Second order optimality conditions with the envelope like effect in nonsmooth multiobjective mathematical programming i l stability and set valued directional derivatives

J. Math. Anal. Appl. 403 (2013) 695–702
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Journal of Mathematical Analysis and
Applications
journal homepage: www.elsevier.com/locate/jmaa
Second-order optimality conditions with the envelope-like
effect in nonsmooth multiobjective mathematical
programming I: l-stability and set-valued directional
derivatives
Phan Quoc Khanh
a,∗
, Nguyen Dinh Tuan
b
a
Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Nam
b
Department of Mathematics and Statistics, University of Economics of Hochiminh City, 59C Nguyen Dinh Chieu, D.3, Hochiminh City, Viet Nam
a r t i c l e i n f o
Article history:
Available online 14 February 2013
Submitted by Heinz Bauschke
Keywords:
Nonsmooth multiobjective programming
Weak solutions
Firm solutions
Set-valued second-order directional
derivatives
Strict differentiability
l-stability
a b s t r a c t
Second-order necessary conditions and sufficient conditions, with the envelope-like effect,

for optimality in nonsmooth multiobjective mathematical programming are established.
We use set-valued second-order directional derivatives and impose strict differentiability
for necessary conditions and l-stability for sufficient conditions, avoiding continuous differ-
entiability. The results improve and sharpen several recent existing ones. Examples are pro-
vided to show advantages of our theorems over some known ones in the literature. In Part I,
we consider l-stability and second-order set-valued directional derivatives of vector func-
tions. Part II is devoted to second-order necessary optimality conditions and sufficient ones.
© 2013 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries
In mathematical programming, and more generally in optimization, second-order optimality conditions occupy an im-
portant place, since they provide significant additional information to the first-order ones. To meet the diversity of practical
applications, the considered optimization problems, and hence the tools and techniques of study, have been becoming more
and more complicated. But we can observe, in most of related contributions in the literature, that the core of the results can
be roughly stated the same as in the classical result of calculus that the second derivative of objective functions (or Lagrange
functions in constrained problems) at minimizers is nonnegative. Kawasaki [14] was the first researcher to reveal that di-
rectional derivatives of Lagrange functions may be negative at minimizers, if the directional derivative of the map composed
from the objective and constraints lies on a particular part of the boundary of the negative composite cone in the product of
the image spaces. He called this phenomenon the envelope-like effect. Kawasaki’s results were developed by several authors
in [5,7,20,21], considering always C
2
scalar programs, like in [14]. In multiobjective programming, the first results of this type
were given in [12,13] also for smooth cases. For nonsmooth (multiobjective) programming, Gutiérrez–Jiménez–Novo [11]
used the set-valued parabolic and Dini second-order directional derivatives to establish second-order optimality conditions
with the envelope-like effect. They considered Fréchet differentiable functions whose derivative is continuous or stable at
the point of study. However, still many authors ignore the envelope-like effect when investigating second-order optimality
conditions. This may lead to unaware mistakes. Furthermore, in the mentioned papers, sometimes it was not made clear
∗
Corresponding author.
E-mail addresses: , (P.Q. Khanh), (N.D. Tuan).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2012.12.076
696 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702
enough when this effect occurs and when it does not. (Even in the careful study of second-order optimality conditions with
the envelope-like effect in [11], there is a confusion with this phenomenon, see Remark 1 (ii) in Part II ([17]).) This observation
motivates our first aim of this paper, which is to make the envelope-like effect clear in second-order optimality conditions.
On the other hand, a major approach to nonsmooth optimization is to propose and apply suitable generalized derivatives
to replace the classical Fréchet and Gateaux derivatives which do not exist in establishing optimality conditions. Various
kinds of derivatives have been employed, each has advantages in some situations but none is universal. Recently, set-valued
derivatives for a single-valued vector function have been used effectively to provide multiplier rules in nonsmooth programs,
see, e.g., [4,8,9,11,15,18] (but in [4,8,9,15,18], the envelope-like effect was not attained). Our second aim in this paper is to
apply the Hadamard second-order directional derivative (we proposed in [15]) together with the l-stability (developed in
[1–3,10]) to obtain new second-order optimality conditions, which improve and sharpen recent existing ones. Namely, since
the value of our Hadamard second-order directional derivative at a point is larger than that of both the parabolic and Dini
second-order directional derivatives, our necessary optimality conditions are stronger than that in [11]. Furthermore, we
relax major assumptions imposed in [11]: replacing continuous differentiability and stability by strict differentiability and
l-stability, respectively (shortly, resp.).
We consider the following multiobjective mathematical programming problem. Let f : R
n
→ R
m
, g : R
n
→ R
p
and
h : R
n
→ R
r
be given. Let C be a closed convex cone in R

m
and K a convex set in R
p
. The problem under our consideration is
(P) min f (x), s.t. g(x) ∈ −K, h(x) = 0.
If K = R
n
+
, the constraint g(x) ∈ −K collapses to the usual inequality constraint.
The organization of the paper is as follows. In the rest of this section we recall some preliminary facts, including those
concerning locally Lipschitz functions and second-order tangency. In Section 2, the notion of l-stability of scalar and vector
functions and some properties are presented. Section 3 is devoted to second-order directional derivatives and properties
which will be in use. In Part II ([17]), we establish in Section 2 necessary optimality conditions, with the envelope-like effect,
for local weak solutions of (P). Section 3 contains sufficient optimality conditions for local firm solutions.
In the forthcoming paper [16], we continue to study problem (P) for the case where the involved maps are between
general (infinite dimensional) Banach spaces. Second-order optimality conditions with the envelope-like effect are obtained
also for local weak and firm solutions, but in terms of other generalized derivatives, with a higher level of nonsmoothness.
Our notations are basically standard. N and R are the sets of natural numbers and real numbers, resp. For a normed
space X, X
∗
stands for the topological dual of X; ⟨., .⟩ is the canonical pairing. ∥.∥ is used for the norm in any normed space
and d(y, S) for the distance from a point y to a set S. B
n
(x, r) = {y ∈ R
n
: ∥x − y∥ < r}; S
n
= {y ∈ R
n
: ∥y∥ = 1};

S
∗
n
= {y ∈ (R
n
)
∗
: ∥y∥ = 1}; L(X, Y) denotes the space of bounded linear mappings from X into Y, where X and Y are
normed spaces. For a cone C ⊂ R
n
, C
∗
= {c
∗
∈ (R
n
)
∗
: ⟨c
∗
, c⟩ ≥ 0, ∀c ∈ C} is the polar cone of C. For A ⊂ R
n
, riA, intA, clA,
bdA, coneA and LinA stand for the relative interior, interior, closure, boundary, cone hull of A and the linear space generated
by A, resp. For t > 0 and r ∈ N, o(t
r
) designates a point depending on t such that o(t
r
)/t
r

→ 0 as t → 0
+
.
Let us recall some definitions and preliminary facts. A map f : R
n
→ X , where X is a normed space, is called strictly
differentiable at x ∈ R
n
if it has Fréchet derivative f
′
(x) at x and
lim
y→x,t→0
+
sup
h∈S
n




1
t
(f (y + th) − f (y)) − f
′
(x)h





= 0.
For locally Lipschitz function f : R
n
→ R
m
, the Clarke generalized Jacobian of f at x is defined by
∂f (x) = conv{limf
′
(x
k
) : x
k
∈ Ω, x
k
→ x},
where f is differentiable in Ω, which is dense by the Rademacher theorem. We collect some basic properties of the Clarke
generalized Jacobian in the following.
Proposition 1.1 ([6]). Let f : R
n
→ R
m
be locally Lipschitz at x. Then,
(i) ∂f (x) is a nonempty, convex, compact subset of L(R
n
, R
m
);
(ii) ∂f (x) is a singleton if and only if f is strictly differentiable at x: ∂f (x) = {f
′
(x)};

(iii) (robustness) ∂f (x) = {lim
k→∞
v
k
: v
k
∈ ∂f (x
k
), x
k
→ x}, in other words (as ∂f (x) is compact), the map ∂f (.) is upper
semicontinuous at x;
(iv) (Lebourg
′
s mean value theorem) if f is locally Lipschitz in a convex neighborhood U of x and a, b ∈ U, then
f (b) − f (a) ∈ conv(∂f ([a, b])(b − a))
and when m = 1, there is some point c ∈ (a, b) such that
f (b) − f (a) ∈ ∂f (c)(b − a).
Now we recall the notions of tangent cones and second-order tangent sets that we will use later.
P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 697
Definition 1.2. Let x
0
, u ∈ R
n
and M ⊂ R
n
.
(a) The contingent cone of M at x
0
is

T (M, x
0
) = {v ∈ R
n
: ∃t
k
→ 0
+
, ∃v
k
→ v, ∀k ∈ N, x
0
+ t
k
v
k
∈ M}.
(b) The interior tangent cone of M at x
0
is
IT (M, x
0
) = {v ∈ R
n
: ∀t
k
→ 0
+
, ∀v
k

→ v, ∀k large enough, x
0
+ t
k
v
k
∈ M}.
(c) The second-order contingent set of M at (x
0
, u) is
T
2
(M, x
0
, u) =

w ∈ R
n
: ∃t
k
→ 0
+
, ∃w
k
→ w, ∀k ∈ N, x
0
+ t
k
u +
1

2
t
2
k
w
k
∈ M

.
(d) The asymptotic second-order tangent cone of M at (x
0
, u) is
T
′′
(M, x
0
, u) =

w ∈ R
n
: ∃(t
k
, r
k
) → (0
+
, 0
+
) : t
k

/r
k
→ 0, ∃w
k
→ w, ∀k ∈ N, x
0
+ t
k
u +
1
2
t
k
r
k
w
k
∈ M

.
(e) The second-order adjacent set of M at (x
0
, u) is
A
2
(M, x
0
, u) =

w ∈ R

n
: ∀t
k
→ 0
+
, ∃w
k
→ w, ∀k ∈ N, x
0
+ t
k
u +
1
2
t
2
k
w
k
∈ M

.
(f) The second-order interior tangent set of M at (x
0
, u) is
IT
2
(M, x
0
, u) =


w ∈ R
n
: ∀t
k
→ 0
+
, ∀w
k
→ w, ∀k large enough, x
0
+ t
k
u +
1
2
t
2
k
w
k
∈ M

.
The following proposition summarizes some basic properties of the above second-order tangent sets (we do not give
references for well-known facts).
Proposition 1.3. Let M ⊂ R
n
, x
0

∈ R
n
and u ∈ R
n
. Then,
(i) IT
2
(M, x
0
, u) ⊂ A
2
(M, x
0
, u) ⊂ T
2
(M, x
0
, u) ⊂ clcone[cone(M − x
0
) − u];
(ii) if u ̸∈ T(M, x
0
), then T
2
(M, x
0
, u) = ∅.
If, in addition, M is convex, intM ̸= ∅ and u ∈ T (M, x
0
), then (see [13,19,22])

(iii) intcone(M − x
0
) = IT(intM, x
0
);
(iv) if A
2
(M, x
0
, u) ̸= ∅, then
IT
2
(M, x
0
, u) = intA
2
(M, x
0
, u), clIT
2
(M, x
0
, u) = A
2
(M, x
0
, u);
(v) if u ∈ cone(M − x
0
), then

(a) IT
2
(M, x
0
, u) = intcone[cone(M − x
0
) − u];
(b) A
2
(M, x
0
, u) = clcone[cone(M − x
0
) − u].
2. l-stable scalar and vector functions
Recall that a function h : R
n
→ R
m
is called stable (or calm) at x ∈ R
n
if there are a neighborhood U of x and a ϑ > 0
such that, for all y ∈ U,
∥h(y) − h(x)∥ ≤ ϑ∥y − x∥.
Definition 2.1 ([1,10]). (i) The lower (resp, upper) directional derivative of a function ϕ : R
n
→ R at x in direction u is
defined by
ϕ
l

(x, u) = lim inf
t→0
+
1
t
(ϕ(x + tu) − ϕ(x))

resp, ϕ
u
(x, u) = lim sup
t→0
+
1
t
(ϕ(x + tu) − ϕ(x))

.
(ii) A function ϕ is called l-stable (resp, u-stable) at x if there exist a neighborhood U of x and a ϑ > 0 such that, for all
y ∈ U and u ∈ S
n
,
|ϕ
l
(y, u) − ϕ
l
(x, u)| ≤ ϑ∥y − x∥ (2.1)
(resp, |ϕ
u
(y, u) − ϕ
u

(x, u)| ≤ ϑ∥y − x∥). (2.2)
Some properties of l-stability of ϕ : R
n
→ R are summarized in the next proposition.
698 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702
Proposition 2.2. (i) [1,3] Any l-stable function is locally Lipschitz and strictly differentiable.
(ii) [10] ϕ is l-stable at x if and only if ϕ is (Fréchet) differentiable at x and there is a neighborhood U of x such that ϕ is
Lipschitz on U, and there is a ϑ > 0 such that
∥ϕ
′
(y) − ϕ
′
(x)∥ ≤ ϑ∥y − x∥ a.e. in U (in the sense of Lebesgue measure).
(iii) [10] The notions of l-stability and u-stability are equivalent; moreover the same neighborhood U and constant ϑ applied
in the inequality (2.1) or in the inequality (2.2) can also be applied in the other one.
Due to Proposition 2.2 (iii), in the sequel we use only the lower directional derivative and l-stability, the upper notions
are mentioned only if necessary.
The notion of l-stability is extended to vector functions as follows.
Definition 2.3 ([2]). The lower (resp, upper) directional derivative of a function Φ : R
n
→ R
m
at x in a direction u with
respect to (shortly wrt) ξ
∗
∈ (R
m
)
∗
is defined by

Φ
l
ξ
∗
(x, u) = lim inf
t→0
+
1
t
⟨ξ
∗
, Φ(x + tu) − Φ(x)⟩

resp, Φ
u
ξ
∗
(x, u) = lim sup
t→0
+
1
t
⟨ξ
∗
, Φ(x + tu) − Φ(x)⟩

.
The following mean value property for continuous vector functions will be needed.
Proposition 2.4 ([2]). Let Φ : R
n

→ R
m
be continuous on an open subset U ⊂ R
n
containing a segment [a, b] and ξ
∗
∈ (R
m
)
∗
.
Then, there are points γ
1
, γ
2
∈ (a, b) such that
Φ
l
ξ
∗
(γ
1
, b − a) ≤ ⟨ξ
∗
, Φ(b) − Φ(a)⟩ ≤ Φ
l
ξ
∗
(γ
2

, b − a).
Definition 2.5. Let Φ : R
n
→ R
m
and Γ = C
∗
∩ S
∗
m
.
(i) [2] Assume that C ⊂ R
m
is a closed, convex and pointed cone with intC ̸= ∅. We say that Φ is l-stable at x in the sense
of Bednařík–Pastor if there are a neighborhood U of x and a ϑ > 0 such that, for all y ∈ U, u ∈ S
n
, and ξ
∗
∈ Γ ,
|Φ
l
ξ
∗
(y, u) − Φ
l
ξ
∗
(x, u)| ≤ ϑ∥y − x∥.
(ii) [10] Φ is said to be l-stable at x in the sense of Ginchev if, for any ξ
∗

∈ (R
m
)
∗
, the scalar function Φ
ξ
∗
(.) := ⟨ξ
∗
, Φ(.)⟩
is l-stable at x.
Of course, if a scalar or vector function f has a Fréchet derivative f
′
which is stable (i.e., calm), then f is l-stable. Note
that in the previous papers (e.g., [2,10]) dealing with l-stability, this notion was defined and applied only for the case of
Euclidean spaces. Here, we consider Banach spaces. It is worth noting that, in many applications, e.g., in economics, Lagrange
multipliers, being elements of the dual spaces involved in the problem under consideration, are prices. Hence, the dual spaces
cannot coincide with the primal ones as for the Euclidean case.
Several properties of l-stable vector functions taken from [2,10], which are valid also for Banach spaces, are collected
below.
Proposition 2.6. (i) [10] Φ : R
n
→ R
m
is l-stable at x in the sense of Ginchev if and only if there exist a neighborhood U of x
and a ϑ > 0 such that, for all y ∈ U, u ∈ S
n
, and ξ
∗
∈ (R

m
)
∗
,
|Φ
l
ξ
∗
(y, u) − Φ
l
ξ
∗
(x, u)| ≤ ϑ∥ξ
∗
∥∥y − x∥.
(ii) [10] Φ : R
n
→ R
m
is l-stable at x in the sense of Ginchev if and only if it is Fréchet differentiable at x and there exist an
open neighborhood U of x and a ϑ > 0 such that Φ is Lipschitz on U, and, for almost every y ∈ U,
∥Φ
′
(y) − Φ
′
(x)∥ ≤ ϑ∥y − x∥.
(iii) [2,10] If function Φ : R
n
→ R
m

is l-stable at x in the sense of Bednařík–Pastor or Ginchev, then Φ is locally Lipschitz at
x and strictly differentiable at x.
Now we prove that the above two definitions of l-stability for vector functions are equivalent in a sense.
Proposition 2.7. If Φ : R
n
→ R
m
is l-stable at x in the sense of Ginchev, then the inequality in the definition of l-stability of
Bednařík–Pastor holds. If C is pointed and intC ̸= ∅, the two definitions are equivalent.
P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 699
Proof. It follows from Proposition 2.6 (i) that Φ is l-stable at x in the sense of Ginchev if and only if there are a neighborhood
U of x and a ϑ > 0 such that, for all y ∈ U, u ∈ S
n
, and ξ
∗
∈ S
∗
m
,
|Φ
l
ξ
∗
(y, u) − Φ
l
ξ
∗
(x, u)| ≤ ϑ∥y − x∥.
Hence, this inequality holds for ξ
∗

∈ Γ ⊂ S
∗
m
as required by Bednařík–Pastor.
Conversely, suppose C is pointed, intC ̸= ∅ and the last inequality holds for ξ
∗
∈ Γ . Because LinΓ = (R
m
)
∗
, for every
i ∈ {1, 2, . . . , m}, there exist ξ
∗
i,1
, . . . , ξ
∗
i,r
i
∈ Γ and α
i,1
, . . . , α
i,r
i
∈ R with r
i
= 1, . . . , m such that e
∗
i
=


r
i
j=1
α
i,j
ξ
∗
i,j
, where
e
∗
i
∈ (R
m
)
∗
is defined by ⟨e
∗
i
, x⟩ = x
i
, for x = (x
1
, x
2
, . . . , x
m
).
Let M = max
i,j

|α
i,j
| and Φ
i
be the ith component of Φ. Observe that, since Φ is l-stable at x in the sense of
Bednařík–Pastor, Φ
l
e
∗
i
(x, u) = Φ
u
e
∗
i
(x, u) = ⟨e
∗
i
, Φ
′
(x)u⟩. For every y ∈ U, u ∈ S
n
, and i ∈ {1, 2, . . . , m}, one has that
Φ
l
i
(y, u) − Φ
l
i
(x, u) = Φ

l
e
∗
i
(y, u) − Φ
l
e
∗
i
(x, u)
= lim inf
t→0
+

e
∗
i
,
1
t
(Φ(y + tu) − Φ(y)) − Φ
′
(x)u

= lim inf
t→0
+
r
i


j=1
α
i,j

ξ
∗
i,j
,
1
t
(Φ(y + tu) − Φ(y)) − Φ
′
(x)u

≥ −
r
i

j=1
|α
i,j
||Φ
l
ξ
∗
i,j
(y, u) − Φ
l
ξ
∗

i,j
(x, u)| ≥ −ϑ
1
∥y − x∥,
where ϑ
1
= mMϑ. Now, by the equivalence of l-stability and u-stability, one has further
Φ
l
i
(y, u) − Φ
l
i
(x, u) ≤ Φ
u
e
∗
i
(y, u) − Φ
u
e
∗
i
(x, u)
= lim sup
t→0
+

e
∗

i
,
1
t
(Φ(y + tu) − Φ(y)) − Φ
′
(x)u

= lim sup
t→0
+
r
i

j=1
α
i,j

ξ
∗
i,j
,
1
t
(Φ(y + tu) − Φ(y)) − Φ
′
(x)u

≤
r

i

j=1
|α
i,j
||Φ
u
ξ
∗
i,j
(y, u) − Φ
u
ξ
∗
i,j
(x, u)| ≤ ϑ
1
∥y − x∥.
The obtained two inequalities say that Φ
i
is l-stable at x in the sense of Ginchev for all i = 1, . . . , m. Hence, using the
technique given in the proof of Theorem 3.3 in [10], also the vector function Φ is l-stable at x in the sense of Ginchev. 
By virtue of this proposition, from now on we mention only one notion of l-stability, namely that of Ginchev.
3. Second-order set-valued directional derivatives
Let us recall the upper limit in the sense of Painlevé–Kuratowski of a set-valued mapping Φ : R
n
⇒ R
m
Limsup
u→u

Φ(u) = {y ∈ R
m
: ∃u
k
→ u, ∃y
k
∈ Φ(u
k
) such that y
k
→ y}.
In this paper, we are concerned with the following three kinds of set-valued derivatives of a single-valued vector function.
Definition 3.1. Let h : R
n
→ R
m
be Fréchet differentiable at x
0
∈ R
n
and u, w ∈ R
n
.
(i) [15] The Hadamard second-order directional derivative of h at x
0
in direction u is
D
2
h(x
0

, u) = Limsup
v→u,t→0
+
h(x
0
+ tv) − h(x
0
) − th
′
(x
0
)u
t
2
/2
.
(ii) [15] The Dini second-order directional derivative of h at x
0
in direction u is
d
2
h(x
0
, u) = Limsup
t→0
+
h(x
0
+ tu) − h(x
0

) − th
′
(x
0
)u
t
2
/2
.
(iii) [11] The parabolic second-order directional derivative of h at x
0
in direction (u, w) is
D
p
2
h(x
0
, u, w) = Limsup
v→w,t→0
+
h(x
0
+ tu +
1
2
t
2
v) − h(x
0
) − th

′
(x
0
)u
t
2
/2
.
700 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702
Note that D
p
2
h(x
0
, u, w) is known in many works also as the contingent derivative of the set-valued map x → {h(x)} at
(x
0
, h(x
0
)) in direction (u, w). It is clear that
d
2
h(x
0
, u) ⊂ D
2
h(x
0
, u), D
p

2
h(x
0
, u, w) ⊂ D
2
h(x
0
, u).
To obtain some more relations between these derivatives we need the following.
Lemma 3.2. Let h : R
n
→ R
m
be l-stable at x
0
∈ R
n
. Then, there is ϑ > 0 such that, for all a, b near x
0
, there exists γ ∈ (a, b)
satisfying
∥h(b) − h(a) − h
′
(x
0
)(b − a)∥ ≤ ϑ∥b − a∥∥γ − x
0
∥.
Proof. By the Hahn–Banach theorem, there exists ξ
∗

∈ S
∗
m
such that
∥h(b) − h(a) − h
′
(x
0
)(b − a)∥ = ⟨ξ
∗
, h(b) − h(a) − h
′
(x
0
)(b − a)⟩.
Proposition 2.4 yields γ ∈ (a, b) fulfilling
⟨ξ
∗
, h(b) − h(a)⟩ ≤ h
l
ξ
∗
(γ , b − a).
Therefore, for ϑ being the l-stability constant of h,
⟨ξ
∗
, h(b) − h(a) − h
′
(x
0

)(b − a)⟩ ≤ h
l
ξ
∗
(γ , b − a) − h
l
ξ
∗
(x
0
, b − a) ≤ ϑ∥γ − x
0
∥∥b − a∥.
Hence,
∥h(b) − h(a) − h
′
(x
0
)(b − a)∥ ≤ ϑ∥b − a∥∥γ − x
0
∥. 
The following relation improves Proposition 2.2 of [15].
Proposition 3.3. If h : R
n
→ R
m
is l-stable at x
0
∈ R
n

with h
′
(x
0
) = 0, then, for all u ∈ R
n
,
d
2
h(x
0
, u) = D
2
h(x
0
, u).
Proof. It suffices to check that D
2
h(x
0
, u) ⊂ d
2
h(x
0
, u) for all u ∈ R
n
. Let y ∈ D
2
h(x
0

, u), i.e., there are t
k
→ 0
+
and u
k
→ u
such that
h(x
0
+ t
k
u
k
) − h(x
0
) − t
k
h
′
(x
0
)u
t
2
k
/2
→ y.
We have that
P :=

h(x
0
+ t
k
u
k
) − h(x
0
) − t
k
h
′
(x
0
)u
t
2
k
/2
−
h(x
0
+ t
k
u) − h(x
0
) − t
k
h
′

(x
0
)u
t
2
k
/2
=
h(x
0
+ t
k
u
k
) − h(x
0
+ t
k
u)
t
2
k
/2
.
Lemma 3.2 with a = x
0
+ t
k
u and b = x
0

+ t
k
u
k
yields, for all k ∈ N large enough, γ
k
∈ (x
0
+ t
k
u, x
0
+ t
k
u
k
) such that
∥P∥ ≤
ϑt
k
∥u
k
− u∥∥γ
k
− x
0
∥
t
2
k

/2
≤
ϑt
k
∥u
k
− u∥t
k
(∥u
k
∥ + ∥u∥)
t
2
k
/2
= 2ϑ∥u
k
− u∥(∥u
k
∥ + ∥u∥) → 0,
which implies that y ∈ d
2
h(x
0
, u). 
Proposition 3.4. Let h : R
n
→ R
m
be l-stable at x

0
∈ R
n
and u, w ∈ R
n
.
(i) If w
k
:= (x
k
− x
0
− t
k
u)/
1
2
t
2
k
→ w with t
k
→ 0
+
, then there is y ∈ D
p
2
h(x
0
, u, w) such that (for a subsequence)

y
k
:=
h(x
k
) − h(x
0
) − t
k
h
′
(x
0
)u
t
2
k
/2
→ y. (3.1)
(ii) D
p
2
h(x
0
, u, w) is nonempty and compact.
(iii) If w
k
:= (x
k
− x

0
− t
k
u)/
1
2
t
k
r
k
→ w with (t
k
, r
k
) → (0
+
, 0
+
) and t
k
/r
k
→ 0, then
y
k
:=
h(x
k
) − h(x
0

) − t
k
h
′
(x
0
)u
t
k
r
k
/2
→ h
′
(x
0
)w.
P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 701
Proof. (i) Suppose w
k
:= (x
k
− x
0
− t
k
u)/
1
2
t

2
k
→ w as t
k
→ 0
+
. Applying Lemma 3.2 to a = x
0
and b = x
k
, we see that, for
all k ∈ N large enough, there are γ
k
∈ (x
0
, x
k
) such that
∥h(x
k
) − h(x
0
) − h
′
(x
0
)(x
k
− x
0

)∥ ≤ ϑ∥x
k
− x
0
∥∥γ
k
− x
0
∥ ≤ ϑ∥x
k
− x
0
∥
2
, (3.2)
which implies that




h(x
k
) − h(x
0
) − t
k
h
′
(x
0

)u
t
2
k
/2
− h
′
(x
0
)w
k




≤ 2ϑ




u +
1
2
t
k
w
k





2
. (3.3)
Hence, {y
k
} is bounded, and therefore, there exists a subsequence converging to some y ∈ D
p
2
h(x
0
, u, w).
(ii) Part (i) shows that D
p
2
h(x
0
, u, w) ̸= ∅. Since D
p
2
h(x
0
, u, w) is closed, we prove that D
p
2
h(x
0
, u, w) is bounded. Let
y ∈ D
p
2

h(x
0
, u, w). Then, y
k
, defined by (3.1) for some (t
k
, w
k
) → (0
+
, w), tends to y. By the argument in (i), we have (3.3)
and, passing it to limit gives ∥y − h
′
(x
0
)w∥ ≤ 2ϑ∥u∥
2
, i.e., D
p
2
h(x
0
, u, w) is bounded.
(iii) Suppose w
k
:= (x
k
− x
0
− t

k
u)/
1
2
t
k
r
k
→ w as (t
k
, r
k
) → (0
+
, 0
+
) with t
k
/r
k
→ 0. According to (i), for all k ∈ N large
enough, we have (3.2) for some γ
k
∈ (x
0
, x
k
). Therefore, y
k
→ h

′
(x
0
)w since




h(x
k
) − h(x
0
) − t
k
h
′
(x
0
)u
t
k
r
k
/2
− h
′
(x
0
)w
k





≤ 2ϑ
t
k
r
k




u +
1
2
r
k
w
k




2
→ 0. 
Proposition 3.4(i), (ii) sharpens Proposition 2 of [11]. Furthermore, part (i) says, more than D
p
2
h(x

0
, u, w) being nonempty,
that any y
k
defined by (3.1) has a subsequence tending to some point of this derivative. Part (iii) of Proposition 3.4 extends
Lemma 3 of [11]. The following fact improves Proposition 3(ii) of [11].
Proposition 3.5. If h : R
n
→ R
m
is strictly differentiable at x
0
∈ R
n
, then, for all u, w ∈ R
n
,
D
p
2
h(x
0
, u, w) = h
′
(x
0
)w + d
2
h(x
0

, u).
Proof. For (t
k
, w
k
) → (0
+
, w), set
y
k
=
h(x
0
+ t
k
u +
1
2
t
2
k
w
k
) − h(x
0
) − t
k
h
′
(x

0
)u
t
2
k
/2
=
h(x
0
+ t
k
u +
1
2
t
2
k
w
k
) − h(x
0
+ t
k
u)
t
2
k
/2
+
h(x

0
+ t
k
u) − h(x
0
) − t
k
h
′
(x
0
)u
t
2
k
/2
:=
ˆ
h
k
+
ˆ
y
k
.
Using Lebourg’s mean value theorem in Proposition 1.1 (iv) for h and [a, b] = [x
0
+ t
k
u, x

0
+ t
k
u +
1
2
t
2
k
w
k
], we have
ˆ
h
k
∈ conv(∂h([a, b])w
k
). The robustness of ∂h implies that
ˆ
h
k
→ h
′
(x
0
)w as k → ∞.
Now, if y ∈ D
p
2
h(x

0
, u, w), then there exists (t
k
, w
k
) → (0
+
, w) such that y
k
→ y. Hence,
ˆ
y
k
= y
k
−
ˆ
h
k
→ y−h
′
(x
0
)w :=
ˆ
y,
i.e.,
ˆ
y ∈ d
2

h(x
0
, u).
Conversely, if we have a
ˆ
y ∈ d
2
h(x
0
, u), then t
k
→ 0
+
exists such that
ˆ
y
k
→
ˆ
y, where
ˆ
y
k
is defined at the beginning of
the proof. For w
k
≡ w and
ˆ
h
k

defined above, we have y ∈ D
p
2
h(x
0
, u, w) since
y
k
:=
ˆ
y
k
+
ˆ
h
k
→
ˆ
y + h
′
(x
0
)w := y. 
We have the following direct implication of Propositions 3.3 and 3.5.
Corollary 3.6. Let h : R
n
→ R
m
and x
0

, u ∈ R
n
.
(i) If h is strictly differentiable at x
0
and h
′
(x
0
) = 0, then, for w ∈ R
n
,
d
2
h(x
0
, u) = D
p
2
h(x
0
, u, w).
(ii) If h is l-stable at x
0
and h
′
(x
0
) = 0, then, for w ∈ R
n

,
D
2
h(x
0
, u) = d
2
h(x
0
, u) = D
p
2
h(x
0
, u, w).
The following example says that the condition h
′
(x
0
) = 0 in Proposition 3.3 is essential, but strict differentiability and
l-stability are only sufficient conditions in Propositions 3.3 and 3.5 and Corollary 3.6.
702 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702
Example 3.1. (a) Let h : R
2
→ R be defined by h(x
1
, x
2
) =
1

2
x
2
1
+ x
2
, x
0
= (0, 0), u = (1, 0), and w = (w
1
, w
2
) ∈ R
2
. Then,
h is l-stable (and so strictly differentiable) at x
0
, h
′
(x
0
) ̸= 0. Direct computations give
d
2
h(x
0
, u) = {1} ̸= D
2
h(x
0

, u) = R, D
p
2
h(x
0
, u, w) = {1 + w
2
} = h
′
(x
0
)w + d
2
h(x
0
, u).
(b) Let u = 1, x
0
= 0, w ∈ R, and h(x) = x
2
sin
1
x
if x ̸= 0, h(0) = 0. Then, h
′
(x
0
) = 0, but h is not strictly differentiable
(and so, not l-stable) at x
0

. However,
d
2
h(x
0
, u) = D
2
h(x
0
, u) = D
p
2
h(x
0
, u, w) = [−2, 2].
Part II will appear in another issue.
Acknowledgments
This research was supported by the grant 101.01-2011-10 of the National Foundation for Science and Technology
Development of Vietnam (NAFOSTED). The final part of working on the paper was completed during a stay by the authors
as research visitors at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefully
acknowledged. The authors are indebted to the Referees for their valuable remarks and suggestions.
References
[1] D. Bednařík, K. Pastor, On second-order conditions in unconstrained optimization, Math. Program. (Ser. A) 113 (2008) 283–298.
[2] D. Bednařík, K. Pastor, Decrease of C
1,1
property in vector optimization, RAIRO Oper. Res. 43 (2009) 359–372.
[3] D. Bednařík, K. Pastor, l-stable functions are continuous, Nonlinear Anal. 70 (2009) 2317–2324.
[4] D. Bednařík, K. Pastor, On second-order optimality conditions in constrained multiobjective optimization, Nonlinear Anal. 74 (2011) 1372–1382.
[5] J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
[6] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.

[7] R. Cominetti, Metric regularity, tangent sets and second order optimality conditions, Appl. Math. Optim. 21 (1990) 265–287.
[8] I. Ginchev, A. Guerraggio, M. Rocca, Second-order conditions for C
1,1
constrained vector optimization, Math. Program. (Ser. B) 104 (2005) 389–405.
[9] I. Ginchev, A. Guerraggio, M. Rocca, From scalar to vector optimization, Appl. Math. 51 (2006) 5–36.
[10] I. Ginchev, On scalar and vector l-stable functions, Nonlinear Anal. 74 (2011) 182–194.
[11] C. Gutiérrez, B. Jiménez, V. Novo, On second order Fritz John type optimality conditions in nonsmooth multiobjective programming, Math. Program.
(Ser. B) 123 (2010) 199–223.
[12] B. Jiménez, V. Novo, Second order necessary conditions in set constrained differentiable vector optimization, Math. Methods Oper. Res. 58 (2003)
299–317.
[13] B. Jiménez, V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets, Appl. Math. Optim. 49 (2004) 123–144.
[14] H. Kawasaki, An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems, Math.
Program. 41 (1988) 73–96.
[15] P.Q. Khanh, N.D. Tuan, Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives, J. Optim. Theory
Appl. 133 (2007) 341–357.
[16] P.Q. Khanh, N.D. Tuan, Second-order optimality conditions with envelope-like effect for nonsmooth vector optimization in infinite dimensions,
Nonlinear Anal. 77 (2013) 130–148.
[17] P.Q. Khanh, N.D. Tuan, Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming II:
Optimality conditions, J. Math. Anal. Appl. Online first 2013, />[18] L. Liu, P. Neittaanmäki, M. Křížek, Second-order optimality conditions for nondominated solutions of multiobjective programming with C
1,1
data,
Appl. Math. 45 (2000) 381–397.
[19] Y. Maruyama, Second-order necessary conditions for nonlinear optimization problems in Banach spaces and their applications to an optimal control
problem, Math. Oper. Res. 15 (1990) 467–482.
[20] J.P. Penot, Optimality conditions in mathematical programming and composite optimization, Math. Program. 67 (1994) 225–245.
[21] J.P. Penot, Second order conditions for optimization problems with constraints, SIAM J. Control Optim. 37 (1999) 303–318.
[22] D.E. Ward, Calculus for parabolic second-order derivatives, Set Valued Anal. 1 (1993) 213–246.
Further reading
[1] K. Allali, T. Amahroq, Second-order approximations and primal and dual necessary optimality conditions, Optimization 40 (1997) 229–246.
[2] A.L. Donchev, R.T. Rockafellar, Implicit Functions and Solution Mappings, Springer, Dordrecht, 2009.

[3] A. Jourani, L. Thibault, Approximations and metric regularity in mathematical programming in Banach spaces, Math. Oper. Res. 18 (1992) 390–400.
[4] P.Q. Khanh, N.D. Tuan, First and second-order optimality conditions using approximations for nonsmooth vector optimization in Banach spaces,
J. Optim. Theory Appl. 130 (2006) 289–308.
[5] P.Q. Khanh, N.D. Tuan, First and second-order approximations as derivatives of mappings in optimality conditions for nonsmooth vector optimization,
Appl. Math. Optim. 58 (2008) 147–166.
[6] P.Q. Khanh, N.D. Tuan, Second-order optimality conditions using approximations for nonsmooth vector optimization problems under inclusion
constraints, Nonlinear Anal. 74 (2011) 4338–4351.
[7] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.