SOOCHOW JOURNAL OF MATHEMATICS
Volume 33, No. 1, pp. 17-31, January 2007
ON HIGHER-ORDER CONDITIONS FOR STRICT
EFFICIENCY
BY
DO VAN LUU AND PHAM TRUNG KIEN
Abstract. Jiménez [14] and Ginchev [7] introduced the notions of a strict local
Pareto minimum of order n and a strict local Pareto minimum, and studied their
some properties. In this paper we investigate further Pareto minima of these types
and derive necessary and sufficient conditions for strict efficiency in terms of the
directional derivatives of higher order in Ginchev’s sense.
1. Introduction
Second and higher-order optimality conditions for mathematical programming problems in abstract spaces with real-valued or vector-valued objective
functions have been developed by many authors (see, e.g., [2]-[22]). The notions
of a strict local minimum and a strict local minimum of order m (called also an
isolated local minimum of order m [2], [21]) are extended to vector optimization
problems in Ginchev [7] and Jiménez [14], and as in [14] these extensions will
be called here respectively strict local Pareto minimum and strict local Pareto
minimum of order m. Jiménez [14], [15] and Jiménez-Novo [16] derive higherorder conditions for strict local Pareto minima of a given order in terms of Studniarski’s higher-order directional derivatives. First and second-order directional
derivatives of Riemann and Dini type have been used to derive first and second
order conditions for C 0,1 and C 1,1 vector optimization problems in several papers of Ginchev-Guerraggio-Rocca [8]-[11], where the strict local Pareto minima
Received July 21, 2005; revised February 24, 2006.
AMS Subject Classification. Primary 90C46; secondary 90C29.
Key words. strict local Pareto minimum, higher-order optimality conditions, lower and
upper Ginchev directional derivatives.
This research was partially supported by the Natural Science Council of Vietnam.
17
18
DO VAN LUU AND PHAM TRUNG KIEN
of orders 1 and 2 take a central place. Ginchev [6] introduces notions of higherorder lower and upper directional derivatives for nonsmooth extended-real-valued
functions and derives higher-order optimality conditions for unconstrained scalar
programs.
Motivated by the results due to Ginchev [6], in this paper we establish higherorder conditions for strict local efficiency of nonsmooth multiobjective programs
with a set constraint in terms of Ginchev’s directional derivatives of higher order. Section 2 presents some preliminaries together with a property ensuring a
point to be not a strict local Pareto minimizer. Section 3 is devoted to derive a
higher-order necessary condition for local efficiency, while Section 4 deals with a
higher-order sufficient condition for strict local Pareto minimizers in case of finite
dimensions. Making use of a result due to Jiménez [14], in Section 5 a higherorder necessary condition for strict local efficiency in terms of higher-order upper
Ginchev directional derivatives is established. Finally, a higher-order sufficient
condition in terms of higher-order lower Ginchev derivatives is also given in this
section.
2. Preliminaries
Let f be an extended-real-valued function defined on a normed space X.
Following [6], for a given positive integer number n, the n-th order lower and
(n)
(n)
upper directional derivatives f− (x; d) and f+ (x; d), respectively, at x ∈ X in a
direction d are defined as
(0)
f− (x; d) = lim inf f (x + td′ ),
(1)
t↓0
d′ →d
(0)
f+ (x; d)
= lim sup f (x + td′ ),
(2)
t↓0
d′ →d
(n)
f− (x; d)
n−1
i
X tj (j)
n! h
′
= lim inf n f (x + td ) −
f− (x; d) ,
t↓0
t
j!
d′ →d
(n)
f+ (x; d)
n−1
i
X tj (j)
n! h
′
= lim sup n f (x + td ) −
f+ (x; d) .
t
j!
t↓0
d′ →d
(3)
j=0
j=0
(4)
ON HIGHER-ORDER CONDITIONS FOR STRICT EFFICIENCY
(n)
19
(n)
Note that in Ginchev’s definitions (1)-(4), he accepted that f− (x; d) and f+ (x;
(i)
(i)
d) exist as elements of R if and only if f− (x; d) and f+ (x; d) (i = 0, 1, . . . , n − 1),
respectively, exist as elements of R, where R denotes the vector space of real
numbers, and R = R ∪ {±∞}. Moreover, the zero-order derivatives of discontinuous functions exist always and are elements of R. In case f is continuous
and n = 1, the lower and upper Studniarski derivatives ([21]) and the lower and
upper Ginchev derivatives are the same, which reduce to the lower and upper
Dini directional derivatives ([2]), respectively, while for n > 1 they are in general
different.
Adapting the definition of Ginchev’s derivatives ([6]), we define the n-th
order directional derivatives to a mapping f from X into another normed space
Y as follows
f (0) (x; d) = lim f (x + td′ ),
(5)
t↓0
d′ →d
f (n) (x; d) = lim
t↓0
d′ →d
n−1
i
X tj
n! h
′
(j)
f
(x
+
td
)
−
f
(x;
d)
,
tn
j!
(6)
j=0
if these limits exist. Let us underline that the derivative f (1) (x; d) appears in
[5] under the name of Hadamard derivative. If f is Fréchet differentiable at x
with the Fréchet derivative f ′ (x), then f (1) (x; d) = f ′ (x)d. In case Y = R, the
(n)
(n)
existence of f (n) (x; d) implies the existence of f+ (x; d) and f− (x; d) and they
are the same.
We also recall [1] that the contingent cone to a set C at x ∈ clC is
n
KC (x) = d ∈ X : there exist sequences tn ↓ 0, and
o
dn → d such that x + tn dn ∈ C, for all n ,
where clC indicates the closure of C.
In this paper, we shall be concerned with the following vector optimization
problem
(P)
min{f (x) : x ∈ C},
where f is a mapping from a normed space X into another normed space Y , C
a subset of X. Let Q be a closed convex cone in Y . A point x ∈ C is said to be
20
DO VAN LUU AND PHAM TRUNG KIEN
a weakly local minimizer of Problem (P) if there exists a neighborhood U of x
such that
f (x) − f (x) 6∈ −int Q
(∀ x ∈ C ∩ U ).
Note that for weakly local minimizers, we suppose that int Q 6= ∅. The point x is
called a local Pareto minimizer of (P) if there exists a neighborhood U of x such
that
f (x) − f (x) 6∈ −Q \ {0}
(∀x ∈ C ∩ U ).
Following [14], x is said to be a strict local Pareto minimizer of (P) if there
exists a neighborhood U of x such that
f (x) − f (x) 6∈ −Q
(∀ x ∈ C ∩ U \ {x}).
For an integer number m > 1, we also recall [14] that x is called a strict local Pareto minimizer of order m of (P) if there exist a constant α > 0 and a
neighborhood U of x such that
(f (x) + Q) ∩ B(f (x); αkx − xkm ) = ∅
(∀ x ∈ C ∩ U \ {x}),
(7)
where B(x; δ) stands for the open ball of radius δ around x. Note that the open
ball B(f (x), αkx − xkm ) in (7) can be replaced by the closed ball B(f (x); αkx −
xkm ) of radius αkx − xkm around f (x), because for α1 ∈ (0, α), (7) implies that
(f (x) + Q) ∩ B(f (x); α1 kx − xkm ) = ∅ (∀ x ∈ C ∩ U \ {x}).
In case Y = R and Q = R+ , (7) becomes
(f (x) + R+ ) ∩ B(f (x); αkx − xkm ) = ∅ (∀ x ∈ C ∩ U \ {x}),
which equivalent to the following
f (x) > f (x) + αkx − xkm
(∀ x ∈ C ∩ U \ {x}),
where R+ is the set of nonegative real numbers. This means that x is a strict
local minimizer of order m.
Notice that every strict local Pareto minimizer of order m is also order k,
for all k > m. For each integer number m > 1, let us remark the following
relationships ([14]):
ON HIGHER-ORDER CONDITIONS FOR STRICT EFFICIENCY
21
Strict local Pareto minimizer of order m =⇒ Strict local Pareto minimizer
=⇒ Local Pareto minimizer =⇒ Weakly local minimizer.
We recall now two results due to Jiménez [14, Proposition 3.4 and Theorem
3.7.(b)], which will be used in the sequel.
Proposition 2.1.([14]) Let x ∈ C. The point x is not a strict local Pareto
minimizer of order m for Problem (P) if and only if there exist sequences xn ∈
C \ {x}, dn ∈ Q such that xn → x, and
lim
n→∞
f (xn ) − f (x) + dn
= 0.
kxn − xkm
Proposition 2.2.([14]) Let x ∈ C, Y = Rr , Q = Rr+ . Then x is a strict
local Pareto minimizer of Problem (P) if and only if there exist a neighborhood U
of x and at most r sets Vi , i ∈ I ⊂ {1, . . . , r} such that {Vi : i ∈ I} is a covering
of C ∩ U \ {x}, and verifying
fi (x) > fi (x)
(∀ x ∈ Ci \ {x}),
where Ci = (C ∩ U ∩ Vi ) ∪ {x}.
For points to be not strict local Pareto minima of (P) we have the following.
Proposition 2.3. Let x ∈ C. Assume that x is not a strict local Pareto
minimizer for Problem (P). Then for each integer number m > 1, there exist
sequences xn ∈ C \ {x} and dn ∈ Q such that xn → x and
lim
n→∞
f (xn ) − f (x) + dn
= 0.
kxn − xkm
(8)
Proof. Assume that x is not a strict local Pareto minimizer for (P). Then
for each integer number m > 1, x is not strict local Pareto minimizer of order m
for (P). Taking account of Proposition 2.1, there exist sequences xn ∈ C \ {x},
xn → x and dn ∈ Q such that (8) holds.
3. A Higher-Order Necessary Condition for Efficiency
The following theorem provides a necessary condition in terms of Ginchev’s
higher-order directional derivatives for weakly local minima, and hence, it is valid
for strict local Pareto minima.
22
DO VAN LUU AND PHAM TRUNG KIEN
Theorem 3.1. Let intQ 6= ∅ and x be a weakly local minimum of Problem
(P). Assume that for each d ∈ KC (x), there exist the directional derivatives
f (j)(x; d) (j = 0, 1, . . . , n) defined by (5) and (6). Then the following optimality
conditions hold:
(i) f (0) (x; d) − f (x) 6∈ −intQ (∀ d ∈ KC (x)).
(ii) If for d ∈ KC (x), f (0) (x; d) = f (x), f (j) (x; d) = 0 (j = 1, . . . , n − 1), then
f (n) (x; d) 6∈ −intQ.
Proof. Since x is a weakly local minimum of (P), there exists a neighborhood
U of x such that
f (x) − f (x) 6∈ −intQ (∀ x ∈ C ∩ U ),
which is equivalent to the following
f (x) − f (x) ∈ −(Y \ intQ) (∀ x ∈ C ∩ U ).
For d ∈ KC (x), there exist sequences tm ↓ 0 and dm → d such that x + tm dm ∈ C
(∀ m). So for sufficiently large m, x + tm dm ∈ C ∩ U , and hence,
f (x + tm dm ) − f (x) ∈ −(Y \ intQ).
(9)
In view of the closedness of Y \ intQ, it follows that
lim f (x + tm dm ) − f (x) = f (0) (x; d) − f (x) ∈ −(Y \ intQ),
m→∞
which implies that
f (0) (x; d) − f (x) 6∈ −intQ.
We thus arrive at (i).
To prove (ii), we observe that by assumption, for d ∈ KC (x), the directional derivatives f (j) (x; d) (j = 0, 1, . . . , n) exist. Assuming f (0) (x; d) = f (x),
f (j)(x; d) = 0 (j = 1, . . . , n − 1), we have
f (n) (x; d) = lim
t↓0
n−1
i
X tj
n! h
′
(j)
f
(x
+
td
)
−
f
(x;
d)
tn
j!
j=0
d′ →d
n−1
i
X tjm
n! h
(0)
(j)
f
(x
+
t
d
)
−
f
(x;
d)
−
f
(x;
d)
m m
m→∞ tn
j!
m
= lim
j=1
= lim
n!
m→∞ tn
m
[f (x + tm dm ) − f (x)].
ON HIGHER-ORDER CONDITIONS FOR STRICT EFFICIENCY
23
Since Y \ intQ is closed, it follows from this and (9) that
f (n) (x; d) ∈ −(Y \ intQ),
which means that
f (n) (x; d) 6∈ −intQ.
We also arrive at the conclusion (ii).
4. A Higher-Order Sufficient Condition for Strict Efficiency
In this section we establish a higher-order sufficient condition for strict local
Pareto minima of order n for Problem (P) in finite dimensions.
Theorem 4.1. Let dim X < +∞. Assume that there is a positive integer
number n such that for every d ∈ KC (x) \ {0}, directional derivatives f (j)(x; d)
(j = 0, 1, . . . , n) exists, and one of the following conditions (Ak )(k = 1, . . . , n)
holds:
(Ak ) f (0) (x; d) = f (x), f (j) (x; d) = 0 (j = 1, . . . , k − 1), f (k) (x; d) 6∈ −Q.
Then x is a strict local Pareto minimum of order n for (P).
Proof. Assume the contrary, that x is not a strict local Pareto of order n
for (P). Then, by Proposition 2.1, there would exist a sequence xm ∈ C, xm 6= x,
xm → x and bm ∈ Q such that
lim
m→+∞
f (xm ) − f (x) + bm
= 0.
kxm − xkn
(10)
xm − x
, we obtain that xm = x +
kxm − xk
tm dm ∈ C. Since dim X < +∞, there is a subsequence of {dm } converging to d
with kdk = 1. Without loss of generality, we can assume that dm → d. Hence
d ∈ KC (x) \ {0}.
Moreover, by virtue of (10), it follows that
Putting tm = kxm − xk and dm =
lim
m→+∞
f (x + tm dm ) − f (x) + bm
= 0.
tnm
(11)
Since tm → 0+ as m → +∞, (11) implies that for k = 1, . . . , n,
lim
m→+∞
f (x + tm dm ) − f (x) + bm
= 0.
tkm
(12)
24
DO VAN LUU AND PHAM TRUNG KIEN
On the other hand, by assumption (Ak ), we have that f (0) (x; d) = f (x) and
f (j)(x; d) = 0 (j = 1, . . . , k − 1). Since f (k) (x; d) exists, it holds that
k−1 j
i
X
k! h
tm (j)
(0)
f
(x
f
+
t
d
)
−
f
(x;
d)
−
(x;
d)
m m
m→+∞ tk
j!
m
j=1
i
k! h
= lim k f (x + tm dm ) − f (x) .
m→+∞ tm
f (k) (x; d) = lim
By assumption, f (k) (x; d) exists (k = 1, . . . , n), and hence, it follows from this
that the following limits exist
f (x + tm dm ) − f (x)
m→+∞
tkm
lim
(k = 1, . . . , n).
(13)
The existence of the limits (13) along with (12) yields the existence of the following limits
bm
lim
(k = 1, . . . , n).
m→+∞ tk
m
Since Q is closed, bm ∈ Q, and tkm > 0, it follows that for k = 1, . . . , n,
bm
lim
m→+∞ tk
m
∈ Q.
This together with (12) yields that for k = 1, . . . , n,
f (k) (x; d) ∈ −Q.
But this conflicts with condition (Ak ).
Example 4.1. Let X = Rk , Y = R2 , Q = R2+ , C = [0, 1]k−1 × [−1, 0], x =
(0, . . . , 0), where Rk+ stands for the nonnegative orthant in Rk , [0, 1]k−1 = [0, 1] ×
· · · × [0, 1] (k − 1 times). Let f be given by
k
X
|xi |k , −|x1 |k+1 ),
f (x1 , . . . , xk ) = (
i=1
where k is a positive even integer number. Then, KC (x) = Rk−1
× R− , where
+
R− = −R+ , and for d = (d1 , . . . , dk ) ∈ KC (x) \ {0},
f
(j)
(0; d) = 0 (j = 1, . . . , k − 1);
f
(k)
k
X
(0; d) = (
|di |k , 0) 6∈ −R2+ .
i=1
ON HIGHER-ORDER CONDITIONS FOR STRICT EFFICIENCY
25
By Theorem 4.1, the point x = 0 is a strict local Pareto minimizer of order k of
the function f over C with respect to the cone R2+ .
In what follows we give a sufficient condition for strict local Pareto minima.
Theorem 4.2. Let dim X < +∞. Assume that for every d ∈ KC (x) \ {0},
there is a positive integer number n (depending on d) such that the directional
derivatives f (j) (x; d) (j = 0, 1, . . . , n) exist, and one of conditions (A1 ), . . . , (An )
holds. Then x is a strict local Pareto minimum for (P).
Proof. Assume the contrary, that x is not a strict local Pareto for (P ).
Then, by Proposition 2.3, for each integer number s > 1, there would exist a
sequence xm ∈ C, xm 6= x, xm → x and bm ∈ Q such that
f (xm ) − f (x) + bm
= 0.
m→+∞
kxm − xks
lim
As also in the proof of Theorem 4.1, putting tm = kxm − xk and dm =
xm − x
, we get that xm = x + tm dm ∈ C, and dm → d with kdk = 1. Hence,
kxm − xk
d ∈ KC (x) \ {0}. For the direction d, there exists a positive integer number n
such that condition (Ak ) is satisfied for some k ∈ [1, . . . , n] and (11) holds. By
an argument to that used for the proof of Theorem 4.1, we arrive at
f (k) (x; d) ∈ −Q,
and this contradicts condition (Ak ).
5. Higher-Order Conditions for Strict Efficiency in Case Q = Rr+
In this section, different higher-order necessary and sufficient conditions for
strict local Pareto minima are provided in case Y = Rr , Q is the nonnegative
orthant Rr+ in Rr , and so f = (f1 , . . . , fr ). As also above, sufficient conditions
for strict efficiency are established in case X is finite dimensional.
Theorem 5.1. Let Q = Rr+ . Assume that x is a strict local Pareto minimum
of Problem (P). Then for every d ∈ KC (x), there exists an index i ∈ {1, . . . , r}
such that the following conditions hold:
(0)
(a) fi,+ (x; d) > fi (x).
26
DO VAN LUU AND PHAM TRUNG KIEN
(0)
(j)
(b) If fi,+ (x; d) = fi (x), fi,+ (x; d) = 0 (j = 1, . . . , n − 1), then
(n)
fi,+ (x; d) > 0,
where
(j)
fi,+ (x; d)
j−1 k
i
X
j! h
t (k)
′
= lim sup j fi (x + td ) −
fi,+ (x; d)
t
k!
t↓0
d′ →d
(0)
fi,+ (x; d)
(j = 1, . . . , n),
k=0
= lim sup fi (x + td′ ).
t↓0
d′ →d
Proof. Assume that x is a strict local Pareto minimum for (P). We invoke Proposition 2.2 to deduce that there exist a neighborhood U of x and sets
V1 , . . . , Vs (s 6 r) such that {Vj , j = 1, . . . , s} is a covering of C ∩ U \ {x} and
verifying
fj (x) > fj (x) (∀ x ∈ Cj \ {x}),
(14)
where Cj = (C ∩ U ∩ Vj ) ∪ {x}.
We now can see that
C∩U =
s
[
Cj .
(15)
j=1
Taking account of a result due to Aubin-Frankowska (see [1, Table 4.1]), it follows
from (15) that
s
[
KC (x) = KC∩U (x) =
KCj (x).
(16)
j=1
Taking d ∈ KC (x), it follows from (16) that there exists i ∈ {1, . . . , s} so that
d ∈ KCi (x). In view of (14), we get that
fi (x) > fi (x) (∀ x ∈ Ci \ {x}).
(17)
Since d ∈ KCi (x), there exist sequences tm ↓ 0 and dm → d such that x + tm dm ∈
Ci \ {x}. Hence, by (17) it holds that
fi (x + tm dm ) > fi (x),
which leads to the following
(0)
fi,+ (x; d) = lim sup fi (x + td′ ) > lim sup fi (x + tm dm ) > fi (x).
t↓0
d′ →d
m→+∞
(18)
ON HIGHER-ORDER CONDITIONS FOR STRICT EFFICIENCY
27
We thus arrive at the conclusion (a).
To prove (b) we take d ∈ KC (x) satisfying
(0)
(j)
fi,+ (x; d) = 0 (j = 1, . . . , n − 1).
fi,+ (x; d) = fi (x),
These along with (18) yields that
(n)
fi,+ (x; d) = lim sup
t↓0
n−1
i
X tj (j)
n! h
′
f
(x
+
td
)
−
f
(x;
d)
i
tn
j! i,+
d′ →d
j=0
n−1
i
X tjm (j)
n! h
> lim sup n fi (x + tm dm ) −
fi,+ (x; d)
j!
m→+∞ tm
j=0
n!
= lim sup n [fi (x + tm dm ) − fi (x)]
m→+∞ tm
> 0,
as was to be shown.
The following example illustrates Theorem 5.1.
Example 5.1. Let X = Y = R2 , Q = R2+ , C = [0, 1] × [−1, 0], x = (0, 0).
Consider on R2 the function:
f (x) = max |x1 |, |x2 | , −|x2 |k ,
where k is a positive integer number. Setting f1 (x) = max |x1 |, |x2 | and f2 (x) =
−|x2 |k , one gets f = (f1 , f2 ). Obviously, KC (x) = R+ × R− , and the point x = 0
is a strict local Pareto minimizer of f over C with respect to the cone R2+ . Then,
for d = (d1 , d2 ) ∈ KC (x),
(0)
(j)
f2,+ (0; d) = f2 (0) = 0; f2,+ (0; d) = 0
(k)
(j = 1, . . . , k − 1); f2,+ (0; d) = |d2 |k > 0.
We close the paper with a high-order sufficient condition in terms of Ginchev’s
lower directional derivatives for strict local Pareto minima of order n.
Theorem 5.2. Let dim X < +∞, Y = Rr and Q = Rr+ , and let n be a
positive integer number. Assume that there exists i0 ∈ {1, . . . , r} such that for
every d ∈ KC (x) \ {0}, one of the following conditions (Bk ) (k = 1, . . . , n) holds:
28
DO VAN LUU AND PHAM TRUNG KIEN
(0)
(j)
(Bk ) fi0 ,− (x; d) = fi0 (x), fi0 ,− (x; d) = 0 (j = 1, . . . , k − 1),
(k)
fi0 ,− (x; d) > 0.
Then x is a strict local Pareto minimum of order n for Problem (P), where
(j)
fi0 ,− (x; d) = lim inf
t↓0
j−1 k
i
X
j! h
t (k)
′
f
(x
+
td
)
−
(x;
d)
f
i
tj 0
k! i0 ,−
k=0
d′ →d
(0)
fi0 ,− (x; d)
(j = 1, . . . , n)
′
= lim inf fi0 (x + td ).
t↓0
d′ →d
Proof. We shall begin with showing that x is a strict local minimum of
order n of the following scalar problem:
(Pi0 )
min{fi0 (x) : x ∈ C},
in which fi0 is a component of the vector f = (f1 , . . . , fr ) as in the statement of
this theorem, and C is as in Problem (P).
Suppose that condition (Bk ) is fulfilled, but x is not a strict local minimum
of order n for (Pi0 ). Then for any integer number m > 1, there would exist
xm ∈ C, xm 6= x, xm → x such that
fi0 (xm ) 6 fi0 (x) +
1
kxm − xkn .
m
(19)
xm − x
yields that xm = x + tm dm ∈ C.
kxm − xk
Since dim X < +∞, there exists a subsequence of {dm } converging to d with
Setting tm = kxm − xk and dm =
kdk = 1. Without loss of generality, we can assume that dm → d. Therefore,
d ∈ KC (x) \ {0}.
Now, in view of (19), it follows that
fi0 (x + tm dm ) 6 fi0 (x) +
tnm
kdkn .
m
Since tm → 0+ as m → +∞, it follows readily from this that for k = 1, . . . , n,
fi0 (x + tm dm ) 6 fi0 (x) +
tkm
kdkn .
m
(20)
ON HIGHER-ORDER CONDITIONS FOR STRICT EFFICIENCY
29
Making use of condition (Bk ), we obtain
(0)
fi0 ,− (x; d) = fi0 (x),
(j)
fi0 ,− (x; d) = 0
(j = 1, . . . , k − 1),
which together with (20) yields that
(k)
fi0 ,− (x; d) = lim inf
t↓0
k−1 j
i
X
k! h
t (j)
′
(x
+
td
)
−
f
(x;
d)
f
i
tk 0
j! i0 ,−
j=0
d′ →d
k−1 j
i
X
k! h
tm (j)
f
(x
+
t
d
)
−
f
(x;
d)
i
m
m
0
m→+∞ tk
j! i0 ,−
m
j=0
i
k! h
= lim inf k fi0 (x + tm dm ) − fi0 (x)
m→+∞ tm
i
k! h tk
6 lim inf k m kdkn
m→+∞ tm m
= 0.
6 lim inf
(k)
But condition (Bk ) gives that fi0 ,− (x; d) > 0. We thus arrive at a contradiction.
Consequently, x is a strict local minimum of order n for Problem (Pi0 ).
We now have to show that x is a strict local Pareto minimum of order n
for Problem (P). If it was not so, then by Proposition 2.1, there would exist a
sequence xm ∈ C, xm 6= x and bm = (bm1 , . . . , bmr ) ∈ Q = Rr+ such that xm → x,
and
f (xm ) − f (x) + bm
lim
= 0,
m→+∞
kxm − xkn
which leads to the following
lim
m→+∞
fi (xm ) − fi (x) + bmi
= 0 (i = 1, . . . , r).
kxm − xkn
In particular,
fi0 (xm ) − fi0 (x) + bmi0
= 0.
m→+∞
kxm − xkn
We again invoke Proposition 2.1 to deduce that x is not a strict local minimum
of order n for (Pi0 ). We thus arrive at a contradiction.
lim
Example 5.2. X = Y = R2 , Q = R2+ , C = [−1, 0] × [0, 1], x = (0, 0). Let f
be defined as
k
(x = (x1 , x2 ) ∈ R2 ),
f (x) = − |x1 |, (x21 + x22 ) 2
30
DO VAN LUU AND PHAM TRUNG KIEN
where k is a positive integer number. Thus f = (f1 , f2 ) with f1 (x) = −|x1 | and
f2 (x) = kxkk . We have that KC (x) = R− × R+ , and the point x = 0 is a strict
local minimizer of order k of the function f2 over C. According to Theorem 5.2, x
is a strict local Pareto minimizer of order k of the function f over C with respect
to the cone R2+ .
Remark 5.1. Theorems 5.1 and 5.2 here are generalizations of Theorems 1
and 2 due to Ginchev [6], respectively.
Discussion 5.1. From the point of view of the “gap” between necessary
and sufficient conditions, we can see that in Theorems 3.1 and 4.1 the gap is the
boundary of the cone −Q, which means that the higher-order derivatives of f
can not be in the boundary of −Q in sufficiency conditions and they may be in
the boundary of −Q in necessary conditions.
In view of working with the coordinate functions f1 , . . . , fr instead of working
directly with the function f as in Sections 3 and 4, the gap between the necessary
and sufficient conditions in Theorems 5.1 and 5.2 becomes smaller than the gap
between Theorems 3.1 and 4.1. The gap between Theorems 5.1 and 5.2 only
is the vertex of the cone Q = −Rr+ , but not the boundary of −Rr+ as the gap
between Theorems 3.1 and 4.1.
Acknowledgments
The authors would like to thank the two anonymous referees for their valuable
comments.
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Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam.
Finalcial Institute, Hanoi, Vietnam.