**Rev. Anal. Num´****er. Th´****eor. Approx., vol. 37 (2008) no. 1, pp. 37–46**
**ictp.acad.ro/jnaat**

SOME CONE SEPARATION RESULTS AND APPLICATIONS

MARIUS DUREA^{∗}

**Abstract.** In this note we present some cone separation results in infinite
dimensional spaces. Our approach is mainly based on two different types
of cone outer approximation. Then we consider an application to vector
optimization.

**MSC 2000.** Primary: 90C29; Secondary: 90C26, 49J52.

**Keywords.** Cone separation results, cone enlargements, vector optimiza-
tion.

1. INTRODUCTION AND PRELIMINARIES

The separation theorems are among the main tools in optimization theory and are especially used to obtain necessary optimality conditions. Starting with the classical convex separation results, a lot of such results have been used in the last decades in order to obtain new optimality results or in order to generalize the existing ones. In general, the functional separation by means of linear continuous maps is, by far, the most common way to separate two sets.

In this work we consider the method of “cone separation” having the same
meaning as in the paper of Henig [6, Definition 2.1, Theorem 2.1]: consider
two cones *P, Q*in a normed vector space with*P* ∩*Q*={0},and find another
cone*K* which strictly separate them, i.e.,*P*∩*K*={0}and*Q*⊂int*K* (where
int denotes the topological interior). Moreover, we want to specify the form
of the cone*K* and we base our analysis on two models of outer approximation
of cones: the Henig dilating cones and the plastering cones in the sense of
Krasnosel’ski˘ı [8]. Let us note that, in fact, the concept of Henig dilating
cones was subsequently extracted by various authors from the proof of the
main result in [6].

The cone separation technique we briefly described above has applications in multicriteria optimization because, once we can apply cone separation, we can treat the Pareto minima as weak Pareto minima and it is well-known that in vector optimization, in general, it is much more easier to work with weak minimality. We shall follow this line in the applications section of this paper.

∗“Al. I. Cuza” University, Faculty of Mathematics, Bd. Carol I, nr. 11, 700506, Ia¸si, Romania, e-mail: [email protected]

The main results of the paper are done in Section 2. Section 3 is dedicated to some applications of the results in Section 2 to the field of vector optimiza- tion. More exactly, we obtain some necessary optimality conditions for Pareto minima in reflexive Banach spaces.

Let us start with some preliminaries. Throughout *Y* is a normed vector
space, unless otherwise stated. If *a* ∈ *Y* and *ρ >* 0, we denote by *B*(a, ρ)
(resp. *D(a, ρ)) the open (resp. closed) ball centered at* *a*with radius *ρ. The*
symbol*S(a, ρ) denotes the sphere centered ata*with radius *ρ. For simplicity*
we shall also use the notation *S** _{Y}* :=

*S(0,*1).If

*A, B*⊂

*Y*are two subsets, the distance from

*A*to

*B*is

*d(A, B) := inf*{ka−

*bk |a*∈

*A, b*∈

*B*}.As usual, for a point

*x*∈

*Y, d(x, A) :=d({x}, A).*The distance function will be denoted as

*d*

*A*:

*Y*→R, d

*A*(x) :=

*d(x, A).*

Let *K*⊂*Y* be a proper cone. The dual cone of the cone*K* ⊂*Y* is defined
as

*K*^{∗}:={y^{∗} ∈*Y*^{∗} |*y*^{∗}(y)≥0,∀y ∈*K}*

and the quasi-interior of *K*^{∗} is

*K** ^{]}*:={y

^{∗}∈

*Y*

^{∗}|

*y*

^{∗}(y)

*>*0,∀y ∈

*K*\ {0}}.

where*Y*^{∗}is the topological dual of*Y.*It is clear that if*K*and*Q*are two cones
with *K* \ {0} ⊂ int*Q* then *Q*^{∗} ⊂ *K*^{∗} and *Q*^{∗}\ {0} ⊂ *K*^{]}*.* A convex set *B* is
said to be a base for the cone *K* if 0∈*/*cl*B* and *K*= cone*B*, where cl denotes
the topological closure and cone*B* := [0,∞)B is the cone generated by *B. A*
cone which admits a base is called based.

2. CONE SEPARATION RESULTS

In the setting of general vector spaces, the following result is well-known (see [2]) and it puts into light the main assumptions one should assume when deal- ing with cone separation results. We present a proof of it for the completeness and for easy comparison with the further results derived in this paper.

Theorem 1. *Let* *Y* *be a locally convex space and* *P, S* ⊂ *Y* *be cones s.t.*

*P* ∩*S* = {0}. *If* *P* *is closed and* *S* *has compact base* *B, then there exists a*
*convex pointed cone* *K* *s.t.* *S*\ {0} ⊂int*K* *andP* ∩*K* ={0}.

*Proof.* The relation*P*∩S ={0}implies that 0∈*/* *P*−B.Since*P*is closed and
*B* is compact, the set*P*−*B* is closed and, taking into account the topological
separation properties of locally convex spaces, there exists a balanced convex
neighborhood *V* of 0 s.t.

(2.1) *V* ∩(P−*B) =*∅.

This means that

0∈*/P* −(B+*V*).

It is easy to see that the cone

*K* := cone(B+*V*)

satisfies the requirements.

The above proof shows that, in fact, for normed vector spaces the cone *K*
can be taken in the form

*K* := cone({y∈*Y* |*d(y, B*)≤*ε}).*

The properties of this theoretical construction are presented in the next result.

Proposition 1. ([5, Lemma 3.2.51]) Let *K* ⊂ *Y* be a closed convex cone
with a base *B* and take *δ* = *d(0, B)* *>* 0. For *ε* ∈(0, δ), consider *B**ε* ={y ∈
*Y* |*d(y, B)*≤*ε}*and *K** _{ε}*= [0,∞)B

_{ε}*,*the cone generated by

*B*

_{ε}*.*Then

(i) *K**ε* is a closed convex cone for every*ε*∈(0, δ);

(ii) if 0*< γ < ε < δ, K*\ {0} ⊂*K** _{γ}*\ {0} ⊂int

*K*

*;*

_{ε}(iii) *K*=∩_{ε∈(0,δ)}*K** _{ε}*=∩

*n∈*N

*K*

_{ε}*where (ε*

_{n}*)⊂(0, δ) converges to 0.*

_{n}The cones*K**ε*constructed in this way are termed as dilating cones or Henig
dilating cones. It is important to note that the cone *K** _{ε}* has a non-empty
interior. Moreover, we can specify the exact form of the elements in the dual
cone of such a dilating cone with respect to the elements in the dual cone of
the original ordering cone. This was done in [3].

Proposition 2. *Let* *K* ⊂ *Y* *be a closed convex cone with a base* *B.* *For*
*every* *ε*∈(0, d(0, B)),

*K*_{ε}^{∗} ={y^{∗}∈*Y*^{∗} | inf

*b∈B**y*^{∗}(b)≥*ε*ky^{∗}k}.

If we analyze the proof of Theorem 1 we observe that one of the essential
facts is that*P*−Bis closed. In order to generalize Theorem 1 we use a criterion
for the closedness of the difference of two non-convex sets derived by Z˘alinescu
in [11]. We present here only the particular case we are interested in and to
this end we need the notion of asymptotically compact subset of *Y* in the
general framework when*Y* is a locally convex space endowed with a topology
*τ* compatible with the duality system (Y, Y^{∗}). A subset *A* of *Y* is called *τ*-
asymptotically compact (τ-a.c. for short) if there exists a neighborhood *U* of
0 in (Y, τ) s.t. *U* ∩[0,1]A is *τ*-relatively compact. In the case when *τ* is the
strong topology, then we shall simply use the term “asymptotically compact”.

If the topology *τ* in question is the weak topology we call it “asymptotically
weakly compact”.

The asymptotic cone of a nonempty set*A*⊂*Y* (with respect of*τ*) is defined
by

*A*^{τ}_{∞}:={u∈*Y* | ∃(t* _{n}*)→0

_{+}

*,*(a

*n*)⊂

*A*:

*t*

*n*

*a*

*n*

*τ*

→*u}*= ^{\}

*t>0*

cl* _{τ}*([0, t]A)
and it is well known that, in general,

*A*

^{τ}_{∞}is a

*τ*-closed cone. When we deal with the norm topology we omit the superscript.

Proposition3. *LetY* *be a normed vector space andC, D*⊂*Y* *be nonempty*
*(weakly) closed sets. If* *C*∞∩*D*∞ = {0} *(C*_{∞}* ^{w}* ∩

*D*

^{w}_{∞}= {0}),

*and*

*C*

*or*

*D*

*is*

*(weakly) a.c., then*

*C*−

*D*

*is a (weakly) closed set.*

The first cone separation result we would like to mention was mainly ob- tained in [3] and reads as follows.

Theorem 2. *Let* *P, Q*⊂*Y* *be (weakly) closed cones such that* *Q* *admits a*
*(weakly) closed base* *B. Moreover, suppose that* *P* *or* *B* *is (weakly) a.c. If*
*P*∩*Q*={0}, then there exists *ε*∈(0, d(0, B)) *s.t.* *P* ∩*Q**ε* ={0}.

*Proof.* Since*P*∩*Q*={0},one deduces that*P*∩*B* =∅,which is equivalent
to 0∈*/P*−B.On the other hand, one has that*P*_{∞}* ^{τ}* ⊂

*P, Q*

^{τ}_{∞}⊂

*Q,*because

*P, Q*are

*τ*-closed cones (in both cases of weak or strong convergence). Therefore,

{0} ⊂*P*_{∞}* ^{τ}* ∩

*B*

_{∞}

*⊂*

^{τ}*P*∩

*Q*

^{τ}_{∞}=

*P*∩

*Q*={0},

whence, from Proposition 3, one deduces that *P*−*B* is a (weakly) closed set,
therefore in any case it is norm closed whence *α* := *d(0, P* −*B)* *>* 0. Let
us take 0 *< ε <* min(α/2, d(0, B)) and we prove that 0 ∈*/* *P* −*B**ε* (in the
notations of Proposition 1). This is obvious, because otherwise there would
exist *b**ε*∈*B**ε*∩*P* and *b*∈*B* s.t. kb* _{ε}*−

*bk<*2ε.Then

*d(0, P* −*B)*≤ kb* _{ε}*−

*bk<*2ε < α,

a contradiction. Consequently,*P*∩ −B* _{ε}*=∅,so

*P*∩ −Q

*={0},and the thesis*

_{ε}is proved.

Since a*τ*-compact set is automatically*τ*-a.c. one has the following corollary.

Corollary 1. *Let* *P, Q* ⊂*Y* *be (weakly) closed cones such that* *Q* *admits*
*a (weakly) closed base* *B. Moreover, suppose that* *P* *or* *B* *is (weakly) a.c. If*
*P*∩*Q*={0} *then there exists* *ε*∈(0, d(0, B)) *s.t.* *P* ∩*Q**ε* ={0}.

If one compares Theorem 1 with Theorem 2 one can observe (apart from
the specific setting) that in the latter result the compactness of the base is
replaced by asymptotically compactness. But, if one looks at the proofs, one
can observe that Theorem 1 works for any closed set *P* (not necessarily a
cone), while in Theorem 2 one uses that*P* is a (τ-closed) cone in the relation
*P*_{∞}* ^{τ}* ⊂

*P.*

Next, we use another method to approximate an original cone by a larger
one (having nonempty interior). For a given cone *K, the following conical*
*ε-enlargement (ε >*0) is studied in the literature (see [8], [1]):

*K** ^{ε}*={u∈

*X*|

*d(u, K)*≤

*ε*kuk}.

It is clear that the so-defined *K** ^{ε}* is a cone (since for every

*x*∈

*X*and

*t*≥0,

*d(tx, K*) =

*td(x, K)) which containsK. It is also clear that forε*≥1, K

*=*

_{ε}*X.*

It is shown in [1, Proposition 3.2.1] that a closed pointed convex cone*K*admits
a convex*ε-enlargement if and only ifK* has a bounded base. Other properties
of this construction are listed below.

Proposition 4. *Let* *K* ⊂*Y* *be a cone and* *K*^{ε}*be the cone defined above.*

*Then:*

(i) *K** ^{ε}*= cone

*B*

^{ε}*,*

*where*

*B*

*:={u∈*

^{ε}*S*

*Y*|

*d(u, K)*≤

*ε};*

(ii) *for everyy*∈*K, D(y,*(1 +*ε)*^{−1}*ε*kyk)⊂*K** ^{ε}*;

(iii) *ify*^{∗} ∈(K* ^{ε}*)

^{∗}

*then for everyy*∈

*K, y*

^{∗}(y)≥(1 +

*ε)*

^{−1}

*ε*kyk ky

^{∗}k

*.*

*Proof.*(i) If

*u*∈

*K*

^{ε}*,*then

*d(u, K)*≤

*ε*kuk

*,*whence

*d(kuk*

^{−1}

*u, K)*≤

*ε.*It is clear that

*v*:= kuk

^{−1}

*u*∈

*B*

*and since*

^{ε}*u*=kuk

*v,*one has that

*u*∈cone

*B*

^{ε}*.*Conversely, if

*u*∈ cone

*B*

*then there exists*

^{ε}*v*∈

*B*

*and*

^{ε}*α*≥ 0 s.t.

*u*=

*αv.*

Therefore,

*d(u, K) =d(αv, K*) =*αd(v, K)*≤*αε*=*ε*kuk
and the first part is proved.

(ii) Let *y*∈*K* and take*v*∈*D(y,*(1 +*ε)*^{−1}*ε*kyk).It is obvious that
kvk ≥ kyk −(1 +*ε)*^{−1}*ε*kyk

= (1 +*ε)*^{−1}kyk
and, on the other hand,

*d(v, K)*≤ kv−*yk ≤*(1 +*ε)*^{−1}*ε*kyk ≤*ε*kvk*.*
This shows that *v*∈*K** ^{ε}*.

(iii) Let *y*^{∗} ∈(K* ^{ε}*)

^{∗}and

*y*∈

*K.*Following the preceding item,

*y*

^{∗}(v)≥0 for every

*v*∈

*D(y,*(1 +

*ε)*

^{−1}

*ε*kyk).In particular,

*y*^{∗}(u)≥(1 +*ε)*^{−1}*ε*kyk*y*^{∗}(u)

for every*u*∈*S*_{Y}*.*This is the conclusion and the proof is complete.

We put into relation the two types of enlargements in a particular situation (see also [1, Propositions 2.1.1, 3.2.1]).

Proposition 5. *Let* *K* ⊂ *Y* *be a closed convex cone with a bounded base*
*B.* *Then for every* *ε >*0 *there is a* *δ >*0 *s.t.* *K** _{δ}* ⊂

*K*

^{ε}*.*

*Proof.* Since 0 ∈*/* cl*B,* there exists *µ* := *d(0, B)* *>* 0 s.t. kbk ≥ *µ* for
every *b* ∈ *B.* Then, from Proposition 4, (ii) we have that for every *b* ∈ *B,*
*D(b,*(1 +*ε)*^{−1}*εµ)* ⊂ *K*^{ε}*,* whence *B*_{(1+ε)}^{−1}* _{εµ/2}* ⊂

*K*

^{ε}*,*i.e. the conclusion for

*δ* = (1 +*ε)*^{−1}*εµ/2.*

Definition 1. *Let* *K* ⊂ *Y* *be a cone. One says that* *K* *has the property*
(S) *if there exist* *x*^{∗}_{1}*, x*^{∗}_{2}*, ..., x*^{∗}* _{n}*∈

*Y*

^{∗}

*(n*∈N\ {0}) s.t.

*K* ⊂ {u∈*Y* | kuk ≤ max

*i=1,n*

{x^{∗}* _{i}*(u)}.

Observe that in the case*n*= 1 in the above definition one gets the definition
of the supernormal cone. It is well known that a cone in a normed vector
space is supernormal if and only if it has a bounded base (see, e.g., [5, p. 37]).

However, is it easy to see that there are cones with the property (S) which are not supernormal: take, for example,

*K*:={(x* _{n}*)∈

*l*

^{2}|(x

*) = (x*

_{n}_{1}

*,*0,0,0...), x

_{1}∈R}.

We are interested in considering this property because if a cone*K* has the
property (S) then it is clear that

(y* _{n}*)⊂

*K, y*

*→*

_{n}*0⇒*

^{w}*y*

*→0.*

_{n}We are now able to present the main result of this note.

Theorem 3. *Let* *Y* *be a reflexive Banach space,* *P, Q* ⊂*Y* *be cones such*
*that* *P, Q* *are weakly closed and* *Q* *has the property* (S). If *P* ∩*Q*={0} *then*
*there exists* *ε >*0 *s.t.* *P*∩*Q** ^{ε}*={0}.

*Proof.* We proceed by contradiction. Suppose that for every *n* ∈ N\ {0}

there exists*u** _{n}*∈

*P*∩

*Q*

^{1/n}

*, u*

*6= 0.Since*

_{n}*u*

*∈*

_{n}*Q*

^{1/n}we have

*d(u*

*n*

*, Q)*≤

*n*

^{−1}ku

*k*

_{n}i.e.

*d(ku** _{n}*k

^{−1}

*u*

*n*

*, Q)*≤

*n*

^{−1}

*.*

By the definition of the distance function, for every *n*∈N\ {0},there exists
*v**n*∈*Q* s.t.

ku* _{n}*k

^{−1}

*u*

*−*

_{n}*v*

_{n}^{}

^{}

_{}≤2n

^{−1}

*.*

Taking into account that *Y* is reflexive and the sequence (ku* _{n}*k

^{−1}

*u*

*n*)

*is bounded there exists a subsequence of it (denoted in the same way) weakly convergent towards an element*

_{n}*u*∈

*Y.*We show that

*u*6= 0. Indeed, in the contrary case ku

*k*

_{n}^{−1}

*u*

*n*

*w*

→ 0 and since ku* _{n}*k

^{−1}

*u*

*n*−

*v*

*n*→ 0 (in norm topol- ogy) we get that

*v*

*n*

*w*

→0.Since*Q*has the property (S),we obtain that (v*n*) is
strongly convergent towards 0,whenceku* _{n}*k

^{−1}

*u*

*n*→0 and this is not possible becauseku

*k*

_{n}^{−1}

*u*

*n*has the norm 1 for every

*n*∈N\ {0}.Consequently,

*u*6= 0.

Moreover, since *P* is weakly closed, *u* ∈ *P* \ {0}. Finally, *v**n* *w*

→ *u, so* *u* ∈ *Q*
and we arrive at a contradiction. The thesis is proved.

Note that a necessary and sufficient condition to have the conclusion of
above theorem (i.e. *P*∩*Q*={0} implies that there is an*ε >*0 s.t. *P*∩*Q** ^{ε}*=
{0}) are given in [10] and it can be written in our notation as: there exists an

*ε >*0 s.t.

*P*

*∩*

^{ε}*Q*

*={0}. In particular, this holds if*

^{ε}*P, Q*are (norm) closed and

*P*is (norm) locally compact. We are interested here to consider somehow different conditions in view of the applications we envisage and in order to avoid the local compactness condition.

We record the following consequences.

Corollary 2. *Let* *Y* *be a reflexive Banach space,P, Q*⊂*Y* *be cones such*
*that* *P, Q* *are weakly closed andQ* *has a bounded base. If* *P*∩*Q*={0} *then:*

(i) *there existsε >*0 *s.t.* *Q*^{ε}*is convex, pointed andP* ∩*Q** ^{ε}*={0}.

(ii) *there existsε >*0 *s.t.* *P* ∩*Q**ε*={0}.

*Proof.* (i) Since *Q* has a bounded base, then it admits a convex *ε-enlar-*
gement. Taking a smaller *ε,* if necessary, and using Theorem 3 we get the
conclusion.

(ii) This part follows easily from Proposition 5. We would like to present
as well a proof based on Corollary 1. First, since *Q* has a base, it is convex.

Moreover, since it is closed and its base *B* is bounded, then cl*B* is a base as
well, so we can consider that*Q*has a closed bounded base. Since in a reflexive
Banach space a bounded weakly closed set is weakly compact we can apply

Corollary 1 to get the conclusion.

In order to illustrate the application field of this corollary let us consider
*Y* =*L*_{2}(Ω) where Ω is a non-empty subset of R* ^{n}* and

*Q*:=

*L*

_{2}(Ω) is the well known space of square Lebesgue-integrable functions

*f*: Ω → R. The space

*L*

_{2}(Ω) is a Hilbert space and hence (L

_{2}(Ω))

^{∗}=

*L*

_{2}(Ω) and the natural ordering cone in

*L*

_{2}(Ω) is given as

*L*^{+}_{2}(Ω) ={f ∈*L*2(Ω)|*f*(x)≥0 almost everywhere on Ω}.

It is interesting to note that (L^{+}_{2}(Ω))^{∗} =*L*^{+}_{2}(Ω) and*L*^{+}_{2}(Ω)* ^{]}* is non-empty (see
for example [7]). Thus by considering an element

*x*

^{∗}∈

*L*

^{+}

_{2}(Ω)

*then*

^{]}*B* ={x∈*L*^{+}_{2}(Ω)|*x*^{∗}(x) = 1}

is a base for *L*^{+}_{2}(Ω) and it is is weakly compact and hence bounded.

At the end of this section, we would like to present another possible en-
largement, smaller than*K** ^{ε}*, but which can be obtained in a similar way. Let

*K*⊂

*Y*be a cone. Based on Proposition 4 let us define

*A** ^{ε}*:={u∈

*S*

*|*

_{Y}*d(u, S*

*∩*

_{Y}*K)*≤

*ε}.*

and

*K*_{1}* ^{ε}*:= cone

*A*

^{ε}*.*

Since*d(u, K)*≤*d(u, S**Y* ∩*K) for everyu*∈*Y,*it is clear that*A** ^{ε}* ⊂

*B*

^{ε}*,*whence

*K*

_{1}

*⊂*

^{ε}*K*

^{ε}*.*

Let us prove now that *K*\ {0} ⊂*K*_{1}^{ε}*.*For this, it is enough to prove that
*K* ∩*S** _{Y}* ⊂

*K*

_{1}

^{ε}*.*Take

*y*∈

*K*∩

*S*

_{Y}*.*Clearly,

*S*

*∩*

_{Y}*D(y, ε)*⊂

*A*

^{ε}*.*Suppose, by way of contradiction that there exists a sequence (y

*)→*

_{n}*y, y*

*∈*

_{n}*/K*

_{1}

*for every*

^{ε}*n*≥1.Therefore,

ky* _{n}*k

^{−1}

*y*

*n*→ kyk

^{−1}

*y*=

*y*

whence *v**n* :=ky* _{n}*k

^{−1}

*y*

*n*∈

*S*

*Y*

*, v*

*n*→

*y.*Since

*S*

*Y*∩

*D(y, ε)*⊂

*A*

*,*

^{ε}*v*

*n*∈

*A*

*for*

^{ε}*n*large enough. This shows that

*y*

*∈*

_{n}*K*

_{1}

*and this is a contradiction. Using these observations and Theorem 3 we get the next result.*

^{ε}Proposition6. *LetY* *be a reflexive Banach space,P, Q*⊂*Y* *be cones such*
*that* *P, Q* *are weakly closed and* *Q* *has the property* (S). If *P* ∩*Q*={0} *then*
*there exists* *ε >*0 *s.t.* *P*∩*Q*^{ε}_{1}={0}.

3. AN APPLICATION

In this section we present some applications to vector optimization, mainly based on Corollary 2 (i),i.e. on the separation with enlargement cones. Note that several applications based on the separation with dilating cones have been done in [3].

Let *K* be a proper closed convex cone. It is well known that such a cone
induces a partial order relation≤* _{K}* on

*Y*by

*y*

_{1}≤

_{K}*y*

_{2}if and only if

*y*

_{2}−y

_{1}∈

*K*and, with respect to this relation, one can consider different types on minimum points. If

*A*⊂

*Y*is a nonempty set, then a point

*y*∈

*A*is called a Pareto minimum point for

*A*with respect to

*K*if (A−

*y)*∩ −K ⊂

*K*. In particular, if

*K*is pointed, this means that (A−

*y)*∩ −K = {0}. Observe that, in fact the condition (A−

*y)*∩ −K ⊂

*K*(resp. (A−

*y)*∩ −K ={0}) is equivalent by cone(A−

*y)*∩ −K ⊂

*K*(resp. cone(A−

*y)*∩ −K ={0}). If int

*K*6=∅, then a point

*y*∈

*A*is called weak Pareto minimum point of

*A*with respect to

*K*if (A−

*y)*∩ −int

*K*=∅, i.e. it is a minimum point for

*A*with respect to the cone int

*K*∪ {0}. One denotes by Min(A |

*K) (WMin(A*|

*K*), respectively) the sets of Pareto (weak Pareto, respectively) minima of

*A*with respect to

*K.*

It is well-known that the main difficulty which arise when dealing with Pareto minima consist of the emptiness of the interior of the underlying ordering cone.

Unfortunately, this situation is a common one for the majority of the natural ordering cones of the usual Banach spaces. In contrast, when the cone has nonempty interior one has several tools to handle the (weak) Pareto minima.

One of this tools is the following scalarizing lemma (see [5, Section 2.3] and
[4]). The symbol*∂* denotes the Fenchel subdifferential of a convex function.

Lemma 1. *Let* *K* ⊂*Y* *be a closed convex cone with nonempty interior and*
*let* *e*∈int*K, M* ⊂*Y, y*∈*M.* *Define the functional* *ϕ** _{e}*:

*Y*→R

*as*

*ϕ** _{e}*(y) = inf{λ∈R|

*y*∈

*λe*−

*K}.*

*This map is continuous, convex, strictly-intK-monotone,d(e,bd(Q))*^{−1}*-Lipschitz*
*(where* bd*Q* *denotes the topological boundary of* *Q) and for every* *λ*∈R

{y |*ϕ**e*(y)≤*λ}*=*λe*−*K,* {y|*ϕ**e*(y)*< λ}*=*λe*−int*K.*

*Moreover, for everyu*∈*Y,*

*∂ϕ**e*(u) ={v^{∗} ∈*K*^{∗} |*v*^{∗}(e) = 1, v^{∗}(u) =*ϕ**e*(u)}.

*The point* *y* *is a weak minimum point for* *M* *with respect to* *K* *if and only if*
*y* *is a minimum point for* *ϕ** _{e}*(· −

*y)*

*onM.*

The other major tools we use in this section are the generalized differentia- tion objects introduced and developed by Mordukhovich and his collaborators (see [9, Vol. I]). We restrict ourselves to the case of Asplund spaces because we shall work in the setting of reflexive Banach spaces, a subset of the collections of Asplund spaces.

Definition 2. *Let* *X* *be an Asplund space and* *S* ⊂ *X* *be a non-empty*
*closed subset ofX* *and let* *x*∈*S.*

(i) *The basic (or limiting, or Mordukhovich) normal cone to* *S* *atx* *is*
*N** _{M}*(S, x) :={x

^{∗}∈

*X*

^{∗}| ∃x

*→*

_{n}

^{S}*x, x*

^{∗}

_{n}*→*

^{w}^{∗}

*x*

^{∗}

*, x*

^{∗}

*∈*

_{n}*N*

*(S, x*

_{F}*)}*

_{n}*whereN**F*(S, z)*denotes the Fr´echet normal cone toS* *at a pointz*∈*S,*
*given as*

*N**F*(S, z) :={x^{∗}∈*X*^{∗}| lim sup

*u∈S,u→z*
*x*^{∗}(u−z)

ku−zk ≤0}.

(ii) *Let* *f* :*X*→R *be finite atx*∈*X;* *the Fr´echet subdifferential of* *f* *atx*
*is the set*

*∂f(x) :=*ˆ {x^{∗} ∈*X*^{∗} |(x^{∗}*,*−1)∈*N** _{F}*(epi

*f,*(x, f(x)))}

*and the basic (or limiting, or Mordukhovich) subdifferential off* *at* *x*
*is*

*∂*_{M}*f*(x) :={x^{∗} ∈*X*^{∗}|(x^{∗}*,*−1)∈*N** _{M}*(epi

*f,*(x, f(x)))},

*where*epi

*f*

*denotes the epigraph of*

*f.*

On the Asplund spaces one has

*∂*_{M}*f(x) = lim sup*

*x*→¯^{f}*x*

*∂f*ˆ (x),

and, in particular, ˆ*∂f(x)*⊂*∂*_{M}*f(x).* Of course, if a function*f* attains a local
minimum at a point *x* then 0∈*∂*_{M}*f*(x).If *δ*_{Ω} denotes the indicator function
associated with a nonempty set Ω⊂*X* (i.e. *δ*_{Ω}(x) = 0 if *x* ∈Ω, δ_{Ω}(x) =∞
if *x /*∈ Ω ), then for any *x* ∈ Ω, ∂_{M}*δ*_{Ω}(x) = *N** _{M}*(Ω, x). In contrast with the
Fr´echet subdifferential, the basic subdifferential satisfies a robust calculus rule:

if *X* is Asplund, *f*_{1} is Lipschitz around *x* and *f*_{2} is l.s.c. around this point,
then

(3.1) *∂**M*(f1+*f*2)(x)⊂*∂**M**f*1(x) +*∂**M**f*2(x).

Theorem 4. *Let* *Y* *be a reflexive Banach space and* *Q* ⊂ *Y* *be a weakly*
*closed cone with bounded base. Let* *A* ⊂ *Y* *be a set and* *y* ∈ Min(A | *K)*
*s.t.* cone(A−*y)* *is weakly closed. Then there exists* *ε >* 0 *such that for*
*every* *e* ∈ *K*\ {0} *there exists* *y*^{∗} ∈ −N* _{M}*(A, y)

*with*

*y*

^{∗}(e) = 1

*and*

*y*

^{∗}(y) ≥ (1 +

*ε)*

^{−1}

*ε*kyk ky

^{∗}k

*for everyy*∈

*Y.*

*Proof.* In our conditions, cone(A−y)∩K ={0},whence, following Corollary
2 (i), there exists a positive *ε* s.t. cone(A−*y)*∩*K** ^{ε}* = {0}, i.e.

*y*∈ Min(A|

*K*

*) ⊂ WMin(A |*

^{ε}*K*

*). We can apply Lemma 1 for the cone*

^{ε}*Q*:=

*K*

*and an element*

^{ε}*e*∈

*K*\ {0} ⊂ int

*K*

_{ε}*.*Then

*y*is a minimum point over

*A*for the functional

*s*

*e*(· −

*y) and then, by the infinite penalization method,*

*y*is a minimum point without constraints for

*s*

*(· −*

_{e}*y) +δ*

_{A}*.*Therefore,

0∈*∂** _{M}*(s

*(· −*

_{e}*y) +δ*

*)(y)*

_{A}and, since the first function is locally Lipschitz and the second one is lower- semicontinuous, we have

0∈*∂**M*(s*e*(· −*y))(y) +∂**M**δ**A*(y).

Moreover, the functional*s**e*(·−y) is sublinear and hence by using again Lemma
1 and Proposition 4 we obtain

*∂** _{M}*(s

*(· −*

_{e}*y))(y) =∂s*

*(0) ={y*

_{e}^{∗}∈(K

*)*

^{ε}^{∗}|

*y*

^{∗}(e) = 1}

={y^{∗}∈*Y*^{∗}|*y*^{∗}(y)≥(1+*ε)*^{−1}*ε*kyk ky^{∗}k*,∀y*∈*Y, y*^{∗}(e) = 1}.

On the other hand, *∂*_{M}*δ** _{A}*(y) =

*N*

*(A, y),whence the conclusion.*

_{M}Note that in this result the original cone *K* can have empty interior. If we
denote by *B* the base of*K, the propertyy*^{∗}(y)≥(1 +*ε)*^{−1}*ε*kyk ky^{∗}k for every
*y*∈*Y* fulfilled by *y*^{∗} yields, in particular,

*b∈B*inf *y*^{∗}(b)≥(1 +*ε)*^{−1}*ε*ky^{∗}k*d(0, B)*
(compare with [3, Theorem 4.1]).

Following the technique developed in [4], one can derive optimality condi- tions for vector optimization problems governed by single-valued or set-valued maps as well. However, in order to keep this note short, we restrict our atten- tion to the above case only.

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Received by the editors: June 25, 2008.