J. Math. Anal. Appl. 403 (2013) 695–702

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Journal of Mathematical Analysis andApplications

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Second-order optimality conditions with the envelope-likeeffect in nonsmooth multiobjective mathematicalprogramming I: l-stability and set-valued directionalderivativesPhan Quoc Khanh a,∗, Nguyen Dinh Tuan b

a Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Namb 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 2013Submitted by Heinz Bauschke

Keywords:Nonsmooth multiobjective programmingWeak solutionsFirm solutionsSet-valued second-order directionalderivatives

Strict differentiabilityl-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 differentiabilityfor 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 practicalapplications, the considered optimization problems, and hence the tools and techniques of study, have been becomingmoreand more complicated. But we can observe, in most of related contributions in the literature, that the core of the results canbe roughly stated the same as in the classical result of calculus that the second derivative of objective functions (or Lagrangefunctions in constrained problems) at minimizers is nonnegative. Kawasaki [14] was the first researcher to reveal that di-rectional derivatives of Lagrange functionsmay be negative atminimizers, if the directional derivative of themap composedfrom the objective and constraints lies on a particular part of the boundary of the negative composite cone in the product ofthe image spaces. He called this phenomenon the envelope-like effect. Kawasaki’s results were developed by several authorsin [5,7,20,21], considering always C2 scalar programs, like in [14]. Inmultiobjective programming, the first results of this typewere 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 conditionswith the envelope-like effect. They considered Fréchet differentiable functions whose derivative is continuous or stable atthe point of study. However, still many authors ignore the envelope-like effect when investigating second-order optimalityconditions. This may lead to unaware mistakes. Furthermore, in the mentioned papers, sometimes it was not made clear

∗ Corresponding author.E-mail addresses: pqkhanh@hcmiu.edu.vn, pqkhanhus@yahoo.com (P.Q. Khanh), ndtuan73@yahoo.com (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 withthe envelope-like effect in [11], there is a confusionwith this phenomenon, seeRemark1 (ii) in Part II ([17]).) This observationmotivates our first aim of this paper, which is to make the envelope-like effect clear in second-order optimality conditions.

On the other hand, amajor approach to nonsmooth optimization is to propose and apply suitable generalized derivativesto replace the classical Fréchet and Gateaux derivatives which do not exist in establishing optimality conditions. Variouskinds of derivatives have been employed, each has advantages in some situations but none is universal. Recently, set-valuedderivatives for a single-valued vector function have beenused effectively to providemultiplier 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 toapply 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, sincethe value of our Hadamard second-order directional derivative at a point is larger than that of both the parabolic and Dinisecond-order directional derivatives, our necessary optimality conditions are stronger than that in [11]. Furthermore, werelax major assumptions imposed in [11]: replacing continuous differentiability and stability by strict differentiability andl-stability, respectively (shortly, resp.).

We consider the following multiobjective mathematical programming problem. Let f : Rn→ Rm, g : Rn

→ Rp andh : Rn

→ Rr be given. Let C be a closed convex cone in Rm and K a convex set in Rp. The problem under our consideration is

(P) min f (x), s.t. g(x) ∈ −K , h(x) = 0.

If K = Rn+, 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 thoseconcerning locally Lipschitz functions and second-order tangency. In Section 2, the notion of l-stability of scalar and vectorfunctions and some properties are presented. Section 3 is devoted to second-order directional derivatives and propertieswhichwill 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 betweengeneral (infinite dimensional) Banach spaces. Second-order optimality conditionswith the envelope-like effect are obtainedalso 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 normedspace X , X∗ stands for the topological dual of X; ⟨., .⟩ is the canonical pairing. ∥.∥ is used for the norm in any normed spaceand d(y, S) for the distance from a point y to a set S. Bn(x, r) = y ∈ Rn

: ∥x − y∥ < r; Sn = y ∈ Rn: ∥y∥ = 1;

S∗n = y ∈ (Rn)∗ : ∥y∥ = 1; L(X, Y ) denotes the space of bounded linear mappings from X into Y , where X and Y arenormed spaces. For a cone C ⊂ Rn, C∗

= c∗∈ (Rn)∗ : ⟨c∗, c⟩ ≥ 0, ∀c ∈ C is the polar cone of C . For A ⊂ Rn, riA, intA, clA,

bdA, coneA and LinA stand for the relative interior, interior, closure, boundary, cone hull of A and the linear space generatedby 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 : Rn→ X , where X is a normed space, is called strictly

differentiable at x ∈ Rn if it has Fréchet derivative f ′(x) at x and

limy→x,t→0+

suph∈Sn

1t(f (y + th) − f (y)) − f ′(x)h

= 0.

For locally Lipschitz function f : Rn→ Rm, the Clarke generalized Jacobian of f at x is defined by

∂ f (x) = convlimf ′(xk) : xk ∈ Ω, xk → x,

where f is differentiable in Ω , which is dense by the Rademacher theorem. We collect some basic properties of the Clarkegeneralized Jacobian in the following.

Proposition 1.1 ([6]). Let f : Rn→ Rm be locally Lipschitz at x. Then,

(i) ∂ f (x) is a nonempty, convex, compact subset of L(Rn, Rm);

(ii) ∂ f (x) is a singleton if and only if f is strictly differentiable at x: ∂ f (x) = f ′(x);(iii) (robustness) ∂ f (x) = limk→∞vk : vk ∈ ∂ f (xk), xk → 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 x0, u ∈ Rn andM ⊂ Rn.(a) The contingent cone ofM at x0 is

T (M, x0) = v ∈ Rn: ∃tk → 0+, ∃vk → v, ∀k ∈ N, x0 + tkvk ∈ M.

(b) The interior tangent cone ofM at x0 is

IT (M, x0) = v ∈ Rn: ∀tk → 0+, ∀vk → v, ∀k large enough, x0 + tkvk ∈ M.

(c) The second-order contingent set ofM at (x0, u) is

T 2(M, x0, u) =

w ∈ Rn

: ∃tk → 0+, ∃wk → w, ∀k ∈ N, x0 + tku +12t2k wk ∈ M

.

(d) The asymptotic second-order tangent cone ofM at (x0, u) is

T ′′(M, x0, u) =

w ∈ Rn

: ∃(tk, rk) → (0+, 0+) : tk/rk → 0, ∃wk → w, ∀k ∈ N, x0 + tku +12tkrkwk ∈ M

.

(e) The second-order adjacent set ofM at (x0, u) is

A2(M, x0, u) =

w ∈ Rn

: ∀tk → 0+, ∃wk → w, ∀k ∈ N, x0 + tku +12t2k wk ∈ M

.

(f) The second-order interior tangent set ofM at (x0, u) is

IT 2(M, x0, u) =

w ∈ Rn

: ∀tk → 0+, ∀wk → w, ∀k large enough, x0 + tku +12t2k wk ∈ M

.

The following proposition summarizes some basic properties of the above second-order tangent sets (we do not givereferences for well-known facts).

Proposition 1.3. Let M ⊂ Rn, x0 ∈ Rn and u ∈ Rn. Then,(i) IT 2(M, x0, u) ⊂ A2(M, x0, u) ⊂ T 2(M, x0, u) ⊂ clcone[cone(M − x0) − u];(ii) if u ∈ T (M, x0), then T 2(M, x0, u) = ∅.

If, in addition, M is convex, intM = ∅ and u ∈ T (M, x0), then (see [13,19,22])(iii) intcone(M − x0) = IT (intM, x0);(iv) if A2(M, x0, u) = ∅, then

IT 2(M, x0, u) = intA2(M, x0, u), clIT 2(M, x0, u) = A2(M, x0, u);

(v) if u ∈ cone(M − x0), then(a) IT 2(M, x0, u) = intcone[cone(M − x0) − u];(b) A2(M, x0, u) = clcone[cone(M − x0) − u].

2. l-stable scalar and vector functions

Recall that a function h : Rn→ Rm is called stable (or calm) at x ∈ Rn 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 : Rn→ R at x in direction u is

defined by

ϕ l(x, u) = lim inft→0+

1t(ϕ(x + tu) − ϕ(x))

resp, ϕu(x, u) = lim supt→0+

1t(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 ally ∈ U and u ∈ Sn,

|ϕ l(y, u) − ϕ l(x, u)| ≤ ϑ∥y − x∥ (2.1)

(resp, |ϕu(y, u) − ϕu(x, u)| ≤ ϑ∥y − x∥). (2.2)

Some properties of l-stability of ϕ : Rn→ 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 appliedin 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 notionsare 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 : Rn→ Rm at x in a direction u with

respect to (shortly wrt) ξ ∗∈ (Rm)∗ is defined by

Φ lξ∗(x, u) = lim inf

t→0+

1t⟨ξ ∗, Φ(x + tu) − Φ(x)⟩

resp, Φuξ∗(x, u) = lim sup

t→0+

1t⟨ξ ∗, Φ(x + tu) − Φ(x)⟩

.

The following mean value property for continuous vector functions will be needed.

Proposition 2.4 ([2]). Let Φ : Rn→ Rm be continuous on an open subset U ⊂ Rn containing a segment [a, b] and ξ ∗

∈ (Rm)∗.Then, there are points γ1, γ2 ∈ (a, b) such that

Φ lξ∗(γ1, b − a) ≤ ⟨ξ ∗, Φ(b) − Φ(a)⟩ ≤ Φ l

ξ∗(γ2, b − a).

Definition 2.5. Let Φ : Rn→ Rm and Γ = C∗

∩ S∗m.

(i) [2] Assume that C ⊂ Rm is a closed, convex and pointed cone with intC = ∅. We say that Φ is l-stable at x in the senseof Bednařík–Pastor if there are a neighborhood U of x and a ϑ > 0 such that, for all y ∈ U , u ∈ Sn, and ξ ∗

∈ Γ ,

|Φ lξ∗(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 ∗∈ (Rm)∗, the scalar function Φξ∗(.) := ⟨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. Notethat in the previous papers (e.g., [2,10]) dealing with l-stability, this notion was defined and applied only for the case ofEuclidean spaces. Here, we consider Banach spaces. It is worth noting that, inmany applications, e.g., in economics, Lagrangemultipliers, being elements of the dual spaces involved in the problemunder consideration, are prices. Hence, the dual spacescannot 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 collectedbelow.

Proposition 2.6. (i) [10] Φ : Rn→ Rm 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 ∈ Sn, and ξ ∗∈ (Rm)∗,

|Φ lξ∗(y, u) − Φ l

ξ∗(x, u)| ≤ ϑ∥ξ ∗∥∥y − x∥.

(ii) [10] Φ : Rn→ Rm 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 : Rn→ Rm 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 : Rn→ Rm 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) thatΦ is l-stable at x in the sense of Ginchev if and only if there are a neighborhoodU of x and a ϑ > 0 such that, for all y ∈ U, u ∈ Sn, and ξ ∗

∈ S∗m,

|Φ lξ∗(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 = (Rm)∗, for everyi ∈ 1, 2, . . . ,m, there exist ξ ∗

i,1, . . . , ξ∗

i,ri∈ Γ and αi,1, . . . , αi,ri ∈ R with ri = 1, . . . ,m such that e∗

i =ri

j=1 αi,jξ∗

i,j, wheree∗

i ∈ (Rm)∗ is defined by ⟨e∗

i , x⟩ = xi, for x = (x1, x2, . . . , xm).Let M = maxi,j |αi,j| and Φi be the ith component of Φ . Observe that, since Φ is l-stable at x in the sense of

Bednařík–Pastor, Φ le∗i

(x, u) = Φue∗i

(x, u) = ⟨e∗

i , Φ ′(x)u⟩. For every y ∈ U, u ∈ Sn, and i ∈ 1, 2, . . . ,m, one has that

Φ li (y, u) − Φ l

i (x, u) = Φ le∗i

(y, u) − Φ le∗i

(x, u)

= lim inft→0+

e∗

i ,1t(Φ(y + tu) − Φ(y)) − Φ ′(x)u

= lim inf

t→0+

rij=1

αi,j

ξ ∗

i,j,1t(Φ(y + tu) − Φ(y)) − Φ ′(x)u

≥ −

rij=1

|αi,j||Φlξ∗i,j(y, u) − Φ l

ξ∗i,j(x, u)| ≥ −ϑ1∥y − x∥,

where ϑ1 = mMϑ . Now, by the equivalence of l-stability and u-stability, one has further

Φ li (y, u) − Φ l

i (x, u) ≤ Φue∗i

(y, u) − Φue∗i

(x, u)

= lim supt→0+

e∗

i ,1t(Φ(y + tu) − Φ(y)) − Φ ′(x)u

= lim sup

t→0+

rij=1

αi,j

ξ ∗

i,j,1t(Φ(y + tu) − Φ(y)) − Φ ′(x)u

≤

rij=1

|αi,j||Φuξ∗i,j(y, u) − Φu

ξ∗i,j(x, u)| ≤ ϑ1∥y − x∥.

The obtained two inequalities say that Φi is l-stable at x in the sense of Ginchev for all i = 1, . . . ,m. Hence, using thetechnique 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 : Rn ⇒ Rm

Limsupu→uΦ(u) = y ∈ Rm: ∃uk → u, ∃yk ∈ Φ(uk) such that yk → y.

In this paper,we are concernedwith the following three kinds of set-valued derivatives of a single-valued vector function.

Definition 3.1. Let h : Rn→ Rm be Fréchet differentiable at x0 ∈ Rn and u, w ∈ Rn.

(i) [15] The Hadamard second-order directional derivative of h at x0 in direction u is

D2h(x0, u) = Limsupv→u,t→0+

h(x0 + tv) − h(x0) − th′(x0)ut2/2

.

(ii) [15] The Dini second-order directional derivative of h at x0 in direction u is

d2h(x0, u) = Limsupt→0+

h(x0 + tu) − h(x0) − th′(x0)ut2/2

.

(iii) [11] The parabolic second-order directional derivative of h at x0 in direction (u, w) is

Dp2h(x0, u, w) = Limsupv→w,t→0+

h(x0 + tu +12 t

2v) − h(x0) − th′(x0)ut2/2

.

700 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702

Note that Dp2h(x0, u, w) is known in many works also as the contingent derivative of the set-valued map x → h(x) at

(x0, h(x0)) in direction (u, w). It is clear that

d2h(x0, u) ⊂ D2h(x0, u), Dp2h(x0, u, w) ⊂ D2h(x0, u).

To obtain some more relations between these derivatives we need the following.

Lemma 3.2. Let h : Rn→ Rm be l-stable at x0 ∈ Rn. Then, there is ϑ > 0 such that, for all a, b near x0, there exists γ ∈ (a, b)

satisfying

∥h(b) − h(a) − h′(x0)(b − a)∥ ≤ ϑ∥b − a∥∥γ − x0∥.

Proof. By the Hahn–Banach theorem, there exists ξ ∗∈ S∗

m such that

∥h(b) − h(a) − h′(x0)(b − a)∥ = ⟨ξ ∗, h(b) − h(a) − h′(x0)(b − a)⟩.

Proposition 2.4 yields γ ∈ (a, b) fulfilling

⟨ξ ∗, h(b) − h(a)⟩ ≤ hlξ∗(γ , b − a).

Therefore, for ϑ being the l-stability constant of h,

⟨ξ ∗, h(b) − h(a) − h′(x0)(b − a)⟩ ≤ hlξ∗(γ , b − a) − hl

ξ∗(x0, b − a) ≤ ϑ∥γ − x0∥∥b − a∥.

Hence,

∥h(b) − h(a) − h′(x0)(b − a)∥ ≤ ϑ∥b − a∥∥γ − x0∥.

The following relation improves Proposition 2.2 of [15].

Proposition 3.3. If h : Rn→ Rm is l-stable at x0 ∈ Rn with h′(x0) = 0, then, for all u ∈ Rn,

d2h(x0, u) = D2h(x0, u).

Proof. It suffices to check that D2h(x0, u) ⊂ d2h(x0, u) for all u ∈ Rn. Let y ∈ D2h(x0, u), i.e., there are tk → 0+ and uk → usuch that

h(x0 + tkuk) − h(x0) − tkh′(x0)ut2k /2

→ y.

We have that

P :=h(x0 + tkuk) − h(x0) − tkh′(x0)u

t2k /2−

h(x0 + tku) − h(x0) − tkh′(x0)ut2k /2

=h(x0 + tkuk) − h(x0 + tku)

t2k /2.

Lemma 3.2 with a = x0 + tku and b = x0 + tkuk yields, for all k ∈ N large enough, γk ∈ (x0 + tku, x0 + tkuk) such that

∥P∥ ≤ϑtk∥uk − u∥∥γk − x0∥

t2k /2

≤ϑtk∥uk − u∥tk(∥uk∥ + ∥u∥)

t2k /2= 2ϑ∥uk − u∥(∥uk∥ + ∥u∥) → 0,

which implies that y ∈ d2h(x0, u).

Proposition 3.4. Let h : Rn→ Rm be l-stable at x0 ∈ Rn and u, w ∈ Rn.

(i) If wk := (xk − x0 − tku)/ 12 t

2k → w with tk → 0+, then there is y ∈ Dp

2h(x0, u, w) such that (for a subsequence)

yk :=h(xk) − h(x0) − tkh′(x0)u

t2k /2→ y. (3.1)

(ii) Dp2h(x0, u, w) is nonempty and compact.

(iii) If wk := (xk − x0 − tku)/ 12 tkrk → w with (tk, rk) → (0+, 0+) and tk/rk → 0, then

yk :=h(xk) − h(x0) − tkh′(x0)u

tkrk/2→ h′(x0)w.

P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 701

Proof. (i) Suppose wk := (xk − x0 − tku)/ 12 t

2k → w as tk → 0+. Applying Lemma 3.2 to a = x0 and b = xk, we see that, for

all k ∈ N large enough, there are γk ∈ (x0, xk) such that

∥h(xk) − h(x0) − h′(x0)(xk − x0)∥ ≤ ϑ∥xk − x0∥∥γk − x0∥ ≤ ϑ∥xk − x0∥2, (3.2)

which implies thath(xk) − h(x0) − tkh′(x0)ut2k /2

− h′(x0)wk

≤ 2ϑu +

12tkwk

2

. (3.3)

Hence, yk is bounded, and therefore, there exists a subsequence converging to some y ∈ Dp2h(x0, u, w).

(ii) Part (i) shows that Dp2h(x0, u, w) = ∅. Since Dp

2h(x0, u, w) is closed, we prove that Dp2h(x0, u, w) is bounded. Let

y ∈ Dp2h(x0, u, w). Then, yk, defined by (3.1) for some (tk, wk) → (0+, w), tends to y. By the argument in (i), we have (3.3)

and, passing it to limit gives ∥y − h′(x0)w∥ ≤ 2ϑ∥u∥2, i.e., Dp2h(x0, u, w) is bounded.

(iii) Suppose wk := (xk − x0 − tku)/ 12 tkrk → w as (tk, rk) → (0+, 0+) with tk/rk → 0. According to (i), for all k ∈ N large

enough, we have (3.2) for some γk ∈ (x0, xk). Therefore, yk → h′(x0)w sinceh(xk) − h(x0) − tkh′(x0)utkrk/2

− h′(x0)wk

≤ 2ϑtkrk

u +12rkwk

2

→ 0.

Proposition 3.4(i), (ii) sharpens Proposition 2 of [11]. Furthermore, part (i) says,more thanDp2h(x0, u, w) being nonempty,

that any yk defined by (3.1) has a subsequence tending to some point of this derivative. Part (iii) of Proposition 3.4 extendsLemma 3 of [11]. The following fact improves Proposition 3(ii) of [11].

Proposition 3.5. If h : Rn→ Rm is strictly differentiable at x0 ∈ Rn, then, for all u, w ∈ Rn,

Dp2h(x0, u, w) = h′(x0)w + d2h(x0, u).

Proof. For (tk, wk) → (0+, w), set

yk =h(x0 + tku +

12 t

2k wk) − h(x0) − tkh′(x0)u

t2k /2

=h(x0 + tku +

12 t

2k wk) − h(x0 + tku)

t2k /2+

h(x0 + tku) − h(x0) − tkh′(x0)ut2k /2

:= hk + yk.

Using Lebourg’s mean value theorem in Proposition 1.1 (iv) for h and [a, b] = [x0 + tku, x0 + tku +12 t

2k wk], we have

hk ∈ conv(∂h([a, b])wk). The robustness of ∂h implies that hk → h′(x0)w as k → ∞.Now, if y ∈ Dp

2h(x0, u, w), then there exists (tk, wk) → (0+, w) such that yk → y. Hence, yk = yk−hk → y−h′(x0)w := y,i.e., y ∈ d2h(x0, u).

Conversely, if we have a y ∈ d2h(x0, u), then tk → 0+ exists such that yk → y, where yk is defined at the beginning ofthe proof. For wk ≡ w and hk defined above, we have y ∈ Dp

2h(x0, u, w) since

yk := yk + hk → y + h′(x0)w := y.

We have the following direct implication of Propositions 3.3 and 3.5.

Corollary 3.6. Let h : Rn→ Rm and x0, u ∈ Rn.

(i) If h is strictly differentiable at x0 and h′(x0) = 0, then, for w ∈ Rn,

d2h(x0, u) = Dp2h(x0, u, w).

(ii) If h is l-stable at x0 and h′(x0) = 0, then, for w ∈ Rn,

D2h(x0, u) = d2h(x0, u) = Dp2h(x0, u, w).

The following example says that the condition h′(x0) = 0 in Proposition 3.3 is essential, but strict differentiability andl-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 : R2→ R be defined by h(x1, x2) =

12x

21 + x2, x0 = (0, 0), u = (1, 0), and w = (w1, w2) ∈ R2. Then,

h is l-stable (and so strictly differentiable) at x0, h′(x0) = 0. Direct computations give

d2h(x0, u) = 1 = D2h(x0, u) = R, Dp2h(x0, u, w) = 1 + w2 = h′(x0)w + d2h(x0, u).

(b) Let u = 1, x0 = 0, w ∈ R, and h(x) = x2 sin 1x if x = 0, h(0) = 0. Then, h′(x0) = 0, but h is not strictly differentiable

(and so, not l-stable) at x0. However,

d2h(x0, u) = D2h(x0, u) = Dp2h(x0, 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 TechnologyDevelopment of Vietnam (NAFOSTED). The final part of working on the paper was completed during a stay by the authorsas research visitors at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefullyacknowledged. The authors are indebted to the Referees for their valuable remarks and suggestions.

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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.