Contents lists available atSciVerse ScienceDirect Journal of Mathematical Analysis and Applications journal homepage:www.elsevier.com/locate/jmaa Second-order optimality conditions with
Trang 1Contents lists available atSciVerse ScienceDirect 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
aDepartment of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Nam
bDepartment 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 didi-rectional 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 C2scalar 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: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.
Trang 2enough 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 : Rn →Rm , g : Rn → Rpand
h:Rn→Rr 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 =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 those concerning locally Lipschitz functions and second-order tangency In Section2, the notion of l-stability of scalar and vector
functions and some properties are presented Section3is devoted to second-order directional derivatives and properties which will be in use In Part II ([17]), we establish in Section2necessary optimality conditions, with the envelope-like effect, for local weak solutions of (P) Section3contains 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 ∈ Rn : ∥x−y∥ < r}; S n = {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 are normed 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 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 : 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
lim
y→x,t→ 0+
sup
h∈S n
1
=0.
For locally Lipschitz function f :Rn→Rm , 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 inΩ, 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 :Rn→Rm be locally Lipschitz at x Then,
(iii) (robustness) ∂f(x) = {limk→∞vk: vk∈ ∂f(x k),x k→x}, in other words (as∂f(x)is compact), the map∂f(.)is upper
(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) ∈ ∂f(c)(b−a).
Now we recall the notions of tangent cones and second-order tangent sets that we will use later
Trang 3Definition 1.2 Let x0,u∈Rn and M⊂Rn.
(a) The contingent cone of M at x0is
T(M,x0) = {v ∈Rn: ∃t k→0+, ∃vk→ v, ∀k∈N,x0+t kvk∈M}
(b) The interior tangent cone of M at x0is
IT(M,x0) = {v ∈Rn: ∀t k→0+, ∀vk→ v, ∀k large enough,x0+t kvk∈M}
(c) The second-order contingent set of M at(x0,u)is
w ∈Rn: ∃t k→0+, ∃wk→ w, ∀k∈N,x0+t k u+1
2t
2
kwk∈M
(d) The asymptotic second-order tangent cone of M at(x0,u)is
w ∈Rn: ∃ (t k,r k) → (0+,0+) :t k/r k→0, ∃wk→ w, ∀k∈N,x0+t k u+ 1
2t k r kwk∈M
(e) The second-order adjacent set of M at(x0,u)is
w ∈Rn: ∀t k→0+, ∃wk→ w, ∀k∈N,x0+t k u+1
2t
2
kwk∈M
(f) The second-order interior tangent set of M at(x0,u)is
2t
2
kwk∈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⊂Rn , x0∈Rn and u∈Rn Then,
(i)IT2(M,x0,u) ⊂A2(M,x0,u) ⊂T2(M,x0,u) ⊂clcone[cone(M−x0) −u];
(iii)intcone(M−x0) =IT(intM,x0);
(iv)if A2(M,x0,u) ̸= ∅, then
(a)IT2(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 aϑ >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 functionϕ : Rn → R at x in direction u is
defined by
ϕl(x,u) =lim inf
t→ 0+
1
t(ϕ(x+tu) − ϕ(x))
resp, ϕu(x,u) =lim sup
t→ 0+
1
t(ϕ(x+tu) − ϕ(x))
(ii) A functionϕis called l-stable (resp, u-stable) at x if there exist a neighborhood U of x and aϑ >0 such that, for all
Some properties of l-stability ofϕ :Rn→R are summarized in the next proposition
Trang 4Proposition 2.2. (i)[1,3] Any l-stable function is locally Lipschitz and strictly differentiable.
∥ 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 constantϑapplied
Due toProposition 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 functionΦ : Rn → Rm at x in a direction u with
respect to (shortly wrt)ξ∗∈ (Rm)∗is defined by
Φl
ξ ∗(x,u) =lim inf
t→ 0+
1
resp, Φu
ξ ∗(x,u) =lim sup
t→ 0+
1
The following mean value property for continuous vector functions will be needed
Proposition 2.4 ([ 2 ]) LetΦ:Rn→Rm be continuous on an open subset U⊂Rn containing a segment[a,b]andξ∗∈ (Rm)∗
.
Φl
ξ ∗(γ1,b−a) ≤ ⟨ξ∗,Φ(b) −Φ(a)⟩ ≤Φl
ξ ∗(γ2,b−a).
Definition 2.5 LetΦ:Rn→RmandΓ =C∗∩S m∗
(i) [2] Assume that C⊂Rm is a closed, convex and pointed cone with intC ̸= ∅ We say thatΦis l-stable at x in the sense
of Bednařík–Pastor if there are a neighborhood U of x and aϑ >0 such that, for all y∈U, u∈S n, andξ∗∈Γ,
|Φl
ξ ∗(y,u) −Φl
ξ ∗(x,u)| ≤ ϑ∥y−x∥ (ii) [10]Φis said to be l-stable at x in the sense of Ginchev if, for anyξ∗∈ (Rm)∗
, the scalar functionΦξ∗(.) := ⟨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 :Rn→Rm is l-stable at x in the sense of Ginchev if and only if there exist a neighborhood U of x
,
|Φl
ξ ∗(y,u) −Φl
ξ ∗(x,u)| ≤ ϑ∥ξ∗∥∥y−x∥
∥Φ′(y) −Φ′(x)∥ ≤ ϑ∥y−x∥
(iii)[2,10] If functionΦ:Rn→Rm is l-stable at x in the sense of Bednařík–Pastor or Ginchev, thenΦis 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
Trang 5Proof It follows fromProposition 2.6(i) thatΦis l-stable at x in the sense of Ginchev if and only if there are a neighborhood
m,
|Φl
ξ ∗(y,u) −Φl
ξ ∗(x,u)| ≤ ϑ∥y−x∥ Hence, this inequality holds forξ∗∈Γ ⊂S m∗as required by Bednařík–Pastor
Conversely, suppose C is pointed, intC̸= ∅and the last inequality holds forξ∗ ∈Γ Because LinΓ = (Rm)∗
, for every
i∈ {1,2, ,m}, there existξ∗
i, 1, , ξ∗
i,r i∈Γ andαi, 1, , αi,r i ∈R with r i=1, ,m such that e∗i = r i
j= 1αi,jξ∗
i,j, where
i ∈ (Rm)∗is defined by⟨e∗
i,x⟩ =x i , for x= (x1,x2, ,x m)
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,Φl
e∗i(x,u) =Φu
e∗i(x,u) = ⟨e∗i,Φ′(x)u⟩ For every y∈U,u∈S n , and i∈ {1,2, ,m}, one has that
Φl(y,u) −Φl(x,u) =Φl
e∗i(y,u) −Φl
e∗i(x,u)
= lim inf
t→ 0+
e∗i,1
= lim inf
t→ 0+
r i
j= 1
αi,j
ξ∗
i,j,1
≥ −
r i
j= 1
| αi,j||Φl
ξ ∗
i,j(y,u) −Φl
ξ ∗
i,j(x,u)| ≥ −ϑ1∥y−x∥ , whereϑ1=mMϑ Now, by the equivalence of l-stability and u-stability, one has further
Φl
i(y,u) −Φl
i(x,u) ≤ Φu
e∗i(y,u) −Φu
e∗i(x,u)
= lim sup
t→ 0+
e∗i,1
= lim sup
t→ 0+
r i
j= 1
αi,j
ξ∗
i,j,1
≤
r i
j= 1
| αi,j||Φu
ξ ∗
i,j(y,u) −Φu
ξ ∗
i,j(x,u)| ≤ ϑ1∥y−x∥ The obtained two inequalities say thatΦi 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 functionΦis 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 mappingΦ:Rn⇒Rm
Limsupu→uΦ(u) = {y∈Rm: ∃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:Rn→Rm be Fréchet differentiable at x0∈Rn and u, w ∈Rn
(i) [15] The Hadamard second-order directional derivative of h at x0in direction u is
D2h(x0,u) =Limsupv→u,t→ 0+
(ii) [15] The Dini second-order directional derivative of h at x0in direction u is
d2h(x0,u) =Limsupt→0+h(x0+tu) −h(x0) −th′(x0)u
(iii) [11] The parabolic second-order directional derivative of h at x0in direction(u, w)is
D p2h(x0,u, w) =Limsupv→w,t→ 0+
2t2v) −h(x0) −th′(x0)u
Trang 6Note that D p2h(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), D p2h(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 isϑ >0 such that, for all a,b near x0, there existsγ ∈ (a,b)
satisfying
∥h(b) −h(a) −h′(x0)(b−a)∥ ≤ ϑ∥b−a∥∥ γ −x0∥
Proof By the Hahn–Banach theorem, there existsξ∗∈S∗msuch that
∥h(b) −h(a) −h′(x0)(b−a)∥ = ⟨ξ∗,h(b) −h(a) −h′(x0)(b−a)⟩.
Proposition 2.4yieldsγ ∈ (a,b)fulfilling
⟨ ξ∗,h(b) −h(a)⟩ ≤h lξ ∗(γ ,b−a).
Therefore, forϑbeing the l-stability constant of h,
⟨ ξ∗,h(b) −h(a) −h′(x0)(b−a)⟩ ≤h lξ ∗(γ ,b−a) −h lξ ∗(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 ,
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 t k→0+and u k→u
such that
We have that
= h(x0+t k u k) −h(x0+t k u)
Lemma 3.2with a=x0+t k u and b=x0+t k u k yields, for all k∈N large enough,γk∈ (x0+t k u,x0+t k u k)such that
∥P∥ ≤ ϑt k∥u k−u∥∥ γk−x0∥
t k2/2
≤ ϑt k∥u k−u∥t k(∥u k∥ + ∥u∥ )
=2ϑ∥u k−u∥ (∥u k∥ + ∥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:= (x k−x0−t k u)/1
(iii)If wk:= (x k−x0−t k u)/1
2t k r k→ wwith(t k,r k) → (0+,0+)and t k/r k→0, then
′(x0)w.
Trang 7Proof (i) Supposewk:= (x k−x0−t k u)/1
2t k2→ was t k→0+ ApplyingLemma 3.2to a=x0and b=x k, we see that, for
all k∈N large enough, there areγk∈ (x0,x k)such that
∥h(x k) −h(x0) −h′(x0)(x k−x0)∥ ≤ ϑ∥x k−x0∥∥ γk−x0∥ ≤ ϑ∥x k−x0∥2, (3.2) which implies that
′(x0)wk
≤2ϑ
u+1
2t kwk
2
Hence,{y k}is bounded, and therefore, there exists a subsequence converging to some y∈D p2h(x0,u, w)
(ii) Part (i) shows that D p2h(x0,u, w) ̸= ∅ Since D p2h(x0,u, w)is closed, we prove that D p2h(x0,u, w)is bounded Let
y∈D p2h(x0,u, w) Then, y k, defined by(3.1)for some(t k, 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., D p2h(x0,u, w)is bounded
(iii) Supposewk:= (x k−x0−t k u)/1
2t k r k→ was(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 someγk∈ (x0,x k) Therefore, y k→h′(x0)wsince
′(x0)wk
≤2ϑt k
r k
u+1
2r kwk
2
→0.
Proposition 3.4(i), (ii) sharpens Proposition 2 of [11] Furthermore, part (i) says, more than D p2h(x0,u, w)being nonempty,
that any y kdefined by(3.1)has a subsequence tending to some point of this derivative Part (iii) ofProposition 3.4extends Lemma 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 ,
D p2h(x0,u, w) =h′(x0)w +d2h(x0,u).
Proof For(t k, wk) → (0+, w), set
2t2wk) −h(x0) −t k h′(x0)u
t k2/2
= h(x0+t k u+1
2t2wk) −h(x0+t k u)
:= ˆh k+ ˆy k.
Using Lebourg’s mean value theorem inProposition 1.1(iv) for h and[a,b] = [x0+t k u,x0+t k u+ 1
2t k2wk], we have ˆ
h k∈conv(∂h([a,b] )wk) The robustness of∂h implies that hˆk→h′(x0)was k→ ∞
Now, if y∈D p2h(x0,u, w), then there exists(t k, wk) → (0+, w)such that y k→y Hence, yˆk=y k− ˆh k→y−h′(x0)w := ˆy,
i.e.,yˆ ∈d2h(x0,u)
Conversely, if we have ayˆ ∈d2h(x0,u), then t k→0+exists such thatyˆk→ ˆy, where yˆkis defined at the beginning of the proof Forwk≡ wandhˆk defined above, we have y∈D p2h(x0,u, w)since
y k:= ˆy k+ ˆh k→ ˆy+h′(x0)w :=y.
We have the following direct implication ofPropositions 3.3and3.5
Corollary 3.6 Let h:Rn→Rm and x0,u∈Rn
d2h(x0,u) =D p2h(x0,u, w).
D2h(x0,u) =d2h(x0,u) =D p2h(x0,u, w).
The following example says that the condition h′(x0) =0 inProposition 3.3is essential, but strict differentiability and
Trang 8Example 3.1 (a) Let h:R2→R be defined by h(x1,x2) =1
2x2+x2, x0= (0,0),u= (1,0), andw = (w1, w2) ∈R2 Then,
d2h(x0,u) = {1} ̸=D2h(x0,u) =R, D p2h(x0,u, w) = {1+ w2} =h′(x0)w +d2h(x0,u).
(b) Let u=1,x0=0, w ∈R, and h(x) =x2sin1x if x̸=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) =D p2h(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 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 C1 , 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 C1 , 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, http://dx.doi.org/10.1016/j.jmaa.2012.12.075
[18] L Liu, P Neittaanmäki, M Křížek, Second-order optimality conditions for nondominated solutions of multiobjective programming with C1 , 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.