File 4 a k-tung 2 radial na printed 12-11

Nonlinear Analysis 74 (2011) 7365–7379
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
Higher-order radial derivatives and optimality conditions in nonsmooth
vector optimization
Nguyen Le Hoang Anha
, Phan Quoc Khanhb
, Le Thanh Tungc,∗
a
Department of Mathematics, University of Science of Hochiminh City, 227 Nguyen Van Cu, District 5, Hochiminh City, Viet Nam
b
Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Nam
c
Department of Mathematics, College of Science, Cantho University, Ninhkieu District, Cantho City, Viet Nam
a r t i c l e i n f o
Article history:
Received 18 February 2011
Accepted 27 July 2011
Communicated by Ravi Agarwal
MSC:
90C46
49J52
46G05
90C26
90C29
Keywords:
Higher-order outer and inner radial
derivatives
Calculus rules
Q -minimality
Ideal and weak efficiency
Various kinds of proper efficiency
Higher-order optimality conditions
Set-valued vector optimization
a b s t r a c t
We propose notions of higher-order outer and inner radial derivatives of set-valued maps
and obtain main calculus rules. Some direct applications of these rules in proving optimality
conditions for particular optimization problems are provided. Then we establish higher-
order optimality necessary conditions and sufficient ones for a general set-valued vector
optimization problem with inequality constraints. A number of examples illustrate both the
calculus rules and the optimality conditions. In particular, they explain some advantages
of our results over earlier existing ones and why we need higher-order radial derivatives.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries
In nonsmooth optimization, a large number of generalized derivatives have been introduced to replace the classical
Fréchet and Gateaux derivatives to meet the continually increasing diversity of practical problems. We can recognize roughly
two approaches: primal space and dual space approaches. Coderivatives, limiting subdifferentials and other notions in the
dual space approach enjoy rich and fruitful calculus rules and little depend on convexity assumptions in applications; see
e.g. the excellent book [1,2]. The primal space approach has been more developed so far, partially since it is more natural and
exhibits clear geometrical interpretations. Most generalized derivatives in this approach are based on linear approximations
and kinds of tangency. Hence, approximating cones play crucial roles. One of the early and most important notions are the
contingent cone and the corresponding contingent derivative; see [3,4]. For a subset A of a normed space X, the contingent
cone of A at ¯x ∈ cl A is
TA(¯x) = {u ∈ X : ∃tn → 0+
, ∃un → u, ∀n, ¯x + tnun ∈ A}.
∗ Corresponding author. Tel.: +84 909353482.
E-mail addresses: nlhanh@hcmuns.edu.vn (N.L.H. Anh), pqkhanh@hcmiu.edu.vn (P.Q. Khanh), lttung@ctu.edu.vn, lttungct@gmail.com (L.T. Tung).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2011.07.055

7366 N.L.H. Anh et al. / Nonlinear Analysis 74 (2011) 7365–7379
However, they capture only the local nature of sets and mappings and are suitable mainly for convex problems. The (closed)
radial cone of A at ¯x ∈ cl A is defined by
RA(¯x) = cone(A − ¯x) = {u ∈ X : ∃tn > 0, ∃un → u, ∀n, ¯x + tnun ∈ A}
and carries global information about A. We have TA(¯x) ⊆ RA(¯x) and this becomes equality if A is convex (in fact, we need
A being only star-shape at ¯x). Hence, the corresponding radial derivative, first proposed in [5], is proved to be applicable to
nonconvex problems and global optimal solutions. In [6,7], radial epiderivatives were introduced, taking some advantages
of other kinds of epiderivatives; see e.g. [4,8]. A modified definition was included in [9], making the radial epiderivative a
notion exactly corresponding to the contingent epiderivative defined in [4,8], to avoid some restrictive assumptions imposed
in [6,7]. Radial epiderivatives were applied in [10] to get optimality conditions for strict minimizers.
To obtain more information for optimal solutions, higher-order (generalized) derivatives and higher-order optimality
conditions have been intensively developed recently; see e.g. [11–15]. However, such contributions are still much fewer
than the first and second-order considerations. Of course, only a number of generalized derivatives may have higher-order
generalizations. As far as we know, the radial derivative has not had higher-order extensions so far. This is a motivation for
our present work.
To meet various practical situations, many optimality (often known also as efficiency) notions have been introduced and
developed in vector optimization. Each above-mentioned paper dealt with only several kinds of optimality. There were also
attempts to classify solution notions in vector optimization. The Q -minimality proposed in [16] subsumes various types of
efficiency, from weak and ideal solutions to many properly efficient solutions. Hence, when applying higher-order radial
derivatives to establish optimality conditions, we start with Q -minimal solutions and then derive results for many other
kinds of efficiency.
The layout of the paper is as follows. In the rest of this section, we recall some definitions and preliminaries for our later
use. Section 2 includes definitions of higher-order outer and inner radial derivatives of set-valued mappings and their main
calculus rules. Some illustrative direct applications of these rules for obtaining optimality conditions in particular problems
are provided by the end of this section. The last section is devoted for establishing higher-order optimality conditions, in
terms of radial derivatives, in a general set-valued vector optimization problem.
In the sequel, let X, Y and Z be normed spaces, C ⊆ Y and D ⊆ Z be pointed closed convex cones with nonempty
interior. BX , BY stands for the closed unit ball in X, Y, respectively. For A ⊆ X, intA, cl A, bdA denote its interior, closure
and boundary, respectively. Furthermore, coneA = {λa | λ ≥ 0, a ∈ A}. For a cone C ⊆ Y, we define:
C∗
= {y∗
∈ Y∗
| ⟨y∗
, c⟩ ≥ 0, ∀c ∈ C},
C∗i
= {y∗
∈ Y∗
| ⟨y∗
, c⟩ > 0, ∀c ∈ C {0}}
and, for u ∈ X, C(u) = cone(C + u). A convex set B ⊂ Y is called a base for C if 0 ̸∈ clB and C = {tb : t ∈ R+, b ∈ B}. For
H : X → 2Y
, the domain, graph and epigraph of H are defined by
dom H = {x ∈ X : H(x) ̸= ∅}, gr H = {(x, y) ∈ X × Y : y ∈ H(x)},
epi H = {(x, y) ∈ X × Y : y ∈ H(x) + C}.
Throughout the rest of this section, let A be a nonempty subset of Y and a0 ∈ A. The main concept in vector optimization
is Pareto efficiency. Recall that a0 is a Pareto minimal point of A with respect to (w.r.t.) C(a0 ∈ Min(A, C)) if
(A − a0) ∩ (−C {0}) = ∅.
In this paper, we are concerned also with the following other concepts of efficiency.
Definition 1.1. (i) a0 is a strong (or ideal) efficient point of A (a0 ∈ StrMin(A, C)) if A − a0 ⊆ C.
(ii) Supposing that int C ̸= ∅, a0 is a weak efficient point of A (a ∈ WMin(A, C)) if (A − a0) ∩ (−int C) = ∅.
(iii) Supposing that C+i
̸= ∅, a0 is a positive-properly efficient point of A (a0 ∈ Pos(A, C)) if there exists ϕ ∈ C+i
such that
ϕ(a) ≥ ϕ(a0) for all a ∈ A.
(iv) a0 is a Geoffrion-properly efficient point of A (a ∈ Ge(A, C)) if a0 ∈ Min(A, C) and there exists a constant M > 0 such
that, whenever there is λ ∈ C+
with norm one and λ(a − a0) > 0 for some a ∈ A, one can find µ ∈ C+
with norm one
such that
⟨λ, a − a0⟩ ≤ M⟨µ, a0 − a⟩.
(v) a0 is a Borwein-properly efficient point of A (a ∈ Bo(A, C)) if
clcone(A − a0) ∩ (−C) = {0}.
(vi) a0 is a Henig-properly efficient point of A (a0 ∈ He(A, C)) if there exists a convex cone K with C {0} ⊆ intK such
that (A − a0) ∩ (−intK) = ∅.
(vii) Supposing that C has a base B, a0 is a strong Henig-properly efficient point of A (a0 ∈ StrHe(A, B)) if there is a scalar
ϵ > 0 such that
clcone(A − a0) ∩ (−clcone(B + ϵBY )) = {0}.

N.L.H. Anh et al. / Nonlinear Analysis 74 (2011) 7365–7379 7367
(viii) a0 is a super efficient point of A (a0 ∈ Su(A, C)) if there is a scalar ρ > 0 such that
clcone(A − a0) ∩ (BY − C) ⊆ ρBY .
Note that Geoffrion originally defined the properness notion in (iv) for Rn
with the ordering cone Rn
+. Hartley extended
it to the case of arbitrary convex ordering cone. The above general definition of Geoffrion properness is taken from [17].
For relations of the above notions and also other kinds of efficiency, see e.g. [16–20]. Some of them are collected in the
proposition below.
Proposition 1.1. (i) StrMin(A) ⊆ Min(A) ⊆ WMin(A).
(ii) Pos(A) ⊆ He(A) ⊆ Min(A).
(iii) Su(A) ⊆ Ge(A) ⊆ Bo(A) ⊆ Min(A).
(iv) Su(A) ⊆ He(A).
(v) Su(A) ⊆ StrHe(A) and if C has a bounded base then Su(A) = StrHe(A).
From now on, unless otherwise specified, let Q ⊆ Y be an arbitrary nonempty open cone, different from Y.
Definition 1.2 ([16]). We say that a0 is a Q -minimal point of A (a0 ∈ Qmin(A)) if
(A − a0) ∩ (−Q ) = ∅.
Recall that an open cone in Y is said to be a dilating cone (or a dilation) of C, or dilating C if it contains C {0}. Let B be
as before a base of C. Setting
δ = inf{‖b‖ : b ∈ B} > 0,
for each 0 < ϵ < δ, we associate to C a pointed open convex cone Cϵ(B), defined by
Cϵ(B) = cone(B + ϵBY ).
For each ϵ > 0, we also associate to C another open cone C(ϵ) defined as
C(ϵ) = {y ∈ Y : dC (y) < ϵd−C (y)}.
The various kinds of efficient points in Definition 1.1 are in fact Q -minimal points with Q being appropriately chosen
cones as follows.
Proposition 1.2 ([16]).
(i) a0 ∈ StrMin(A) if and only if a0 ∈ Qmin(A) with Q = Y (−C).
(ii) a0 ∈ WMin(A) if and only if a0 ∈ Qmin(A) with Q = int C.
(iii) a0 ∈ Pos(A) if and only if a0 ∈ Qmin(A) with Q = {y ∈ Y | ϕ(y) > 0}, ϕ being some functional in C+i
.
(iv) a0 ∈ Ge(A) if and only if a0 ∈ Qmin(A) with Q = C(ϵ) for some ϵ > 0.
(v) a0 ∈ Bo(A) if and only if a0 ∈ Qmin(A) with Q being some open cone dilating C.
(vi) a0 ∈ He(A) if and only if a0 ∈ Qmin(A) with Q being some open pointed convex cone dilating C.
(vii) a0 ∈ StrHe(A) if and only if a0 ∈ Qmin(A) with Q = int Cϵ(B), ϵ satisfying 0 < ϵ < δ.
(viii) (supposing that C has a bounded base) a0 ∈ Su(A) if and only if a0 ∈ Qmin(A) with Q = intCϵ(B), ϵ satisfying 0 < ϵ < δ.
2. Higher-order radial derivatives
We propose the following higher-order radial derivatives.
Definition 2.1. Let F : X → 2Y
be a set-valued map and u ∈ X.
(i) The mth-order outer radial derivative of F at (x0, y0) ∈ gr F is
D
m
R F(x0, y0)(u) = {v ∈ Y : ∃tn > 0, ∃(un, vn) → (u, v), ∀n, y0 + tm
n vn ∈ F(x0 + tnun)}.
(ii) The mth-order inner radial derivative of F at (x0, y0) ∈ gr F is
Dm
R F(x0, y0)(u) = {v ∈ Y : ∀tn > 0, ∀un → u, ∃vn → v, ∀n, y0 + tm
n vn ∈ F(x0 + tnun)}.
Remark 2.1. Let us discuss some ideas behind this definition. The term ‘‘radial’’ in the definitions of the radial set or
derivative of a map means taking directions which give points having global properties related to the set or graph of the
map, not only local ones around the considered point (as in usual notions like the contingent derivative, where tn → 0+
appears). Here we propose higher-order notions based on this idea. Moreover, the higher-order character in our definition

is different from that of many known notions. For instance, the well-known mth-order contingent derivative of F : X → 2Y
at (x0, y0) ∈ gr F with respect to (w.r.t.) (u1, v1), . . . , (um−1, vm−1) is defined as
Dm
F(x0, y0, u1, v1, . . . , um−1, vm−1)(u) = {v ∈ Y : ∃tn → 0+
, ∃(un, vn) → (u, v),
∀n, y0 + tnv1 + · · · + tm−1
n vm−1 + tm
n vn ∈ F(x0 + tnu1 + · · · + tm−1
n um−1 + tm
n un)}
(and similarly for the m-order adjacent and Clarke derivatives). Another (recent) notion with some similarity is the mth-
order variational set (see [12,13]), which is defined as
Vm
(F, x0, y0, v1, . . . , vm−1) = lim sup
x
F
−→x0,t→0+
1
tm
(F(x) − y0 − tv1 − · · · − tm−1
vm−1).
In these definitions, say that of the contingent derivative, a direction of the m-order derivative continues to improve
the approximating point based on the given m − 1 lower-order directions (u1, v1), . . . , (um−1, vm−1) with an mth-order
rate to get closer to the graph. In our definition of the mth-order outer radial derivative, the direction is not based on given
information of lower-order approximating directions, but also gives an approximation of mth-order rate. Furthermore, the
graph of our derivative is not a corresponding tangent set of the graph of the map, because the rates of change of the point
under consideration in X and Y are different (tn and tm
n ). Note also that it is reasonable to develop also higher-order radial
derivatives, based on the encountered given information. We carried out this task in [21]. The definition is
Dm
R F(x0, y0, u1, v1, . . . , um−1, vm−1)(u) = {v ∈ Y : ∃tn > 0, ∃(un, vn) → (u, v),
∀n y0 + tnv1 + · · · + tm−1
n vm−1 + tm
n vn ∈ F(x0 + tnu1 + · · · + tm−1
n um−1 + tm
n un)}.
The following example highlights detailed differences between the above-mentioned three derivatives.
Example 2.1. Let X = Y = R and F(x) = {x2
} and (x0, y0) = (0, 0). Direct calculations yield
D
1
RF(x0, y0)(x) = D1
RF(x0, y0)(x) = R+,
D1
F(x0, y0)(x) = {0}.
Without any information, we have
D
2
RF(x0, y0)(x) = {x2
}.
Now let (u1, v1) = (0, 0) be given. Then
D2
RF(x0, y0, u1, v1)(x) = R+,
D2
F(x0, y0, u1, v1)(x) = {0}.
For another given direction (u1, v1) = (1, 0), these two second-order derivatives alter as follows
D2
RF(x0, y0, u1, v1)(x) = {1 + a2
x2
+ 2ax : a ≥ 0},
D2
F(x0, y0, u1, v1)(x) = {1}.
Remark 2.2. We collect here several simple properties of D
m
R F(x0, y0) and Dm
R F(x0, y0)(u) for (x0, y0) ∈ gr F.
(i) For m = 1, D
m
R F(x0, y0) is just the radial derivative defined in [5].
(ii) For all m ≥ 1 and u ∈ X, the following properties hold true
(a) Dm
R F(x0, y0)(u) ⊆ D
m
R F(x0, y0)(u);
(b) (0, 0) ∈ grD
m
R F(x0, y0);
(c) if u ∈ dom D
m
R F(x0, y0) then, for any h ≥ 0, hu ∈ dom D
m
R F(x0, y0);
(d) dom D
(m+1)
R F(x0, y0) ⊆ dom D
m
R F(x0, y0).
Proposition 2.1. Let (x0, y0) ∈ grF. Then, for all x ∈ X and m ≥ 1,
F(x) − y0 ⊆ D
m
R F(x0, y0)(x − x0).
Proof. Let x ∈ X, y ∈ F(x)−y0. Then y0+y ∈ F(x). Hence, for tn = 1, yn = y and xn = x−x0, one has y0+tm
n yn ∈ F(x0+tnxn)
for all n. So, y ∈ D
m
R F(x0, y0)(x − x0).
Definition 2.2. Let F : X → 2Y
, (x0, y0) ∈ gr F. If D
m
R F(x0, y0)(u) = Dm
R F(x0, y0)(u) for any u ∈ dom [D
m
R F(x0, y0)], then we
call D
m
R F(x0, y0) an mth-order proto-radial derivative of F at (x0, y0).

Proposition 2.2. Let F1, F2 : X → 2Y
, x0 ∈ int(dom F1)∩dom F2 and yi ∈ Fi(x0) for i = 1, 2. Suppose that F1 has an mth-order
proto-radial derivative at (x0, y1). Then, for any u ∈ X,
D
m
R F1(x0, y1)(u) + D
m
R F2(x0, y2)(u) ⊆ D
m
R (F1 + F2)(x0, y1 + y2)(u).
Proof. Of course we need consider only u ∈ dom D
m
R F1(x0, y1) ∩ dom D
m
R F2(x0, y2). Let vi ∈ D
m
R Fi(x0, yi)(u) for i = 1, 2.
Because v2 ∈ D
m
R F2(x0, y2)(u), there exist tn > 0, un → u, v2
n → v2 such that, for all n,
y2 + tm
n v2
n ∈ F2(x0 + tnun).
Since D
m
R F1(x0, y1) is an mth-order proto-radial derivative, with tn and un above, there exists v1
n → v1 such that, for all n
y1 + tm
n v1
n ∈ F1(x0 + tnun).
Therefore,
(y1 + y2) + tm
n (v1
n + v2
n ) ∈ (F1 + F2)(x0 + tnun),
i.e., v1 + v2 ∈ D
m
R (F1 + F2)(x0, y1 + y2)(u).
The following example shows that the assumption about the proto-radial derivative in Proposition 2.2 cannot be dropped.
Example 2.2. Let X = Y = R, C = R+ and F1, F2 : X → 2Y
be given by
F1(x) =

{1}, if x =
1
n
, n = 1, 2, . . . ,
{0}, if x = 0,
F2(x) =

{0}, if x =
1
n
, n = 1, 2, . . . ,
{1}, if x = 0.
It is easy to see that F1 and F2 do not have proto-radial derivatives of order 1 at (0, 0) and (0, 1), respectively, and
D
1
RF1(0, 0)(0) = R+,
D
1
RF2(0, 1)(0) = R−.
We have
(F1 + F2)(x) =

{1}, if x =
1
n
, n = 1, 2, . . . ,
{1}, if x = 0
and
D
1
R(F1 + F2)(0, 1)(0) = {0}.
Thus,
D
1
RF1(0, 0)(0) + D
1
RF2(0, 1)(0) ̸⊆ D
1
R(F1 + F2)(0, 1)(0).
We cannot reduce the condition x0 ∈ int(dom F1) ∩ dom F2 to x0 ∈ dom F1 ∩ dom F2 as illustrated by the following
example.
Example 2.3. Let X = Y = R, C = R+, x0 = y1 = y2 = 0 and
F1(x) =

R+, if x ≥ 0,
∅, if x < 0,
F2(x) =

R−, if x < 0,
{0}, if x = 0,
∅, if x > 0.
It is easy to see that F1 has the proto-radial derivative of order 1 at (0, 0), dom F1 = dom D1
RF1(0, 0) = R+, dom F2 =
dom D1
RF2(0, 0) = R−. Then, x0 ∈ dom F1 ∩ dom F2. For u = 0 ∈ dom D1
RF1(0, 0) ∩ dom D1
RF2(0, 0), we have
D
1
RF1(0, 0)(0) = R+,
D
1
RF2(0, 0)(0) = R−.

Since
(F1 + F2)(x) =

∅, if x ̸= 0,
R+, if x = 0,
we get
D
1
R(F1 + F2)(0, 0)(0) = R+.
Thus,
D
1
RF1(0, 0)(0) + D
1
RF2(0, 0)(0) ̸⊆ D
1
R(F1 + F2)(0, 0)(0).
Proposition 2.3. Let F : X → 2Y
, G : Y → 2Z
with Im F ⊆ dom G, (x0, y0) ∈ gr F and (y0, z0) ∈ grG.
(i) Suppose that G has an mth-order proto-radial derivative at (y0, z0). Then, for any u ∈ X,
D
m
R G(y0, z0)(D
1
RF(x0, y0)(u)) ⊆ D
m
R (G ◦ F)(x0, z0)(u).
(ii) Suppose that G has a proto-radial derivative of order 1 at (y0, z0). Then, for any u ∈ X,
D
1
RG(y0, z0)(D
m
R F(x0, y0)(u)) ⊆ D
m
R (G ◦ F)(x0, z0)(u).
Proof. By the similarity, we prove only (i). Let u ∈ X, v1 ∈ D
1
RF(x0, y0)(u) and v2 ∈ D
m
R G(y0, z0)(v1). There exist tn > 0,
un → u, v1
y0 + tnv1
n ∈ F(x0 + tnun).
Since v2 ∈ D
m
R G(y0, z0)(v1) = Dm
R G(y0, z0)(v1), with tn and v1
n above, there exists v2
z0 + tm
n v2
n ∈ G(y0 + tnv1
n ).
So we get
z0 + tm
n v2
n ∈ G(y0 + tnv1
n ) ⊆ (G ◦ F)(x0 + tnun)
and hence v2 ∈ D
m
R (G ◦ F)(x0, z0)(u).
The following example shows that the assumption about the proto-radial derivative cannot be dispensed from
Proposition 2.3.
Example 2.4. Let X = Y = R, C = R+ and F1, F2 : X → 2Y
be defined by
F1(x) =

{0}, if x = 1,
{1}, if x = 0,
F2(x) =

{1, 2}, if x = 1,
{0}, if x = 0.
It is easy to see that F1 does not have a proto-radial derivative of order 2 at (0, 1) and
(F1 ◦ F2)(x) =

{0}, if x = 1,
{1}, if x = 0.
Direct calculations yield
D
2
R(F1 ◦ F2)(0, 1)(1/2) = {−1/4},
D
1
RF2(0, 0)(1/2) = {1/2, 1},
D
2
RF1(0, 1)(1/2) = {−1/4},
D
2
RF1(0, 1)(1) = {−1}.
So
D
2
RF1(0, 1)[D
1
RF2(0, 0)(1/2)] = {−1/4, −1}
and
D
2
RF1(0, 1)[D
1
RF2(0, 0)(1/2)] ̸⊆ D
2
R(F1 ◦ F2)(0, 1)(1/2).

We now investigate the sum M +N of two multimaps M, N : X → 2Y
. To express M +N as a composition so that we can
apply a chain rule, we define F : X → 2X×Y
and G : X × Y → 2Y
by, for I being the identity map on X and (x, y) ∈ X × Y,
F = I × M and G(x, y) = N(x) + y. (1)
Then clearly M + N = G ◦ F. However, the rule given in Proposition 2.3, though simple and relatively direct, is not suitable
for dealing with these F and G, since the intermediate space (Y there and X × Y here) is little involved. Inspired by [11,22],
we develop another composition rule as follows. Let general multimaps F : X → 2Y
and G : Y → 2Z
be considered now.
The so-called resultant multimap R : X × Z → 2Y
is defined by
R(x, z) := F(x) ∩ G−1
(z).
Then
dom R = gr(G ◦ F).
We define another kind of radial derivative of G ◦ F with a significant role of intermediate variable y as follows.
Definition 2.3. Let (x, z) ∈ gr(G ◦ F) and y ∈ clR(x, z).
(i) The mth-order y-radial derivative of the multimap G ◦ F at (x, z) is the multimap D
m
R (G ◦y F)(x, z) : X → 2Z
given by
D
m
R (G ◦y F)(x, z)(u) := {w ∈ Z : ∃tn > 0, ∃(un, yn, wn) → (u, y, w), ∀n ∈ N, yn ∈ R(x + tnun, z + tm
n wn)}.
(ii) The mth-order quasi-derivative of the resultant multimap R at ((x, z), y) is defined by, for (u, w) ∈ X × Z,
D
m
q R((x, z), y)(u, w) := {y ∈ Y : ∃hn → 0+
, ∃(un, yn, wn) → (u, y, w),
∀n ∈ N, y + hm
n yn ∈ R(x + hnun, z + hm
n wn)}.
One has an obvious relationship between D
m
R (G ◦y F)(x, z) and D
m
R (G ◦ F)(x, z) as noted in the next proposition.
Proposition 2.4. Given (x, z) ∈ gr(G ◦ F), y ∈ clR(x, z) and u ∈ X, one always has
D
m
R (G ◦y F)(x, z)(u) ⊆ D
m
R (G ◦ F)(x, z).
Proof. This follows immediately from the definitions.
Proposition 2.5. Let (x, z) ∈ gr(G ◦ F), y ∈ clR(x, z) and u ∈ X.
(i) If for all w ∈ Z one has
D
1
RF(x, y)(u) ∩ D
m
R G−1
(z, y)(w) ⊆ D
m
q R((x, z), y)(u, w), (2)
then
D
m
R G(y, z)[D
1
RF(x, y)(u)] ⊆ D
m
R (G ◦y F)(x, z)(u);
(ii) If (2) holds for all y ∈ clR(x, z), then

y∈clR(x,z)
D
m
R G(y, z)[D
1
RF(x, y)(u)] ⊆ D
m
R (G ◦ F)(x, z)(u).
Proof. (i) If the left side of the conclusion of (i) is empty we are done. Let v ∈ D
m
R G(y, z)[D
1
RF(x, y)(u)], i.e., there exists
some y ∈ D
1
RF(x, y)(u) such that y ∈ D
m
R G−1
(z, y)(v). Then (2) ensures that y ∈ D
m
q R((x, z), y)(u, v). This means the
existence of tn → 0+
, (un, yn, vn) → (u, y, v) satisfying
y + tm
n yn ∈ R(x + tnun, z + tm
n vn).
With yn := y + tm
n yn, we have
yn ∈ R(x + tnun, z + tm
n vn).
So v ∈ D
m
R (G ◦y F)(x, z)(u) and we are done.
(ii) Is immediate from (i) and Proposition 2.4.

Now we apply the preceding composition rules to establish sum rules for M, N : X → 2Y
. For this purpose, we use
F : X → 2X×Y
and G : X × Y → 2Y
defined in (1). Then, M + N = G ◦ F. For (x, z) ∈ X × Y, following [23] we set
S(x, z) := M(x) ∩ (z − N(x)).
Then the resultant multimap R : X × Y → 2X×Y
associated to these F and G is
R(x, z) = {x} × S(x, z).
We also modify the definition of y-contingent derivative D(M +y N) in [23] to obtain a kind of radial derivatives as follows.
Definition 2.4. Given (x, z) ∈ dom S and y ∈ clS(x, z), the mth-order y-radial derivative of M +y N at (x, z) is the multimap
D
m
R (M +y N)(x, z) : X → 2Y
given by
D
m
R (M +y N)(x, z)(u) := {w ∈ Y : ∃tn > 0, ∃(un, yn, wn) → (u, y, w), ∀n, yn ∈ S(x + tnun, z + tm
n wn)}.
Observe that
D
m
R (M +y N)(x, z)(u) = D
m
R (G ◦y F)(x, z)(u).
One has a relationship between D
m
R (M +y N)(x, z)(u) and D
m
R (M + N)(x, z)(u) as follows.
Proposition 2.6. Given (x, z) ∈ gr(M + N) and y ∈ clS(x, z), one always has
D
m
R (M +y N)(x, z)(u) ⊆ D
m
R (M + N)(x, z)(u).
Proof. This is an immediate consequence of the definitions.
Proposition 2.7. Let (x, z) ∈ gr(M + N) and u ∈ X.
(i) Let y ∈ clS(x, z). If for all v ∈ Y, one has
D
m
R M(x, y)(u) ∩ [v − D
m
R N(x, z − y)(u)] ⊆ D
m
q S((x, z), y)(u, v), (3)
then
D
m
R M(x, y)(u) + D
m
R N(x, z − y)(u) ⊆ D
m
R (M +y N)(x, z)(u).
(ii) If (3) holds for all y ∈ clS(x, z), then

y∈clS(x,z)
D
m
R M(x, y)(u) + D
m
R N(x, z − y)(v) ⊆ D
m
R (M + N)(x, z)(u).
Proof. (i) If the left side of the conclusion of (i) is empty, nothing is to be proved. If v ∈ D
m
R M(x, y)(u) + D
m
R N(x, z − y)(u),
there exists some y ∈ D
m
R M(x, y)(u) such that y ∈ v−D
m
R N(x, z −y)(u). Hence, (3) ensures that y ∈ D
m
q S((x, z), y)(u, v).
Therefore, there exist tn → 0+
, (un, yn, vn) → (u, y, v) such that
y + tm
n yn ∈ S(x + tnun, z + tm
n vn).
Setting yn := y + tm
n yn, we have
yn ∈ S(x + tnun, z + tm
n vn).
Consequently, v ∈ D
m
R (M +y N)(x, z)(u).
(ii) This follows from (i) and Proposition 2.6.
The following example shows that assumption (3) cannot be dispensed and is not difficult to check.
Example 2.5. Let X = Y = R and M, N : X → 2Y
be given by
M(x) =

{1}, if x =
1
n
, n = 1, 2, . . . ,
{0}, if x = 0,
N(x) =

{0}, if x =
1
n
, n = 1, 2, . . . ,
{1}, if x = 0.

Then
S(x, z) = M(x) ∩ (z − N(x)) =



{0}, if (x, z) = (0, 1),
{1}, if (x, z) =

1
n
, 1

, n = 1, 2, . . . ,
∅, otherwise.
Choose x = 0, z = 1, y = 0 ∈ clS(x, z) and u = v = 0. Then,
D
1
RM(x, y)(u) = R+,
D
1
RN(x, z − y)(u) = R−,
D
1
qS((x, z), y)(u, v) = {0}.
Thus,
D
1
RM(x, y)(u) ∩ [v − D
1
RN(x, z − y)(u)] ̸⊆ D
1
qS((x, z), y)(u, v).
Direct calculations show that the conclusion of Proposition 2.7 does not hold, since
D
1
R(M +y N)(x, z)(u) = {0}
and hence
D
1
RM(x, y)(u) + D
1
RN(x, z − y)(u) ̸⊆ D
1
R(M +y N)(x, z)(u).
Proposition 2.8. Let F : X → Y, (x0, y0) ∈ gr F, λ > 0 and β ∈ R. Then
(i) D
m
R (βF)(x0, βy0)(u) = βD
m
R F(x0, y0)(u);
(ii) D
m
R (F)(x0, y0)(λu) = λm
D
m
R F(x0, y0)(u).
In the remaining part of this section, we apply Propositions 2.2 and 2.3 to establish necessary optimality conditions
for several types of efficient solutions of some particular optimization problems. (Optimality conditions, using higher-order
radial derivatives, for general optimization problems are discussed in Section 3.) As Q -minimality includes many other kinds
of solutions as particular cases, we first prove a simple characterization of this notion.
Proposition 2.9. Let X, Y and Q be as before, F : X → 2Y
and (x0, y0) ∈ gr F. Then y0 is a Q -minimal point of F(X) if and
only if
D
m
R F(x0, y0)(X) ∩ (−Q ) = ∅. (4)
Proof. ‘‘Only if’’ suppose to the contrary that y0 is a Q -minimal point of F(X) but there exist x ∈ X and y ∈ D
m
R (F, x0, y0)(x)∩
(−Q ). There exist sequences tn > 0, xn → x and yn → y such that, for all n,
y0 + tm
n yn ∈ F(x0 + tnxn).
Since the cone Q is open, we have tm
n yn ∈ −Q for n large enough. Therefore,
tm
n yn ∈ (F(x0 + tnxn) − y0) ∩ (−Q ),
a contradiction.
‘‘If’’ assume that (4) holds. From Proposition 2.1 one has, for all x ∈ X,
(F(x) − y0) ∩ (−Q ) ⊆ D
m
R F(x0, y0)(x − x0) ∩ (−Q ) = ∅.
This means that y0 is a Q -minimal point of F(X).
Let X and Y be normed spaces, Y being partially ordered by a pointed closed convex cone C with nonempty interior,
F : X → 2Y
and G : X → 2X
. Consider the problem
(P1) min F(x′
) subject to x ∈ X and x′
∈ G(x).
This problem can be restated as the following unconstrained problem: min(F ◦ G)(x). Recall that (x0, y0) is said to be a
Q -minimal solution of (P2) if y0 ∈ (F ◦ G)(x0) and ((F ◦ G)(X) − y0) ∩ (−Q ) = ∅.
Proposition 2.10. Let Im G ⊆ dom F, (x0, z0) ∈ grG and (z0, y0) ∈ gr F. Assume that (x0, y0) is a Q -minimal solution of (P1).
(i) If F has an mth-order proto-radial derivative at (z0, y0) then, for any u ∈ X,
D
m
R F(z0, y0)(D
1
RG(x0, z0)(u)) ∩ (−Q ) = ∅. (5)

(ii) If F has a proto-radial derivative of order1 at (z0, y0) then, for any u ∈ X,
D
1
RF(z0, y0)(D
m
R G(x0, z0)(u)) ∩ (−Q ) = ∅. (6)
Proof. By the similarity, we prove only (i). From Proposition 2.9, we have
D
m
R ((F ◦ G))(x0, y0) ∩ (−Q ) = ∅.
Proposition 2.3(i) says that
D
m
R F(z0, y0)(D
1
RG(x0, z0)(u)) ⊆ D
m
R (F ◦ G)(x0, y0)(u).
So
D
m
R F(z0, y0)(D
1
RG(x0, z0)(u)) ∩ (−Q ) = ∅.
Later on we adopt the usual convention that for x0 being feasible, (x0, y0) is a solution in a sense of a vector optimization
problem if and only if y0 is the efficient point in this sense of the image of the feasible set, in the objective space. Then, from
Propositions 2.1 and 2.10 we obtain the following theorem.
Theorem 2.11. Let the assumptions of Proposition 2.10 be satisfied and (x0, y0) ∈ grF. Then (5) and (6) hold in each of the
following cases
(i) (x0, y0) is a strong efficient solution of (P1) and Q = Y (−C);
(ii) (x0, y0) is a weak efficient solution of (P1) and Q = int C;
(iii) (x0, y0) is a positive-properly efficient solution of (P1) and Q = {y : ϕ(y) > 0} for some functional ϕ ∈ C+i
;
(iv) (x0, y0) is a Geoffrion-properly efficient solution of (P1) and Q = C(ϵ) for some ϵ > 0 (C(ϵ) = {y ∈ Y : dC (y) <
ϵd−C (y)});
(v) (x0, y0) is a Borwein-properly efficient solution of (P1) and Q = K for some open cone K dilating C;
(vi) (x0, y0) is a Henig-properly efficient solution of (P1) and Q = K for some open convex cone K dilating C;
(vii) (x0, y0) is a strong Henig-properly efficient solution of (P1) and Q = int Cϵ(B) for some ϵ satisfying 0 < ϵ < δ (B is a base
of C, Cϵ(B) = cone(B + ϵBY ) and δ = inf{‖b‖ : b ∈ B});
(viii) (x0, y0) is a super efficient solution of (P1) and Q = int Cϵ(B) for ϵ satisfying 0 < ϵ < δ.
To compare with a result in [22], we recall the definition of contingent epiderivatives. For a multimap F between normed
spaces X and Y, Y being partially ordered by a pointed convex cone C and a point (¯x, ¯y) ∈ gr F, a single-valued map
EDF(¯x, ¯y) : X → Y satisfying epi(EDF(¯x, ¯y)) = TepiF (¯x, ¯y) ≡ Tgr F+ (¯x, ¯y) is said to be the contingent epiderivative of F
at (¯x, ¯y).
Example 2.6. Let X = Y = R, C = R+, G(x) = {−|x|} and F be defined by
F(x) =

R−, if x ≤ 0,
∅, if x > 0.
Since G is single-valued, we try to make use of Proposition 5.2 of [22]. By a direct computation, we have DG(0, G(0))(h) =
{−|h|} for all h ∈ X and TepiF (G(0), 0) = R2
and hence the contingent epiderivative EDF(G(0), 0)(h) does not exist for any
h ∈ X. Therefore, the necessary condition in the mentioned proposition of [22] cannot be applied. However, F has an mth-
order proto-radial derivative at (G(0), 0) and D
1
R(0, G(0))(0) = {0}, D
m
R F(G(0), 0)[D
1
RG(0, G(0))(0)] = R−, which meets
−int C and hence Proposition 2.10 above rejects the candidate for a weak solution.
To illustrate sum rules, we consider the following problem
(P2) min F(x) subject to g(x) ≤ 0,
where X, Y are as for problem (P2), F : X → 2Y
and g : X → Y. Denote S = {x ∈ X | g(x) ≤ 0} (the feasible set). Define
G : X → 2Y
by
G(x) =

{0}, if x ∈ S,
{g(x)}, otherwise.
Consider the following unconstrained set-valued optimization problem, for an arbitrary positive s,
(PC) min(F + sG)(x).
In the particular case, where Y = R and F is single-valued, (PC) is used to approximate (P2) in penalty methods (see [24]).
Optimality conditions for this general problem (PC) are obtained in [11] using calculus rules for variational sets and in [22] by
using such rules for contingent epiderivatives. Here we will apply Propositions 2.2 and 2.8 for mth-order radial derivatives
to get the following necessary condition for several types of efficient solutions of (PC).

Proposition 2.12. Let dom F ⊆ dom G, x0 ∈ S, y0 ∈ F(x0) and either F or G has an mth-order proto-radial derivative at
(x0, y0) or (x0, 0), respectively. If (x0, y0) is a Q -minimal solution of (PC) then, for any u ∈ X,
(D
m
R F(x0, y0)(u) + sD
m
R G(x0, 0)(u)) ∩ −Q = ∅. (7)
Proof. We need to discuss only u ∈ dom D
m
R F(x0, y0)∩dom D
m
R G(x0, 0). By Proposition 2.9, one gets D
m
R (F +sG)(x0, y0)(u)∩
−Q = ∅. According to Proposition 2.8, sD
m
R G(x0, 0)(u) = D
m
R (sG)(x0, 0)(u). Then, Proposition 2.2 yields
D
m
R F(x0, y0)(u) + sD
m
R G(x0, 0)(u) ⊆ D
m
R (F + sG)(x0, y0 + 0)(u).
The proof is complete.
Similarly as stated in Theorem 2.11, formula (8) holds true for each of our eight kinds of efficient solutions of (P2), with
Q being chosen correspondingly. The next example illustrates a case, Proposition 2.12 is more advantageous than earlier
existing results.
Example 2.7. Let X = Y = R, C = R+, g(x) = x4
− 2x3
and
F(x) =

R−, if x ≤ 0,
∅, if x > 0.
Then S = [0, 2] and G(x) = {max{0, x4
− 2x3
}}. Furthermore, since TepiF (0, 0) = R2
, TepiG(0, 0) = {(x, y), y ≥ 0},
the contingent epiderivative DF(0, 0)(h) does not exist for any h ∈ X and Proposition 5.1 of [22] cannot be employed.
But F has a proto-radial derivative of order 1 at (0, 0), D
1
RF(0, 0)(0) = R− and {0} ⊆ D
1
RG(0, 0)(0) ⊆ R+. So,
D
1
RF(0, 0)(0) + sD
1
RG(0, 0)(0) ∩ (−int C) ̸= ∅. By Proposition 2.12, (x0, y0) is not a weak efficient solution of (PC). This
fact can be checked directly too.
3. Optimality conditions
Let X and Y be normed spaces partially ordered by pointed closed convex cones C and D, respectively, with nonempty
interior. Let S ⊆ X, F : X → 2Y
and G : X → 2Z
. In this section, we discuss optimality conditions for the following general
set-valued vector optimization problem with inequality constraints
(P) min F(x), subject to x ∈ S, G(x) ∩ (−D) ̸= ∅.
Let A := {x ∈ S : G(x) ∩ (−D) ̸= ∅} and F(A) :=

x∈A F(x). We assume that F(x) ̸= ∅ for all x ∈ X.
Proposition 3.1. Let dom F ∪ dom G ⊆ S and (x0, y0) is a Q -minimal solution for (P). Then, for any z0 ∈ G(x0) ∩ (−D) and
x ∈ X,
D
m
R (F, G)(x0, y0)(x) ∩ (−Q × −int D) = ∅. (8)
Proof. We need to investigate only x ∈ dom D
m
R (F, G)(x0, y0). For all x ∈ A,
(F(x) − y0) ∩ (−Q ) = ∅.
Suppose (8) does not hold. Then, there exists x ∈ dom D
m
R (F, G)(x0, y0) such that
(y, z) ∈ D
m
R (F, G)(x0, y0)(x) (9)
and
(y, z) ∈ −Q × −int D. (10)
It follows from (9) that there exist {tn} with tn > 0 and {(xn, yn, zn)} in X × Y × Z with (xn, yn, zn) → (x, y, z) such that
(y0, z0) + tm
n (yn, zn) ∈ (F, G)(x0 + tnxn).
Hence, xn := x0 + tnxn ∈ dom (F, G) ⊆ S and there exists (yn, zn) ∈ (F, G)(xn) such that
(y0, z0) + tm
n (yn, zn) = (yn, zn).
As (yn, zn) → (y, z), this implies that
yn − y0
tm
n
→ y, (11)

and
zn − z0
tm
n
→ z. (12)
Combining (10)–(12), one finds N > 0 such that, for n ≥ N,
yn − y0
tm
n
∈ −Q , (13)
and
zn − z0
tm
n
∈ −int D.
Thus, zn ∈ −D. Hence, xn ∈ A for large n.
On the other hand, by (13) we get, for n ≥ N,
yn − y0 ∈ −Q .
This is a contradiction. So, (8) holds.
From Propositions 3.1 and 1.2, we obtain immediately the following result.
Theorem 3.2. Assume that dom F ∪ dom G ⊆ S. Then (8) holds in each of the following cases
(i) (x0, y0) is a strong efficient solution of (P) and Q = Y −C;
(ii) (x0, y0) is a weak efficient solution of (P) and Q = int C;
(iii) (x0, y0) is a positive-properly efficient solution of (P) and Q = {y : ϕ(y) > 0} for some functional ϕ ∈ C+i
;
(iv) (x0, y0) is a Geoffrion-properly efficient solution of (P) and Q = C(ϵ) for some scalar ϵ > 0;
(v) (x0, y0) is a Borwein-properly efficient solution of (P) and Q = K for some open cone K dilating C;
(vi) (x0, y0) is a Henig-properly efficient solution of (P) and Q = K for some open convex cone K dilating C;
(vii) (x0, y0) is a strong Henig-properly efficient solution of (P) and Q = int Cϵ(B) for some ϵ satisfying 0 < ϵ < δ;
(viii) (x0, y0) is a super efficient solution of (P) and Q = int Cϵ(B) for some ϵ satisfying 0 < ϵ < δ.
For the comparison purpose, we recall from [12] that the (first-order) variational set of type 2 of F : X → 2Y
at (x0, y0) is
W1
(F, x0, y0) = lim sup
x
F
−→x0
cone+(F(x) − y0),
where x
F
−→ x0 means that x → x0 and x ∈ dom F. A multimap H : X → 2Y
is called pseudoconvex at (x0, y0) ∈ grH if
epi H ⊆ (x0, y0) + Tepi H (x0, y0).
The following example shows a case where many existing theorems using other generalized derivatives do not work,
while Theorem 3.2 rejects a candidate for a weak efficient solution.
Example 3.1. Let X = Y = R, C = R+ and F be defined by
F(x) =

[1, +∞), if x = 0,
R+, if x = 1,
∅, otherwise.
Let (x0, y0) = (0, 1) and u = 1. Then
W1
(F+, x0, y0) = R+.
Hence, Theorem 3.2 of [12] says nothing about (x0, y0). From Remark 2.2(i) and Proposition 4.1 of [12], we see that Theorem
7 of [8], Theorem 5 of [25], Theorem 4.1 of [26], Proposition 3.1–3.2, and Theorem 4.1 of [5] cannot be applied to reject
(x0, y0) as a candidate for a weak efficient solution.
On the other hand,
D
1
RF(x0, y0)(u) = [−1, +∞).
Then, D
1
RF(x0, y0)(u) ∩ (−int C) ̸= ∅. It follows from Theorem 3.2(ii) that (x0, y0) is not a weak efficient solution.
The following example explains that we have to use higher-order radial derivatives instead of lower-order ones when
applying Theorem 3.2 in some cases.

F(x) =



{0}, if x = 0,
{|x|}, if x = −
1
n
, n = 1, 2, . . . ,
{−1}, if x =
1
n
, n = 1, 2, . . . ,
∅, otherwise.
Let (x0, y0) = (0, 0) and u = 0. Then
D
1
RF(x0, y0)(u) = {0},
D
2
RF(x0, y0)(u) = R.
Because D
2
RF(x0, y0)(u) ∩ (−int C) ̸= ∅, (x0, y0) is not a weak efficient solution. (But D
1
RF(x0, y0) cannot be used here.)
We have seen that our necessary optimality conditions using radial derivatives are stronger than many existing
conditions using other generalized derivatives, since images of a point or set through radial derivatives are larger than the
corresponding images through other derivatives. The next proposition gives a sufficient condition, which has a reasonable
gap with the corresponding necessary condition in Proposition 3.1.
Proposition 3.3. Let dom F ∪ dom G ⊆ S, x0 ∈ A, y0 ∈ F(x0) and z0 ∈ G(x0) ∩ (−D). Then (x0, y0) is a Q -minimal solution
of (P) if the following condition holds
D
m
R (F, G)(x0, y0)(A − x0) ∩ −(Q × D(z0)) = ∅. (14)
Proof. From Proposition 2.1, for x ∈ A one has
(F, G)(x) − (y0, z0) ⊆ D
m
R (F, G)(x0, y0)(x − x0).
Then
(F, G)(x) − (y0, z0) ∩ −(Q × D(z0)) = ∅. (15)
Suppose the existence of x ∈ A and y ∈ F(x) such that y − y0 ∈ −Q . For any z ∈ G(x) ∩ (−D) one has z − z0 ∈ −D(z0) and
hence (y, z) − (y0, z0) ∈ −(Q × D(z0)), contradicting (15).
A natural question now arises: can we replace D by D(z0) in the necessary condition given by Proposition 3.1 to obtain
a smaller gap with the sufficient one expressed by Proposition 3.3? Unfortunately, a negative answer is supplied by the
following example.
Example 3.3. Suppose that X = Y = Z = R, S = X, C = D = R+ and F : X → 2Y
and G : X → 2Z
are given by
F(x) =

{y : y ≥ x2
}, if x ∈ [−1, 1],
{−1}, if x ̸∈ [−1, 1],
G(x) = {z ∈ Z : z = x2
− 1}.
We have A = [−1, 1] in problem (P). It easy to see that (x0, y0) = (0, 0) is a weak efficient solution of (P). So the condition
of Theorem 3.2(ii) is satisfied. Take z0 = −1 ∈ G(x0) ∩ (−D). Since for all x ∈ X, D
2
R(F, G)(x0, y0, z0)(x) ⊆ R × R+, we have
D
2
R(F, G)(x0, y0, z0)(x) ∩ −int(C × D) = ∅.
On the other hand, D(z0) = R. We claim the existence of x ∈ X such that
D
2
R(F, G)(x0, y0, z0)(x) ∩ −int(C × D(z0)) ̸= ∅.
Indeed, choose x > 0, xn = x and tn = 2
xn
. Then
(y0, z0) + t2
n (vn, wn) ∈ (F, G)(x0 + tnxn)
means that
(0, −1) +
4
x2
(vn, wn) ∈ (F, G)(2),

i.e.,
4
x2
vn = −1,
4
x2
wn = 4.
So, there exist vn = −x2
4
→ −x2
4
∈ −int C and wn = x2
→ x2
∈ −intD(z0). Thus,

−x2
4
, x2

∈ D
2
R(F, G)(x0, y0, z0)(x).
Similarly as before, from an assertion for Q -minimal solutions we always obtain the corresponding ones for our eight
kinds of efficient solutions. Hence we arrive at the following sufficient conditions.
Theorem 3.4. Assume that dom F ∪ dom G ⊆ S, x0 ∈ A, y0 ∈ F(x0) and z0 ∈ G(x0) ∩ (−D). Let condition (14) hold. Then
(i) (x0, y0) is a strong efficient solution of (P) if Q = Y −C;
(ii) (x0, y0) is a weak efficient solution of (P) if Q = int C;
(iii) (x0, y0) is a positive-properly efficient solution of (P) if Q = {y : ϕ(y) > 0} for some functional ϕ ∈ C+i
;
(iv) (x0, y0) is a Geoffrion-properly efficient solution of (P) if Q = C(ϵ) for some scalar ϵ > 0;
(v) (x0, y0) is a Borwein-properly efficient solution of (P) if Q = K for some open cone K dilating C;
(vi) (x0, y0) is a Henig-properly efficient solution of (P) if Q = K for some open convex cone K dilating C;
(vii) (x0, y0) is a strong Henig-properly efficient solution of (P) if Q = int Cϵ(B) for some scalar ϵ satisfying 0 < ϵ < δ;
(viii) (x0, y0) is a super efficient solution of (P) if Q = int Cϵ(B) for some scalar ϵ satisfying 0 < ϵ < δ.
We illustrate advantages of Theorem 3.4 by showing a case where it works while many earlier results do not in the
following example.
F(x) =



{0}, if x = 0,
1
n2

, if x = n for n = 1, 2, . . . ,
∅, otherwise.
Let (x0, y0) = (0, 0) and u ∈ dom D
1
RF(x0, y0) = R+. Then, for n = 1, 2 . . .,
D
1
RF(x0, y0)(u) =
 u
n3
 
{0}.
It follows from Theorem 3.4(ii) that (x0, y0) is a weak efficient solution. It is easy to see that dom F = {0, 1, 2, . . .} is not
convex and F is not pseudoconvex at (x0, y0). So Theorem 8 of [8], Theorem 6 of [25] and Theorem 3.3 of [12] cannot be
used.
Acknowledgments
This work was supported by National Foundation for Science and Technology Development of Viet Nam. The authors are
grateful to a referee for valuable remarks helping them to improve the paper.
References
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