t) ={x∈X: g(x)≤t}
is calm at (0,x) not to be confused with the concept of metric regularity.¯
We recall that following [26],M is calm at (0,x) of its graph if there exist neighborhoods¯ V andW of 0 and ¯x respectively, and someL >0 such that the corresponding distance functions satisfy
d(x, M(0))≤L|t| ∀x∈M(t)∩W,∀t∈V.
Obviously, calmness is also weaker than the well-known Aubin property of multifunctions d(x, M(t))≤Ld(t, M−1(x))∀t∈V,∀x∈W.
This last one is equivalent to saying thatM−1 is metrically regular at (¯x,0) in the Robinson’s sense [25] (see [26, 25, 14, 15] and references therein for more studies on these concepts including necessary and sufficient conditions).
The study of error bounds has received a lot of attention in the mathematical programming literature during the last decades (see [4, 17, 18, 20, 21, 5, 22, 23, 19, 16] and references therein). Note that a simple condition ensuring error bound for the system (8) is Slater condition, that is there exists u ∈ X such that g(u) < 0.
Indeed, for all ¯x∈S and allx /∈S, the convexity ofS ensures thatv:=x+g(x)−g(u)g(x) (u−x)∈S and d(x, S)≤ kx−vk= g(x)
g(x)−g(u)kx−uk (10)
which implies that the local error bound holds for the system (8).
In this paper, we are also concerned with the following concepts of error bound which give characterization of Guignard’s qualification condition in some special situations.
Definition 1.2 (Bounded error bound) We say that the system (8) satisfies bounded error bound if for all r >0 there existsαr>0 such that
d(x, S)≤αrg+(x) ∀x∈rB. (11)
Definition 1.3 (Global error bound) We say that the system (8) satisfies global error bound if there exists α >0such that
d(x, S)≤αg+(x) ∀x∈X. (12)
It is easy to see that
Global error bound =⇒Bounded error bound =⇒Local error bound =⇒Guignard condition.
The aim of the present work is to characterize Guignard’s condition in terms of these error bound concepts.
More precisely, we will show that this characterization holds in the following situations:
1. g is the maximum of a finite number of differentiable convex functions.
2. S is closed convex and polyhedral.
3. The dimension of the subspace lin(S) is less than 2 andg is positively homogeneous.
The first item has been studied in the paper [19] in finite dimensional spaces by using an euclidean approach.
Note that the third item is included in the second one and contains the situations whereS is a singleton, a ray or affine subspace.
We will show that there is no way to get a characterization outside of the three situations above. We will give examples showing the limit of the cited cases. The first example shows that the thirth item is no longer true if g is not assumed positively homogeneous. The second example shows the loss of the characterization in spaces of dimension greater than 3 even ifg is positively homogeneous.
We will show that we need more to charaterize error bounds by introducing a new condition in terms of the gauge function. This last one allows us to obtain as a direct concequence of our results different characteriza- tions of the concept of strong conical hull intersection property (SCHIP) for a finite collection of convex sets in Banach spaces.
The paper is organized as follows: Section 2 presents the basic notation and concepts used in this paper, namely tools from convex analysis. Section 3 is devoted to the equivalence of Guignard’s and Abadie’s conditions under a closedness assumption as well as to an elementary characterization of Guignard’s condition by mean of the concept of calmness in the Clarke’s sense of the value function.
Different error bound characterization of the Guignard’s condition are established in Section 4 for differentiable convex inequality systems, namely the equivalence between this condition as well as the bounded and local error bounds. Section 5 contains different characterizations of Guignard’s condition in special cases for nondif- ferentiable convex systems. Section 6 is devoted to three technichal examples showing that the equivalence of Guignard’s condition and error bound is limited to the situations above. This allows us to introduce in Section 7 a new condition in terms of the gauge function implying Guignard’s condition in order to characterize error bounds. Finally, Section 8 provides an illustration of this new condition in the characterization of the SCHIP property.
2 Preliminaries
Otherwise stated, the spaceX will be a Banach space equipped with a norm k · k, X∗ is its topological dual with a pairing h·,·i. The closed and the open unit ball of X (resp. X∗) are identified by B and ˚B(resp. B∗ and ˚B∗), respectively. The closure (resp. w∗-closure) and the convex hull of a setA⊂X (resp. A∗⊂X∗) are denoted by clA and coA (resp. cl∗A∗), respectively. Let ˚C be the interior of a set C ⊂X and lin(C) be the smallest subspace ofX containingC.
For an extended-real valued functionf :X 7→X∪ {+∞}, the Fenchel subdifferential is defined by
∂f(x) ={x∗∈X∗: hx∗, u−xi ≤f(u)−f(x)∀u∈X}
iff(x)<+∞and∅ iff(x) = +∞.
The directional derivative off atx, withf(x)<+∞, is given by f0(x, h) = lim
t→0+
f(x+th)−f(x)
t .
So that
∂f(x) ={x∗∈X∗: hx∗, hi ≤f0(x, h)∀h∈X}.
Whenf is locally Lipschitz continuous aroundx, then f0(x, h) = max
x∗∈∂f(x)hx∗, hi ∀h∈X.
The tangent coneT(C, x) to a closed setC⊂X atx∈C is defined by T(C, x) = cl(R+(C−x)) or equivalently
T(C, x) ={h∈X : d0(·, C)(x, h) = 0}.
The normal coneN(C, x) toC atx∈C is given by
N(C, x) ={x∗∈X∗: hx∗, hi ≤0∀h∈T(C, x)}.
We have also the following characterization of the normal cone N(C, x) =R+∂d(x, C).
Lemma 2.1 Let C⊂X be a closed convex set and let x /∈C. Then for allε >0there exist uε∈C,x∗ε∈X∗ andb∗ε∈B∗ such that
1. kuε−xk ≤d(x, C) +ε2, 2. x∗ε+εb∗ε∈(1 +ε)∂d(uε, C), 3. hx∗ε, x−uεi=kuε−xk.
Moreover, if eitherS is included in a finite dimensional subspace ofX orX is a Hilbert space, then there exist u∈S andx∗∈X∗ such that
1. d(x, C) =kx−uk, 2. x∗∈∂d(u, C),
3. hx∗, x−ui=ku−xk.
Proof. Letvε∈Csuch thatkx−vεk ≤d(x, C) +ε2. Define the functionf onX byf(u) =ku−xk. Then f(vε)≤ inf
u∈Cf(u) +ε2. By Ekeland’s variational principle [8], there existsuε∈C such that
f(uε)≤f(vε), kuε−vεk ≤ε, f(uε)≤f(u) +εku−uεk ∀u∈C.
This last inequality is equivalent to saying thatuεminimizes the functionu7→f(u) +εku−uεk+ (1 +ε)d(u, C) or equivalently
0∈∂f(uε) +εB∗+ (1 +ε)∂d(uε, C).
So that there exist−x∗ε∈∂f(uε) and b∗ε ∈B∗ such thatx∗ε+εb∗ε∈(1 +ε)∂d(uε, C). To conclude, it remains to see that∂f(uε) ={x∗∈X∗: hx∗, uε−xi=kuε−xk}.
Lemma 2.2 Let C⊂X be a closed convex set and letx∈C. Then
∂d(x, C) =N(C, x)∩B∗=∂d(0, T(C, x)).
Lemma 2.3 Let K ⊂X be a closed convex cone with negative polar K0(:={x∗∈X∗ : hx∗, hi ≤0∀h∈K}).
Then
d(x, K) = sup
x∗∈K0∩B∗
hx∗, xi ∀x∈X.
The following lemma establishes a subdifferential formula of homogeneous and supremum functions.
Lemma 2.4 (Subdifferential of the supremum of homogeneous functions) Let h: Rm → R andhk : Rm→R,k∈N, be homogeneous convex functions. Then
1. For allx∈Rm, we have
x∗∈∂h(x) ⇐⇒ hx∗, xi= 0, x∗∈∂h(0).
2. Ifh= sup
k∈N
hk, then
∂h(0) =clco [
k∈N
∂hk(0)
! .
Proof. Item 1 is obvious. Let us establish the second one. Using the definition of h, we obtain that for all k∈N,∂hk(0)⊂∂h(0) and hence clco [
k∈N
∂hk(0)
!
⊂∂h(0). Proposition 5.2 in [9] asserts that
∂h(0) =∩ε>0clco [
k∈N
∂εhk(0)
!
where ∂εhk(0) ={x∗ ∈ Rm : hx∗, xi ≤ hk(x) +ε∀x∈ Rm} is the ε−subdifferential of hk at 0. Since hk is homogeneous, we have∂εhk(0) =∂hk(0) +εB∗Rm. So that
∂h(0) =∩ε>0clco [
k∈N
(∂hk(0) +εB∗Rm)
!
⊂clco [
k∈N
∂hk(0)
! .
3 Some elementary characterizations of Guignard’s constraint qual- ification for nondifferentiable convex systems
In this section, we give two elementary characterisations of Guignard’s condition. The first one concerns its equivalence with that of Abadie and the second one with the concept of calmness in the Clarke’s sense of the value function. We state them without proof.
As we saw in the introduction (see Example 1.1) that Guignard’s and Abadie’s constraint qualification are not equivalent for nondifferentiable convex systems. The following result shows that both Guignard’s and Abadie’s constraint qualifications for nondifferentiable convex systems are equivalent under an aditional closedeness hypothesis.
Proposition 3.1 The following assertions are equivalent for x∈S, withg(x) = 0:
1. Guignard’s constraint qualification (6) holds atx;
2. Abadie’s constraint qualification (2) holds atxand the setR+∂g(¯x)is weak-star closed.
For the second characterization, consider convex continuous functions f, gi : X → R, i = 1,· · · , m and the optimization problem
(Pf)
ß minf(x)
gi(x)≤0i= 1,· · ·, m (13)
To this problem, we associate the following perturbed one (Py)
ß minf(x)
gi(x)≤yii= 1,· · ·, m (14) wherey= (y1,· · ·, ym)∈Rmis the perturbation parameter. The value function associated to (P) is given by
vf(y) = inf{f(x) : gi(x)≤yii= 1,· · ·, m}.
It is easy to see thatvf is convex. Following Clarke [6],vf is calm at 0, wherevf(0)∈R, if lim inf
y→0
vf(y)−vf(0)
kyk >−∞.
In the convex setting, this definition is equivalent to say that
∂vf(0)6=∅.
Then, we have:
Proposition 3.2 Suppose that the solution setSf of the problemPf is nonempty. Then 1. −λ∈∂vf(0)IFF λis a KKT multiplier forPf associated to allx¯∈Sf.
2. Guignard’s condition holds for the system (8), withg= max
i=1,···,mgi, IFF for any locally Lipschitz function f :X→Rfor whichSf 6=∅,vf is calm at0.
4 Error bound characterization of Guignard’s constraint qualifica- tion for differentiable convex inequality systems
Recall that a constraint{x∈X : g(x)≤0}, or simply g, satisfies Slater condition if there exists u∈X such that
g(u)<0 and thatg(x) = max
i=1,···,pgi(x).
Consider the setI :={J ⊂I: gJ:= max
i∈J gi satisfies Slater condition}.
The following result states error bound characterizations of Guignard’s constraint qualification under the differ- entialbility of the data, especialy the equivalence between this last one and the bounded and local error bounds in Banach spaces.
Theorem 4.1 Suppose that the function g is a maximum of finite numberpof convex differentiable functions gi:X→R. Then the following assertions are equivalent:
i) Guignard’s constraint qualification holds for system (8), that is, , for allx¯∈S, N(S,x) =¯ R+co{∇gi(¯x)i∈I(¯x)}
where I(¯x) :={i∈ {1,· · ·, p}: gi(¯x) = 0} is the index set of active constraints atx.¯
ii) The system (8) satisfies error bound. More precisely, I 6=∅ and there exists(xJ)J∈I ⊂X such that gJ(xJ)<0 andd(x, S)≤g+(x) max
J∈I
Å kx−xJk gJ+(x)−gJ(xJ)
ã
∀x∈X.
iii) The system (8) satisfies bounded error bound. More precisely, there exists c >0 such that for all r >0 d(x, S)≤c(r+ 1)g+(x)∀x∈X, with kxk ≤r.
iv) The system (8) satisfies local error bound.
Proof. iv) =⇒i) : This implication is obvious and is based on the formula∂d(x, S) =N(S, x)∩B∗, and the subdifferential calculus of the maximum of convex functions.
ii) =⇒iii) : It is enough to takec= max
J∈I
1
−gJ(xJ)max(1,kxJk).
i) =⇒ii) : This implication will be established in three steeps.
Steep 1: We start by the following lemma whose proof can be deduced from that of Theorem 4.1 in [13]. We give a proof to make the paper self-contained.
Lemma 4.1 Let x∗, x∗1,· · · , x∗m∈X∗\{0}. Suppose there existµ1,· · ·, µm∈R+ such that x∗=
m
X
i=1
µix∗i.
Then there exist J ⊂ {1,· · · , m} and (βi)i∈J, with βi ≥ 0 for all i ∈ J and not all equal to zero, such that (x∗i)i∈J are linearly independent and
x∗=X
i∈J
µix∗i.
Proof. It is included for completeness. SetI0 ={1,· · · , m}. If (x∗i)i∈I0 are not linearly independent, then there is non thing to prove. Suppose the contrary and so there existγi ∈R, i∈I0, not all equal to zero such that
X
i∈I0
γix∗i = 0.
Without loss of generality, we can assume that there is at leasti∈I0such thatγi<0. Hence for allt∈R x∗= X
i∈I0
(µi+tγi)x∗i.
Settmax = max{t: µi+tγi ≥0, ∀i∈ I0}. Thentmax = min
i∈I0{−µi γi
:γi <0}. Let then i0 ∈I0 be such that
−µγi0
i0
=tmax, that is,µi0+tmaxγi0= 0. Hence settingI1=I0\{i0}andµ(1)i =µi+tmaxγ, ∀i∈I1, we have x∗=X
i∈I1
µ(1)i x∗i, withµ(1)i ≥0,∀i∈I1
By induction we show that there existI⊂ {1,· · ·, m}and (βi≥0)i∈I such that (x∗i)i∈I is linearly independent and
x∗=X
i∈I
βix∗i.
Steep 2: Suppose first thatI 6=∅. Then for all J ∈ I there existsxJ∈X such that gJ(xJ)<0 and (by (10))d(x, SJ)≤ gJ+(x)
gJ+ (x)−gJ(xJ)kx−xJk ∀x∈X (15) whereSJ :={x∈X : gJ(x)≤0}. Hence
d(x, SJ)≤ g+(x)
gJ+(x)−gJ(xJ)kx−xJk ∀x∈X. (16) We will prove that the setI is in fact not empty.
Lemma 4.2 Suppose i) holds. Let x /∈ S and x∗ ∈ ∂d(x, S). Then kx∗k = 1 and there exist sequences (vn)n∈N⊂S,v∗n∈N(S, vn), for alln∈N,(un)n∈N⊂X and(Jn)n∈N⊂ I such that
1. kx∗−v∗nk →0, 2. kun−xk →0, 3. gJn:= max
i∈Jn
gi satisfies Slater condition and
4. d(un, S)≤ 1 +n1
1−n1d(un, SJn).
Proof. Fix x /∈ S and x∗ ∈ ∂d(x, S). It is easy to see thatkx∗k = 1. For each integern > 0, there exists wn∈S such that
kx−wnk ≤d(x, S) + 1 n2. So
hx∗, u−xi ≤d(u, S)−d(x, S)≤ ku−vk − kx−wnk+ 1
n2∀u∈X,∀v∈S.
So that the Lipschitz functiong:X×X 7→Rdefined by
g(u, v) =ku−vk − hx∗, ui satisfies
g(x, wn)≤ inf
(u,v)∈X×Sg(u, v) + 1 n2.
So, endowingX×Xwith the normk(x, y)k=kxk+kyk, Ekeland’s variational principle [8] ensures the existence ofun∈X andvn∈S such that
kx−unk+kwn−vnk ≤ 1
n, g(un, vn)≤g(u, v) +1
n[ku−unk+kv−vnk] ∀u∈X,∀v∈S or equivalently
(x∗,0)∈∂k · − · k(un, vn) +{0} ×N(S, vn) + 1
nBX∗×BX∗.
Due to the fact thatx /∈S, un 6=vn fornlarge enough, there exist u∗n ∈∂k · k(un−vn), withku∗nk= 1, and b∗n ∈ 1nBX∗ such that
kx∗−u∗nk ≤ 1
n, vn∗ :=u∗n+b∗n∈N(S, vn).
By our hypothesisi) there areµn1,· · ·, µnm∈R+, not all equal to zero such that v∗n=
m
X
i=1
µni∇gi(vn).
Lemma 4.2 ensures the existence ofJn⊂ {1,· · · , p}such that (∇gi)i∈Jn are linearly independent and vn∗ = X
i∈Jn
µni∇gi(vn).
So that Jn ∈ I and affirms thatI 6=∅and v∗n ∈(1 + 1n)∂d(vn, SJn) (becausekv∗nk ≤1 +n1 and ∂d(vn, SJn) = N(SJn)∩BX∗). Sinceu∗n∈∂k · k(un−vn), we have
d(un, S)≤ kun−vnk=hu∗n, un−vni=hvn∗, un−vni − 1
nhb∗n, un−vni
≤ hvn∗, un−vni+ 1
nkun−vnk Then
(1− 1
n)kun−vnk ≤ hv∗n, un−vni ≤(1 + 1
n)d(un, SJn) and the result follows.
Steep 3: Now, using the previous steeps and relation (16), we get
d(un, S)≤ 1 +n1
1−n1d(un, SJn)
≤ 1 +n1 1−n1
gJn+(un)
gJn+(un)−gJn(xJn)kun−xJnk
≤ 1 +n1
1−n1g+(un) max
J∈I
Å kun−xJk gJ+(un)−gJ(xJ)
ã . Now passing to the limit onn, we obtain
d(x, S)≤g+(x) max
J∈I
Å kx−xJk g+(x)−gJ(xJ)
ã .
5 Error bound characterization of Guignard’s constraint qualifica- tion for nondifferentiable convex systems : Special cases
The situation of nondifferentiable systems is quitte different and involves additionnal hypothesis execpt in the following special situations:
1. g is a polyhedral function and X is a Banach space. In this case both Guignard’s condition and error bound are satisfied.
2. S is a closed convex polyhedral set andX is a Banach space.
3. The dimension of the subspace lin(S) is less than 2 andg is positively homogeneous.
Remark 5.1 Unfortunately, when ”g is not positively homogeneous and dimlin(S) ≥ 2” or ”g is positively homogeneous but dimlin(S) ≥ 3”, the condition of Guignard is not sufficient to guarantee the existence of an error bound concept. In this respect, we shall give counterexamples in Section 6 showing the limit of this characterization.
