Optimality conditions for vector optimization problems with generalized order

Noname manuscript No.
(will be inserted by the editor)

Optimality conditions for vector optimization
problems with generalized order
N. V. Tuyen

Received: date / Accepted: date

Abstract The aim of this paper is to present new optimality conditions for
vector optimization problems with generalized order by using the extremal
principle.
Keywords Vector optimization · Generalized order · Optimality conditions
Mathematics Subject Classification (2000) 90C29 · 90C46 · 49J53

1 Introduction
Kruger and Mordukhovich [16, Definition 5.53] have introduced the new concept of the locally (f ; Θ)-optimal solution, where f is a single-valued mapping
between Banach spaces and the ordering set Θ (may not be convex and/or
conic) containing the origin. This notion is directly induced by the concept of
local extremal points for systems of sets and covers all the traditional notions
of optimality in vector optimization. To the best of our knowledge, there are a
few works studying the necessary and sufficient optimality conditions for optimality solutions to vector optimization problems with generalized order (see,
e.g., [8, 16, 17]). In [16, Theorem 5.59], Mordukhovich established some preliminary necessary conditions to vector optimization problems with geometric
contraints. Bao [8] used subdifferentials of set-valued mapping to establish
This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2014.39. A part of this work was done when
the author was working at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for providing a fruitful research environment and
working condition.
N. V. Tuyen
Department of Mathematics, Hanoi Pedagogical Institute No. 2, Xuan Hoa, Phuc Yen, Vinh
Phuc, Vietnam.
E-mail:

2

N. V. Tuyen

some necessary conditions to set-valued optimization problems with equilibrium contraints. In [17], Tuyen and Yen established some sufficient conditions
for a point satisfying the necessary optimality condition [16, Theorem 5.59] to
be a generalized order solution of the vector optimization problem under convexity assumptions. Recently, Bao and Mordukhovich [3, 4, 7] gave some new
sufficient conditions for global weak Pareto (global Pareto) solutions to setvalued optimization problems. However, we are not familiar with any sufficient
optimality conditions for nonconvex vector optimization problems with generalized order. This motivates us to study the sufficient optimality conditions
for vector optimization problems with respect to generalized order optimality.
The rest of this paper is organized as follows. Section 2 investigates the
notion of generalized order optimality and preliminaries from variational analysis. In Section 3, we establish some necessary and sufficient conditions for
global generalized solutions in vector optimization.

2 Preliminaries
Let Z be a Banach space and Θ ⊂ Z be a set containing origin. The topological
interior, topological closure, the relative interior and the affine hull of Θ are
denoted respectively by intΘ, cl Θ (or A), ri Θ, and aff Θ. The dual space of
Z is denoted by Z ∗ . The weak∗ topology in Z ∗ is denoted by w∗ . The closed
unit ball in Z is abbreviated to B. Let A ⊂ Z be given arbitrarily. A point z
is said to be a boundary point of A if every neighborhood U of z, we have
U ∩ A = ∅ and U ∩ AC = ∅,
with AC := Z\A. The set of all the boundary points of A denoted by bd (A).
Definition 1 A point z¯ ∈ A is said to be a generalized efficient point of A
with respect to Θ, if there is a sequence {zk } ⊂ Z with zk → 0 as k → ∞
such that
A ∩ (Θ + z¯ − zk ) = ∅, ∀k ∈ N.
(1)

The set of all the generalized efficient points of A with respect to Θ is
denoted by GE(A | Θ).
In Definition 1 we don’t assume that Θ is a convex cone, and we also
don’t require that the interior of Θ is nonempty. If Θ is a convex cone with
riΘ = ∅, then the above optimality concept covers the conventional concept of
optimality.
Definition 2 Suppose that Θ is a convex cone with riΘ = ∅. A point z¯ ∈ A
is said to be a (Slater) relative efficient point of A with respect to the order
generated by cone Θ, if
A ∩ (¯
z + ri Θ) = ∅.
(2)
The set of all the relative efficient points of A is denoted by RE(A | Θ).


Optimality conditions for vector optimization problems with generalized order

3

Let Θ be a convex cone in Z. The cone Θ induces a partial order in Z as
follows: z1 , z2 ∈ Z, z1 ≤ z2 if z1 − z2 ∈ Θ.
Definition 3 Let A be a nonempty subset in Z.
(a) Suppose that intΘ = ∅. A point z¯ ∈ A is said to be a weak efficient point
of A with respect to Θ, if
A ∩ (¯
z + intΘ) = ∅.
The set of weak efficient points of A is denoted by W E(A | Θ).
(b) A point z¯ ∈ A is said to be an efficient point of A with respect to Θ, if
(¯
z ≥ y, for some y ∈ A) ⇒ (y ≥ z¯).

The set of efficient points of A is denoted by E(A | Θ).
Proposition 1 (see [17, Proposition 2.11]) If Θ is a convex cone, then the
following holds:
(i) If intΘ = ∅, then GE(A | Θ) ⊂ W E(A | Θ).
(ii) If intΘ = ∅, then W E(A | Θ) ⊂ RE(A | Θ).
(iii) If riΘ = ∅, then RE(A | Θ) ⊂ GE(A | Θ).
Thus, if Θ is a convex cone with nonempty interior, then
W E(A | Θ) = RE(A | Θ) = GE(A | Θ).

(3)

Proposition 2 (see [11, Proposition 2]) Suppose that Θ is a convex cone with
Θ\l(Θ) = ∅. If z¯ is an efficient point of A with respect to Θ, then z¯ is a
generalized efficient point of A with respect to Θ, or
E(A | Θ) ⊂ GE(A | Θ)

(4)

Proposition 3 (see [11, Theorem 2.3]) Let Z be a Banach space, A be a
nonempty set in Z, and 0 ∈ Θ ⊂ Z. Then
GE(A | Θ) = A ∩ bd (A − Θ).

(5)

Proof Let z¯ ∈ GE(A | Θ). Then, there exists (zk ) ⊂ Z with zk → 0 as
k → ∞ such that z¯ − zk ∈
/ (A − Θ) for all k ∈ N. Thus z¯ − zk ∈ (A − Θ)C for
all k ∈ N. Let U be an arbitrary neighborhood of z¯. From z¯ ∈ A and 0 ∈ Θ
we have z¯ ∈ (A − Θ). Thus U ∩ (A − Θ) = ∅. Since lim (¯
z − zk ) = z¯, we

k→∞

have z¯ − zk ∈ U for large enough k. Thus z¯ − zk ∈ U ∩ (A − Θ)C for large
enough k. It follows that U ∩ (A − Θ)C = ∅. Therefore z¯ ∈ bd (A − Θ). This
shows that GE(A | Θ) ⊂ A ∩ bd (A − Θ). To prove the converse inclusion, let
z¯ ∈ A ∩ bd (A − Θ). Since z¯ ∈ bd (A − Θ), we have
B z¯,

1
k

∩ (A − Θ)C = ∅ ∀k ∈ N,

where B z¯, k1 := z ∈ Z | z − z¯ ≤

1
k

. For each k ∈ N, let xk ∈ B z¯, k1 ∩


4

N. V. Tuyen

(A − Θ)C . We have lim xk = z¯ and {xk } ⊂ (A − Θ)C . For each k ∈ N, put
k→∞

zk = z¯ − xk , then
lim zk = 0 and z¯ − zk = xk ∈ (A − Θ)C ∀k ∈ N,

k→∞

or z¯ − zk ∈
/ (A − Θ) ∀k ∈ N. This shows that z¯ is a generalized efficient point
of A with respect to Θ. Therefore A ∩ bd (A − Θ) ⊂ GE(A | Θ). The proof is
complete.
✷
Next, we recall some basic notions that will be used later. Let F : Z ⇒ Z ∗ be
a multifunction from Z to Z ∗ . The Painlev´e-Kuratowski upper limit at z¯ of
F with respect to the norm topology of Z and the weak∗ topology of Z ∗ is
defined by
w∗

Lim sup F (z) := {z ∗ ∈ Z ∗ | ∃ zk → z¯, zk∗ −→ z ∗ , zk∗ ∈ F (zk ) ∀k ∈ N}.
z→¯
z

Definition 4 (see [15, Difinition 1.1]) Let Ω ⊂ Z, z¯ ∈ Ω, and
(i) The set of -normals of Ω at z¯ is defined by
ˆ (¯
N
z ; Ω) :=

z ∗ ∈ Z ∗ | lim sup
Ω

z →¯
z

z ∗ , z − z¯
z − z¯

0.

,

Ω

where the notation z −→ z¯ means that z → z¯ and z ∈ Ω. We call the closed
ˆ (¯
ˆ0 (¯
convex cone N
z ; Ω) := N
z ; Ω) the Fr´echet normal cone of Ω at z¯.
(ii) The Mordukhovich normal cone or the limiting normal cone of Ω at z¯ is
the set
N (¯
z , Ω) = Lim sup N (x, Ω),
z→¯
z
↓0

that is,
N (¯
z ; Ω) = z ∗ ∈ Z ∗ | ∃

∗

k

Ω
w
ˆ (zk ; Ω) ∀k .
→ 0+ , zk → z¯, zk∗ → z ∗ , zk∗ ∈ N
k

(6)
Definition 5 (see [15, Difinition 1.20]) A set Ω ⊂ Z is said to be sequentially
Ω
normally compact (SNC) at z¯ if for any sequences k ↓ 0, zk −→ z¯ and zk∗ ∈
ˆ (zk , Ω) it holds
N
k
w∗

zk∗ −−→ 0 as k → 0 =⇒
where

k

zk∗ → 0 as k → 0 ,

can be omitted if Z is Asplund and Ω is locally closed around z¯.

Recall that a Banach space is Asplund if every convex continuous function
ϕ : U → R defined on an open convex subset U of Z is Fr´echet differentiable
on a dense subset of U . The class of Asplund spaces is quite broad including
every reflexive Banach space and every Banach space with a separable dual
are Asplund spaces. It is known from [15, Theorem 1.21] that a convex set

Optimality conditions for vector optimization problems with generalized order

5

is fulfilled the SNC condition if it has a nonempty relative interior and the
closure of affine hull of Θ has a finite-codimensions. The codimension of aff Θ
is defined as the dimension of the quotient space X/(aff Θ −θ), for some θ ∈ Θ,
and is denoted by codim aff Θ.
Definition 6 (see [15, Definition 2.1]) Let Ω1 , Ω2 be nonempty subsets of a
Banach space Z and z¯ ∈ Ω1 ∩ Ω2 . We say that z¯ is a global extremal point of
the system {Ω1 , Ω2 } in Z if there exists a sequence (ak ) such that ak → 0 as
k → ∞ and
Ω1 ∩ (Ω2 − ak ) = ∅ ∀k ∈ N.
(7)
In this case {Ω1 , Ω2 , z¯} is said to be an extremal system in Z.
Theorem 1 (The Extremal Principle, see [15, Theorem 2.20]) If z¯ is an
extremal point of the closed set system {Ω1 , Ω2 } in the Asplund space Z, then
it satisfies the following relationships: for every > 0, there are xi ∈ Z and
x∗i ∈ Z ∗ satisfying
ˆ (xi , Ωi ) f or i = 1, 2,
xi ∈ Ωi ∩ (¯
z + B), x∗i ∈ N
x∗1 + x∗2 ≤ , and 1 − ≤ x∗1 + x∗2 ≤ 1 + .

3 Main results
Theorem 2 (Necessary condition) Let Z be an Asplund space, and let
∅ = A ⊂ Z, 0 ∈ Θ ⊂ Z be closed subsets. If z¯ ∈ GE(A | Θ), then there exists
0 = z ∗ ∈ Z ∗ such that

−z ∗ ∈ N (¯
z , A) ∩ (−N (0; Θ))

(8)

provided that either A is SNC at z¯ or Θ is SNC at 0.
Proof Put Ω1 := Z × (A − Θ), Ω2 := A × {¯
z }. Obviously, (¯
z , z¯) ∈ Ω1 ∩ Ω2 . We
claim that {Ω1 , Ω2 , (¯
z , z¯)} is an extremal system. Indeed, from z¯ ∈ GE(A | Θ)
it follows that there exists a sequence (zk ) ⊂ Z such that zk → 0, and
A ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N,
or, equivalent to
(A − Θ) ∩ (¯
z − zk ) = ∅ ∀k ∈ N.

(9)

Put ak := (0, zk ). We have ak → 0 as k → ∞ and
Ω1 ∩ (Ω2 − ak ) = ∅ ∀k ∈ N.

(10)

Arguing by contradiction, suppose that (10) does not hold for some k0 ∈ N.
This mean that
Ω1 ∩ (Ω2 − ak0 ) = ∅.


6

N. V. Tuyen

Then, there exist a ∈ A and θ ∈ Θ such that
a − θ = z¯ − zk0 ,
contrary to (9). Thus {Ω1 , Ω2 , (¯
z , z¯)} is an extremal system. Employing the
extremal principle from Theorem 1 to the system {Ω1 , Ω2 , (¯
z , z¯)} shows that for
∗
each k ∈ N, there are elements (uik , zik ) and (u∗ik , −zik
) for i = 1, 2 satisfying
the relationships

z , z¯),

(uik , zik ) ∈ Ωi with (uik , zik ) → (¯

 and (u∗ , −z ∗ ) ∈ N
ˆ ((uik , zik ), Ωi ) , i = 1, 2,
ik
ik
(11)
∗
∗
 with (u∗1k , z1k
) + (u∗2k , z2k
) → 1,




∗
∗
) + (u∗2k , z2k
) → 0.
and (u∗1k , z1k
From
∗
ˆ ((u2k , z2k ), Ω2 )
(u∗2k , −z2k
)∈N
ˆ ((u2k , z2k ), A × {¯
∈N
z })

ˆ (u2k , A) × N
ˆ (¯
=N
z , {z})
∗
ˆ
= N (u2k , A) × Z

(12)

ˆ (u2k , A). Since
it follows that u∗2k ∈ N
∗
ˆ ((u1k , z1k ), Ω1 )
(u∗1k , −z1k

)∈N
ˆ ((u1k , z1k ), Z × (A − Θ)),
=N

(13)

we have
∗
∗
ˆ ((u1k , 0), Z × (−Θ))
(u∗1k − z1k
, −z1k
)∈N
ˆ (u1k , Z) × N
ˆ (0, −Θ)
=N

ˆ (0, Θ)).
= {0} × (−N

(14)

∗
∗
ˆ (0, Θ). The Asplund property of
Hence, we get u∗1k − z1k
= 0 and z1k
∈ N
∗
∗

∗
, u∗2k , z2k
) in (11) allow us to
Z and the boundedness of the sequence (u1k , z1k
∗ ∗
∗ ∗
find a quadruple (u1 , z1 , u2 , z2 ) such that
w∗

∗
(u∗ik , −zik
) → (u∗i , −zi∗ )

for i = 1, 2 along some subsequences. Employing (11)–(14)gives us
(u∗1 , −z1∗ ) = (−u∗2 , z2∗ ) = (z ∗ , −z ∗ ), −z ∗ ∈ N (¯
z , A) and z ∗ ∈ N (0, Θ).
To complete the proof of the theorem, it remains to show that z ∗ = 0 in (8)
under assumeed SNC. Arguing by contradiction, suppose that z ∗ = 0. The
imposed SNC assumptions give us two cases:


Optimality conditions for vector optimization problems with generalized order

7

∗

ˆ (u2k , A), u2k → z¯ and u∗ w
Case1 : A is SNC at z¯. From u∗2k ∈ N
2k → 0 it follows

that u∗2k → 0. Since
u∗1k ≤ u∗1k + u∗2k + u∗2k
∗
∗
≤ (u∗1k , z1k
) + (u∗2k , z2k
) + u∗2k

for all k ∈ N,

(15)
(16)

∗
∗
we have u∗1k → 0 as k → ∞. Thus z1k
→ 0 and z2k
→ 0. This contradicts the nontriviality in the third line of (11).
∗
∗
ˆ (0, Θ) for all k ∈ N,
Case2 : Θ is SNC at 0. From u∗1k = z1k
, z1k
∈ N
w∗

∗
∗
and z1k
→ 0 imply that z1k

→ 0 and u∗1k
∗
∗
(u2k , z2k ) → 0, a contrary again.

→ 0. Since (11), we have
✷

Corollary 1 Let Z be an Asplund space, and let ∅ = A ⊂ Z be a closed subset.
Suppose that Θ is a closed convex cone and ri Θ = ∅. If z¯ ∈ RE(A | Θ) then
there exists 0 = z ∗ ∈ Z ∗ such that
−z ∗ ∈ N (¯
z , A) ∩ (−N (0; Θ))

(17)

provided that either A is SNC at z¯ or codim aff Θ < ∞.
Corollary 2 Let Z be an Asplund space, and let ∅ = A ⊂ Z be a closed subset.
Suppose that Θ is a closed convex cone and intΘ = ∅. If z¯ ∈ W E(A | Θ) then
there exists 0 = z ∗ ∈ Z ∗ such that
−z ∗ ∈ N (¯
z , A) ∩ (−N (0; Θ))

(18)

provided that either A is SNC at z¯ or codim aff Θ < ∞.
Corollary 3 Let Z be an Asplund space, and let ∅ = A ⊂ Z be a closed subset.
Suppose that Θ is a closed convex cone and Θ\l(Θ) = ∅. If z¯ ∈ E(A | Θ) then
there exists 0 = z ∗ ∈ Z ∗ such that
−z ∗ ∈ N (¯

z , A) ∩ (−N (0; Θ))

(19)

provided that either A is SNC at z¯ or Θ is SNC at 0.
Proof We have E(A | Θ) ⊂ GE(A | Θ) by Proposition 2. Assertion (19) is immediate from Theorem 2.
✷
Corollary 4 Let Z be an Asplund space and ∅ = A ⊂ Z, 0 ∈ Θ ⊂ Z,
z¯ ∈ GE(A | Θ). Suppose that A − Θ and Θ are closed subsets and the following
condition
Θ+Θ =Θ
(20)
holds true. Then, there exists z ∗ ∈ Z ∗ such that
0 = −z ∗ ∈ N (¯
z , A − Θ) ∩ (−N (0, Θ))
provided that either A − Θ is SNC at z¯ or Θ is SNC at 0.

(21)


8

N. V. Tuyen

Proof We first show that if the condition (20) is satisfied, then
GE(A | Θ) ⊂ GE(A − Θ | Θ).

(22)

Indeed, suppose that z¯ ∈ GE(A | Θ). Then, there exists a sequence (zk ) ∈ Z

such that zk → 0 as k → ∞ and
A ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N.
From this and (20) imply that
A ∩ (Θ + Θ + z¯ − zk ) = ∅ ∀k ∈ N,
or, equivalent to
(A − Θ) ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N.
Thus z¯ ∈ GE(A − Θ | Θ). By Theorem 2, there exists 0 = z ∗ ∈ Z ∗ such that
−z ∗ ∈ N (¯
z , A − Θ) ∩ (−N (0, Θ)) .
✷

The proof is complete.

Theorem 3 (Necessary and sufficient condition) Let Z be an Asplund
space, ∅ = A ⊂ Z, 0 ∈ Θ ⊂ Z and z¯ ∈ A. Assume that A − Θ is a closed subset
in Z and either A − Θ is SNC at z¯ or dim Z < ∞. Then, z¯ ∈ GE(A | Θ) if
and only if there exists z ∗ ∈ Z ∗ satisfying
0 = −z ∗ ∈ N (¯
z , A − Θ) ∩ (−N (0, Θ)) .

(23)

Proof (⇒): Suppose that z¯ ∈ GE(A | Θ). Then, there exists a sequence zk → 0
as k → ∞ satisfying
A ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N,
or
(A − Θ) ∩ (¯
z − zk ) = ∅ ∀k ∈ N.

(24)

The equation (24) implies that {A − Θ, {¯
z }, z¯} is an extremal system in Z.
By [15, Theorem 1.21], {¯
z } is SNC if and only if dim Z < ∞. Thus A − Θ or
{¯
z } is SNC at z¯. Clearly, the singleton set {¯
z } is a closed subset in Z. Theorem
2.22 [15] now shows that the exact extremal principle holds for {A−Θ, {¯
z }, z¯}.
Thus there exists 0 = z ∗ ∈ Z ∗ such that
−z ∗ ∈ N (¯
z , A − Θ) ∩ N (¯
z , {¯
z }),

(25)

−z ∗ ∈ N (¯
z , A − Θ).

(26)

or, equivalent to
As in the proof of [5, Theorem 3.1], equation (26) gives
−z ∗ ∈ (−N (0, Θ)).


Optimality conditions for vector optimization problems with generalized order

9

Hence, there exists z ∗ ∈ Z ∗ such that
0 = −z ∗ ∈ N (¯
z , A − Θ) ∩ (−N (0, Θ)) .
(⇐): Arguing by contradiction, assume that there is z ∗ ∈ Z ∗ such that
0 = −z ∗ ∈ N (¯
z , A − Θ) ∩ (−N (0, Θ)) ,

(27)

but z¯ ∈
/ GE(A | Θ). Therefore z¯ ∈
/ bd (A − Θ) by Lemma 3. From this and
z¯ ∈ (A − Θ) imply that z¯ ∈ int(A − Θ). Thus N (¯
z , A − Θ) = {0}, contrary to
(27). The proof is complete.
✷
Example 1 Let Z = R2 , A = {z = (z1 , z2 ) ∈ R2 | z2 = −z1 , 0 ≤ z1 ≤ 1},
Θ = {z = (z1 , z2 ) ∈ R2 | z2 = −z1 , z1 ≤ 0} ∪ {z = (z1 , z2 ) ∈ R2 | z2 ≤
− |z1 | , −1 ≤ z1 ≤ 1}, and z¯ = (0, 0) ∈ A. It is easy to see that Θ is neither
convex nor conic. We have
A − Θ = {z ∈ R2 | z2 ≥ |z1 |, −1 ≤ z1 ≤ 1}
∪ {z ∈ R2 | z1 − 2 ≤ z2 ≤ z1 , z1 ≤ 1}
∪ {z ∈ R2 | z1 ≤ −|z2 | + 2, 1 ≤ z1 ≤ 2}.
From this we obtain
N (¯
z ; A − Θ) = {z = (z1 , z2 ) ∈ R2 | z1 = −|z2 |},
and
N (¯

z ; Θ) = {z = (z1 , z2 ) ∈ R2 | z2 = z1 }
∪ {z = (z1 , z2 ) ∈ R2 | z2 = −z1 , z1 ≤ 0}.
We have N (¯
z , A − Θ) ∩ (−N (0, Θ)) = {z = (z1 , z2 ) ∈ R2 | z1 = −|z2 |}. Thus
z¯ ∈ GE(A | Θ) by Theorem 3.
Now we compare Theorem 3 with [16, Theorem 5. 89], which characterizes
the linear suboptimality of set systems via the relations of the exact extremal
principle. Given two subsets Ω1 and Ω2 of a Banach space Z. Put
ϑ(Ω1 , Ω2 ) := sup{υ ≥ 0 | υB ⊂ Ω1 − Ω2 }.

(28)

The constant ϑ(Ω1 , Ω2 ) describing the measure of overlapping for these sets
Ω1 and Ω2 . Note that one has ϑ(Ω1 , Ω2 ) = −∞ if Ω1 ∩ Ω2 = ∅.
It is easy to observe that a point z¯ ∈ Ω1 ∩ Ω2 is locally extremal for the
set system {Ω1 , Ω2 } if and only if
ϑ(Ω1 ∩ Br (¯
z ), Ω2 ∩ Br (¯
z )) = 0 for some r > 0.

(29)


10

N. V. Tuyen

Definition 7 (see [16, Definition 5.87]) Given Ω1 , Ω2 ⊂ Z and z¯ ∈ Ω1 ∩ Ω2 .
We say that the set system {Ω1 , Ω2 } is linearly subextremal around the point
z¯ if ϑlin (Ω1 , Ω2 , x

¯) = 0, where
ϑlin (Ω1 , Ω2 , z¯) := lim inf
Ω

1
x1 −→¯
z

ϑ ([Ω1 − x1 ] ∩ rB, [Ω2 − x2 ] ∩ rB)
,
r

(30)

Ω

2
x2 −→¯
z
r↓0

where the measure of overlapping ϑ(·, ·) is defined in (28).
The next result characterizes the linear suboptimality of set systems.
Theorem 4 (see [16, Theorem 5. 89]) Let Ω1 and Ω2 be two subsets in a
Asplund space Z. Assume that {Ω1 , Ω2 } ⊂ Z is a linearly subextremal around
z¯ ∈ Ω1 ∩ Ω2 , that the sets Ω1 , Ω2 are locally closed around z¯, and that one of
them is SNC at this point. Then there is z ∗ ∈ Z ∗ satisfying
0 = z ∗ ∈ N (¯
z ; Ω1 ) ∩ (−N (¯
z ; Ω2 )) .

(31)

Furthermore, condition (31) is necessary and sufficient for the linear subextremality of {Ω1 , Ω2 } around z¯ if dim Z < ∞.
The following examples show that the sufficient condition in Theorem 4 is not
sufficient for an extremal system even in the finite dimensional case.
Example 2 Let X = R2 , A = (1, 0) + B, Θ = (−2, 0) + 2B, z¯ = (0, 0). Clearly,
Θ is not a cone in R2 . An easy computation shows that
N ((0, 0); A) = R− × {0}, N ((0, 0); Θ) = R+ × {0}.
This implies that
N ((0, 0); A) ∩

− N ((0, 0); Θ) = R− × {0} = {0}.

Thus the system {A, Θ} is linearly subextremal around the point z¯ by Theorem
4. However, z¯ is not an extremal point of the system {A, Θ}. Indeed, we have
A − Θ = (1, 0) + 3B. Thus N (¯
z , A − Θ) = {0}. Theorem 3 now shows that z¯
is not an extremal point of the system {A, Θ}.
Example 3 Let Z = R2 , A = {z = (z1 , z2 ) ∈ R2 | z1 = 0, −1 ≤ z2 ≤ 0}, and
Θ = {z = (z1 , z2 ) ∈ R2 | z2 = z1 } ∪ {z = (z1 , z2 ) ∈ R2 | z2 = −z1 } ∪ {z =
(z1 , z2 ) ∈ R2 | z2 < − |z1 |}. It is easy to see that Θ is a nonconvex cone, and
z¯ = (0, 0) ∈ A ∩ Θ. We have
N (¯
z ; A) = {z = (z1 , z2 ) ∈ R2 | z2 ≥ 0},
and
N (¯
z ; Θ) = {z = (z1 , z2 ) ∈ R2 | z2 = z1 } ∪ {z = (z1 , z2 ) ∈ R2 | z2 = −z1 }.

Optimality conditions for vector optimization problems with generalized order

11

Thus N (¯
z ; A) ∩ (−N (¯
z ; Θ)) = {z = (z1 , z2 ) ∈ R2 | z2 = |z1 |}. By Theorem
4, {A, Θ} is linearly subextremal around the point z¯. However, z¯ is not an
extremal point of the system {A, Θ}. Indeed, we have
A−Θ = {z ∈ R2 | z2 = z1 −1}∪{z ∈ R2 | z2 = −z1 −1}∪{z ∈ R2 | z2 > |z1 |−1}.
It is easy to check that z¯ ∈ int(A − Θ). This implies that N (¯
z , A − Θ) = {0}.
Thus z¯ ∈
/ GE(A | Θ) by Theorem 3.
Theorem 5 (Sufficient condition under convexity assumption, see [17, Theorem 4.3]) Let Z be a Banach space, ∅ = A ⊂ Z, 0 ∈ Θ ⊂ Z. Suppose that
Θ, A − Θ are convex subsets and intΘ = ∅. If there exists 0 = z ∗ ∈ Z ∗ such
that
−z ∗ ∈ N (¯
z , A − Θ),
(32)
then z¯ ∈ GE(A | Θ).
Proof Since A − Θ is a convex set in Z, the Mordukhovich normal cone of
A − Θ at z¯ coincides with the normal cone of A − Θ in the sense of convex
analysis; that is, −z ∗ ∈ N (¯
z , A − Θ) if and only if
z ∗ , z − z¯ ≥ 0 ∀z ∈ A − Θ.

(33)

From definition of the Mordukhovich normal cone and (33) imply that z ∗ ∈

N (0; Θ). We have to prove that z¯ is a generalized efficient point of A with
respect to Θ. Arguing by contradiction, assume that z¯ ∈
/ GE(A | Θ). Since
z0
for all k ∈ N. Since zk → 0 as
intΘ = ∅, let z0 ∈ intΘ and put zk := −
k
k → ∞ and z¯ ∈
/ GE(A | Θ) imply that there exists k0 ∈ N such that
A ∩ (Θ + z¯ − zk0 ) = ∅.
Thus there are z ∈ A and θ ∈ Θ satisfying a = θ + z¯ − zk0 . Put z := a − θ, we
have z ∈ A − Θ and
z0
(34)
z − z¯ = .
k0
Substituting (34) into the left side of (33) we obtain
z ∗ , z0 ≥ 0.

(35)

Since z0 ∈ intΘ, it follows that there exists > 0 such that B(z0 , ) ⊂ Θ,
where B(z0 , ) := {z ∈ Z | z − z0 ≤ }. From z ∗ ∈ N (0, Θ) and the convexity
of Θ imply that
z ∗ , z0 + v ≤ 0 ∀v ∈ B.
This means that
z ∗ , z0 + z ∗ , v ≤ 0 ∀v ∈ B.
Thus
z ∗ , z0 + sup z ∗ , v ≤ 0,
v∈B

or
z ∗ , z0 + z ∗ ≤ 0,
which contradicts the fact that z ∗ , z0 ≥ 0 and z ∗ = 0. Hence z¯ ∈ GE(A | Θ).
The proof is complete.
✷


12

N. V. Tuyen

References
1. Bao, T. Q. and MordukhovichB. S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Math. Program. 122 (2010), 101–138.
2. Bao, T. Q. and MordukhovichB. S.: Extended Pareto optimality in multiobjective problems, Chapter 13 of the book Recent Advances in Vector Optimization (Q. H. Ansari and
J.-C. Yao, eds.), pp. 467-516, Springer, Berlin, 2011.
3. Bao, T. Q. and MordukhovichB. S.: Sufficient conditions for global weak Pareto solutions
in multiobjective optimization, Positivity 16 (2012), 579–602.
4. Bao, T. Q. and MordukhovichB. S.: To dual-space theory of set-valued optimization,
Vietnam J. Math. 40 (2012), 131–163.
5. Bao, T. Q., Tammer, C.: Lagrange necessary conditions for Pareto minimizers in Asplund
spaces and applications, Nonlinear Anal. 75 (2012), 1089–1103.
6. Bao, T. Q. and MordukhovichB. S.: Necessary nondomination conditions in set and
vector optimization with variable ordering structures, J. Optim. Theory Appl. 162 (2014),
350–370.
7. Bao, T. Q. and MordukhovichB. S.: Sufficient optimality conditions for global Pareto
solutions to multiobjective problems with equilibrium constraints, J. Nonlinear Convex
Anal. 15 (2014), 105–127.
8. Bao, T. Q.: Subdifferential necessary conditions in set-valued optimization problems with
equilibrium constraints, Optimization 63 (2014), 181–205.

9. Bao, T. Q., Pattanaik, S. R.: Necessary conditions for εe-minimizers in vector optimization with empty interior ordering sets, Optimization (2014), DOI:
10.1080/02331934.2014.926358.
10. Ha, T. X. D.: Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems, Nonlinear
Anal. 75 (2012), 1305–1323.
11. Huy, N. Q., Kim, D. S., Tuyen, N. V.: Existence theorems in vector optimization with
generalized order, submited.
12. Kruger, A. Y.: Generalized differentials of nonsmooth functions and necessary conditions
for an extremum, Siberian Math. J. 26 (1985), 370–379.
13. Luc, D. T.: Theory of Vector Optimization, Lecture Notes in Econom. and Math. Systems 319, Springer-Verlag, Berlin, Heidelberg, 1989.
14. Mordukhovich, B.S.: An abstract extremal principle with applications to welfare economics, J. Math. Anal. Appl. 251(2000), 187–216.
15. Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic
Theory, Springer, Berlin, 2006.
16. Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, Vol. II:
Applications, Springer, Berlin, 2006.
17. Tuyen, N. V., Yen, N. D.: On the concept of generalized order optimality, Nonlinear
Analysis 75 (2012), 1592–1601.