Locating minimal sets using polyhedral cones

Operations Research Letters 39 (2011) 466–470

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

Locating minimal sets using polyhedral cones Elvira Hernández ∗ , Luis Rodríguez-Marín Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Juan del Rosal 12, Madrid 28.040, Spain

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Article history: Received 15 April 2011 Accepted 16 August 2011 Available online 6 September 2011

abstract In this paper, we develop a theory of localization for minimal sets of a family S of nonempty subsets of Rn by considering polyhedral cones. To this end, we consider the first method to locate all efficient points of a nonempty set A ⊂ Rn introduced by Yu (1974) [10]. © 2011 Elsevier B.V. All rights reserved.

Keywords: Set optimization Global optimization Localization

1. Introduction Set-valued optimization problems are very interesting in several areas of applied mathematics (see for instance [6]). A general set-valued optimization problem is

(P)

Min F (x) subject to x ∈ M ,



where M is a nonempty set and F is a set-valued map from M to a topological vector space Y ordered by a convex cone K ⊂ Y . To solve problem (P), we can consider vector solutions (based on  the efficient line of F (M ) = x∈M F (x)) or set solutions (based on set relations between all image sets {F (x): x ∈ M }). This problem has been studied, and among the various publications we cite [9,6,8,1,4]. On the other hand, the methods to locate the efficient solutions are very interesting from a practical point of view. The important fact to note here is the relationship between solutions of vector type and solutions of set type, since the latter can be useful to find the former (see [5]). In this work, we shall employ solutions of set type for problem (P) and extend the first theorem for locating the set of all efficient points of a set through ordinary mathematical programming introduced by Yu [10].

We denote by K ⊂ Rn a solid (int(K ) ̸= ∅) closed convex cone and pointed (K ∩ (−K ) = {0}). The ordering defined by K on Rn is defined as follows: if x, y ∈ Rn , we denote x ≤ y if and only if y − x ∈ K. We say that a ∈ A is a minimal (respectively, weakly minimal) point of A with respect to the ordering ≤ defined by K on Rn if (a − K ) ∩ A = {a} (respectively, (a − int(K )) ∩ A = ∅). The set of minimal (respectively, weakly minimal) elements of A is denoted by Min A (respectively, WMin A). We will need the following properties for a nonempty set A ⊂ Rn .

• A is minorized if there exists x ∈ Rn such that A ⊂ x + K . It is said that x is a dominating point of A.

• A is K -bounded if there exists λ > 0 such that A ⊂ λB + K , where B is the unit ball.

• A has the K -domination (respectively, weak K -domination) property if A ⊂ Min A + K (respectively, A ⊂ WMin A + K ).  The asymptotic cone of A is defined by A∞ = ε>0 cl([0, ε]A). Following the notation used in [2], the set of all dominating points of A is denoted by AD = {x ∈ Y : x ≤ a ∀a ∈ A}; that is, AD =

 (a − K ).

(1)

a∈A

Thus, A is minorized if and only if AD ̸= ∅. 2. Notation and previous results In what follows, our working space is Rn . Given a nonempty set A ⊂ Rn , we denote by int(A), cl(A), and conv(A) the interior, the closure, and the convex hull of A, respectively.

∗

Corresponding author. E-mail address: [email protected] (E. Hernández).

0167-6377/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2011.08.004

Lemma 2.1. Let A ⊂ Rn . A is minorized if and only if A is K -bounded. Note that, in terms of topological vector spaces, each minorized set is a K -bounded set but, in general, the converse is false. From now on, we also assume that K is a polyhedral cone and that Γ = {h1 , . . . , hm } ⊂ Rn is a minimum generator set of K − ; that is, we cannot eliminate any hj ∈ Γ to generate K − . Thus, K = {x ∈ Rn : ⟨hj , x⟩ ≤ 0 ∀hj ∈ Γ }

E. Hernández, L. Rodríguez-Marín / Operations Research Letters 39 (2011) 466–470

and K − = {h ∈ Rn : ⟨h, x⟩ ≤ 0 ∀x ∈ K }

=

 m −

 λ j hj : λ j ≥ 0 ,

j =1

where ⟨·, ·⟩ denotes the Euclidean scalar product on Rn . The index set of Γ is denoted by J = {1, . . . , m}. For any nonzero hj ∈ Rn and r ∈ R, we consider the hyperplanes: Hj = {x ∈ Rn : ⟨hj , x⟩ = 0} and Hj,r = {x ∈ Rn : ⟨hj , x⟩ = r }; − n the closed half-spaces: Hj+ ,r = {x ∈ R : ⟨hj , x⟩ ≥ r }, Hj,r =

{x ∈ Rn : ⟨hj , x⟩ ≤ r }; and the open half-spaces: Hj++ = {x ∈ ,r n Rn : ⟨hj , x⟩ > r } and Hj−− ,r = {x ∈ R : ⟨hj , x⟩ < r }. We recall that by the Bipolar Theorem the expression x ≤ y is equivalent to ⟨hj , x⟩ ≥ ⟨hj , y⟩ for all hj ∈ Γ . From this, we obtain the following.

Lemma 2.2. Let {b1 , . . . , br } ⊂ Rn and let K1 be the polyhedral cone such that {l1 , . . . , ls } ⊂ Rn generate its negative polar cone K1− . Then

  (bi − K1 ) = Hj+ ,βj , i

j

where βj = maxi ⟨lj , bi ⟩ for every j ∈ {1, . . . , s}.

(2)

Proposition 2.3. Suppose that A ⊂ R and Γ = {h1 , h2 }. If x, y ∈ A satisfy that A1 (x) = {x} and A2 (y) = {y}, then x, y ∈ Min A and (x − K ) ∩ (y − K ) = AD . 2

Proof. Since (x − K ) ∩ A ⊂ Ai (x) for i = 1, 2, we have x, y ∈ Min A. We see that (x − K ) ∩ (y − K ) ⊂ AD . Suppose that z ∈ (x − K ) ∩ (y − K ). Then z = x − k = y − k′ for some k, k′ ∈ K . Let a ∈ A be such that y ̸= a ̸= x. Then ⟨h1 , a − z ⟩ = ⟨h1 , a − x + k⟩ = ⟨h1 , a⟩ − ⟨h1 , x⟩ + ⟨h1 , k⟩ < 0, since A1 (x) = {x} and k ∈ K . Similar arguments can be applied to prove that ⟨h2 , a − z ⟩ < 0. Hence, a − z ∈ K.  As the example below shows, if n > 2, the above result could be false. Example 2.4. Let R3 be ordered by K = R3− and A = {(x, y, z ) ∈ R3 : x2 + y2 + z 2 = 1; 0 ≤ z ≤ 1}. Then, if h1 = (1, 0, 0), h2 = (0, 1, 0), and h3 = (0, 0, 1), wehave A1 (0, 1, 0)= {(0, 1, 0 )}, A2 (1, 0, 0) = {(1, 0, 0)}, and A3

√1 , √1 , 0 , √12 , 0 = . 2 2 It is clear  that (100  , 100, 0) ∈ ((0, 1, 0) − K ) ∩ ((1, 0, 0) − √1 , √1 , 0 − K K) ∩ and (0, 0, 1) ∈ A but (0, 0, 1) ̸∈ 2 2 (100, 100, 0) − K . √1

2

From [10, Theorem 4.2], we obtain the following characterization of minimal points of A. Theorem 2.5. x0 ∈ Min A if and only if, for any arbitrary j ∈ J, ⟨hj , x0 ⟩ > ⟨hj , x⟩ for all x ∈ Aj (x0 ). Yu [10] established a method to compute Min A based on the above result. Our purpose is to give a version of Theorem 2.5 in terms of minimal sets of a family. To get it, we need to consider a set relation between two nonempty sets. In this paper, we consider the following one denoted by ≤l . Let A, B ⊂ Rn be nonempty sets: A ≤l B if and only if B ⊂ A + K .

It is easy to check that the set relation ∼l defined as A ∼l B ⇔ A ≤l B and B ≤l A is an equivalence relation on Rn . We denote by [·]l the equivalence classes via ∼l . Obviously, A ∈ [B]l if and only if A + K = B + K . In what follows, S means a family of nonempty subsets of Rn . We say that A ∈ S is a minimal set with respect to ≤l , and we write A ∈ l-Min S , if B ∈ S and B ≤l A imply that B ∈ [A]l . The family Sl (A) = {B ∈ S : B ≤l A} is called the l-section of S at A. More details of ≤l and ∼l can be found in [6,1,3,7], and the references therein. We emphasize that other set relations on Rn which also generalize the ordering defined by K have recently been presented in [7]. Such set relations could be more appropriate in practice. Our aim is to locate the family l-Min S . To this end, first we extend the sufficient condition of Theorem 2.5 in a natural way. Then, by introducing the notion of K -essential points of a set, we present the necessary condition. The main motivation of this work is practical, since the assumptions of Theorem 2.5 can easily be computed and, on the other hand, the investigation of the orderings on set space play an important role in several fields of research. 3. The sufficient condition

In [10], the following subsets are considered for hj ∈ Γ and a ∈ A: Aj (a) = {y ∈ A: ⟨hk , y⟩ ≥ ⟨hk , a⟩ ∀k ∈ J , k ̸= j}.

467

In this section, we assume that S is a family of K -bounded sets of Rn . We introduce the operator ⋆ as follows. Definition 3.1. Let A ⊂ Rn be a K -bounded set. Given h ∈ K − , we define h ⋆ A: = sup{⟨h, a⟩: a ∈ A}. Note that, since A is a K -bounded set, we have h⋆A < ∞. Indeed, by Proposition 3.4 in [9], we obtain that h(A) = {h(a): a ∈ A} is h(K )-bounded. Since h ∈ K − , h(K ) is either {0} or R− . So, if h(K ) = {0}, h(A) ⊂ R is a bounded set, and, if h(K ) = R− , h(A) ⊂ R is an upper bounded set. The following properties are easily checked. Proposition 3.2. Let h ∈ K − and A, B ∈ S . Then the following statements hold. (i) If A ≤l B, then h ⋆ B ≤ h ⋆ A (monotonicity). (ii) h ⋆ A = h ⋆ (A + K ) (invariant by the equivalence class). For A ∈ S and hj ∈ Γ , we denote

Sj (A) = {B ∈ S : hk ⋆ B ≥ hk ⋆ A ∀k ∈ J , k ̸= j}.

(3)

It is clear that Sj (A) ̸= ∅ for all j and A ∈ S . In addition, by Proposition 3.2, we can prove the following properties. Proposition 3.3. Let A ∈ S and hj ∈ Γ . (i) Sl (A) ⊂ Sj (A). (ii) If B ∈ Sj (A), then [B]l ⊂ Sj (A). (iii) If Sj (A) = [A]l , then A ∈ l-Min S . The operator ⋆ and the family defined in (3) allow us to extend, in a natural way, the sufficient condition of Theorem 2.5 in terms of l-minimal sets. Theorem 3.4. Let S be a family of K -bounded sets of Rn and A0 ∈ S . If there exists hj ∈ Γ such that hj ⋆ B < hj ⋆ A0 for all B ∈ Sj (A0 ) with B ̸∈ [A0 ]l , then A0 ∈ l-Min S . Proof. Suppose that A0 ̸∈ l-Min S . Then there exists B ∈ S such that B ≤l A0 and B ̸∈ [A0 ]l . By Proposition 3.2(i), for every hk ∈ Γ ,

468

E. Hernández, L. Rodríguez-Marín / Operations Research Letters 39 (2011) 466–470

we have

In contrast,

hk ⋆ A0 ≤ hk ⋆ B.



Thus, B ∈ Sj (A0 ), which is a contradiction.

i̸=1



In general, the converse condition does not hold, as the following example shows. Example 3.5. Consider R ordered by K = R+ . For x, y ∈ R , we define [x, y] = {λx + (1 − λ)y: λ ∈ [0, 1]}. Suppose that S is defined by A = [(0, 1), (2, 0)] and B = [(0, 2), (1, 0)]. Under these assumptions, A ̸∈ [B]l , both A and B are l-minimal sets of S , and hj ⋆ B = hj ⋆ A for every hj ∈ Γ . 2

2

2

Consequently, it is necessary to assume any stronger condition on the family S . 4. The necessary condition

Definition 4.1. Let A ⊂ Rn be a nonempty set. {a1 , . . . , ar } ⊂ A is said to be a K -essential set of A if

 (ai − K ) = AD where I = {1, . . . , r }. i∈I

In addition, if @k ∈ I such that i∈I , i̸=k (ai − K ) = i∈I (ai − K ), then {a1 , . . . , ar } is called a strongly K -essential set of A.





We emphasize that if x, y ∈ Rn we have (x − K ) ∩ (y − K ) ̸= ∅, since int(K ) ̸= ∅. Thus, a necessary condition for the existence of K -essential sets of A is that A is K -bounded. However, we will check that such a condition is not sufficient, since each K -essential set of A is contained in cl(A) (Proposition 4.3). As Cambini et al. state in [2], in general, it is difficult to compute AD . We show that the notion of a K -essential set of A could help in such a computation since, under certain conditions, AD is determined by finite points of A which are easier to compute. By Lemma 2.2, we obtain a characterization of AD through a K essential set {a1 , . . . , ar } of A,



Hj+ ,αj ,

(4)

j∈J

where αj = maxi ⟨hj , ai ⟩ for every j ∈ Γ . So, A is a polyhedral set. As Example 4.2 shows, in general, the K -essential points may be outside of Min A. D

Example 4.2. Consider R2 ordered by a Pareto cone, K = R2+ . If A = {conv([1 + α, 0], [0, 1 + α]): α ∈ (0, 1]}, then Min A = ∅, AD ̸= ∅, and {(2, 0), (0, 2)} is a K -essential set of A. The following result shows that the K -essential set of A is contained in the weakly minimal points of A. Proposition 4.3. Let {a1 , . . . , ar } be a strongly K -essential of A ⊂ Rn . Then {a1 , . . . , ar } ⊂ WMin A. Proof. Suppose that a1 ̸∈ WMin A. Then there exists x ∈ A such that x ∈ a1 − int(K ). Moreover, (a1 − int(K )) ∩ (x − K )c is a nonempty open set. We see that

[(a1 − int K ) ∩ (x − K )c ] ∩

 (ai − K ) ̸= ∅. i̸=1

(5)

(6)

Since x ∈ a1 − int(K ), we obtain that x − K  and (a1 − int K )c are disjoint. From (6), the nonempty convex set i̸=1 (ai − K ) satisfies



(ai − K ) ⊂ x − K ⊂ a1 − K ,

i̸=1

which contradicts that {a1 , . . . , ar } is a strongly K -essential set of A.   By (5), there exists z ∈ (a1 − int(K )) ∩ ( a − K ) such i i̸=1 that z ̸∈ x − K . Hence,

   (ai − K ) ⊂ (ai − K ) = AD , z ∈ (a1 − int(K )) ∩ 

i̸=1

As in the previous section, S denotes a family of K -bounded sets of Rn . For fixed A, B ∈ S and h ∈ K − , the relationship between h ⋆ A and h ⋆ B is close to the set of all dominating points of A and B, respectively. To prove this, we introduce the notion of a K -essential set.

AD =

(ai − K ) ⊂ (a1 − int(K ))c ∪ (x − K ).

i

which is not possible since z ∈ AD and x ∈ A imply that z ∈ x − K.  Example 4.4. (i) Consider R2 ordered by a Pareto cone, K = R2+ . The circle of radius 1 and center (0, 0) has a unique strongly R2+ -essential set {(0, −1), (−1, 0)}. (ii) If A ⊂ Rn has the K -domination property and Min A = {a1 , . . . , ar }, then A has a K -essential set of A determined by minimal points of A. (iii) If A = conv({a1 , . . . , ar }), then there exists a subset of {a1 , . . . , ar } which is a K -essential set of A. Proposition 2.3 gives a sufficient condition for the existence of a K -essential set of a subset A ⊂ R2 . Moreover, by considering the supporting points of A, we can find K -essential sets of A. Proposition 4.5. Let A ⊂ Rn . Suppose that for every hj ∈ Γ there exists a supporting point aj ∈ A, that is, hj ⋆ A = ⟨hj , aj ⟩; then {a1 , . . . , am } is a K -essential set of A. Lemma 4.6. Let {a1 , . . . , ar } be a K -essential set of A. Then, for every hj ∈ Γ , it holds that Hj,αj ∩ AD ̸= ∅,

(7)

where αj = maxi ⟨hj , ai ⟩. Proof. We only prove (7) for j = 1. Conversely, we have H1,α1 ∩ + AD = ∅. From (4), AD = H1+,α1 ∩ H1+,α2 ∩ · · · ∩ Hm ,αm ; then + H1,α1 ∩ H2+,α2 ∩ · · · ∩ Hm ,αm = ∅.

Thus, only one of the following cases holds. −− + Case 1. H2+,α2 ∩ · · · ∩ Hm ,αm ⊂ H1,α1 . + + Case 2. H2,α2 ∩ · · · ∩ Hm,αm ⊂ H1++ ,α1 .

By hypothesis, Case 1 cannot happen, since AD = H1+,α1 ∩ · · · ∩ + Hm ,αm ̸= ∅. Case 2. Thus, + + + AD = H1+,α1 ∩ · · · ∩ Hm ,αm = H2,α2 ∩ · · · ∩ Hm,αm .

(8)

+ Taking into account Lemma 2.2, we obtain H2+,α2 ∩ · · · ∩ Hm ,αm =  i (ai − K1 ), K1 being the cone such that {h2 , . . . , hm } is a generator set of K1− . Hence, by considering the asymptotic cone in (8), it follows that K = K1 (Remark 2.6 in [9]), which is a contradiction, since, by hypothesis, Γ is a minimum generating K − . 

E. Hernández, L. Rodríguez-Marín / Operations Research Letters 39 (2011) 466–470

From (4) and (7), every hj ∈ Γ is a supporting functional of AD . Lemma 4.7. Let A ⊂ Rn and S = {x1 , . . . , xr } ⊂ Rn . If z ∈ Rn satisfies that ⟨hk , z ⟩ > maxxi ∈S ⟨hk , xi ⟩ for some k ∈ J, then

(z − K )



(xi − K ) ̸=

xi ∈S



(xi − K ).

(⇐). For every hj ∈ Γ , we denote hj ⋆ A = ⟨hj , ai(j) ⟩ = αj and hj ⋆ B = ⟨hj , bi′ (j) ⟩ = βj , with ai(j) ∈ {a1 , . . . , ar } and bi′ (j) ∈ {b1 , . . . , bs } (which exist by Lemma 4.8). + Let us consider Hj+ ,αj and Hj,βj . Thus, AD =

xi ∈S

469



Hj+ ,αj

and BD =

j



Hj+ ,βj .

j

Proof. For every hj ∈ Γ , we denote xi(j) ∈ S such that

Suppose that there exists hk ∈ Γ such that hk ⋆ B < hk ⋆ A; that is,

⟨hj , xi(j) ⟩ = max⟨hj , xi ⟩ = rj .

βk < αk .

Thus, xi ∈S (xi −K ) = {x ∈ Rn : x ≤ xi ∀xi ∈ S } = {x ∈ Rn : ⟨hj , x⟩ ≥ ⟨hj , xi(j) ⟩ = rj ∀hj ∈ Γ }. Therefore, if j = k, there exists

Applying Lemma 4.6, we obtain Hk,βk ∩ B ̸= ∅. Let z ∈ Hk,βk ∩ BD . Then, by (10), we deduce that z ̸∈ Hk+,αk . Hence, z ̸∈ AD which

xi ∈S



 x∈

 

(xi − K ) ∩ Hk,rk .

But x ̸∈ z − K , because ⟨hk , x⟩ = ⟨hk , xi(k) ⟩ < ⟨hk , z ⟩. Hence,

(z − K )

(xi − K ) ̸=

xi ∈S



(xi − K ), 

Lemma 4.8. Let A ⊂ Rn and let S = {a1 , . . . , ar } be a K -essential set of A. Then, for every hj ∈ Γ , there exists ai ∈ S such that hj ⋆ A = ⟨hj , ai ⟩. Proof. Conversely, there would be hj ∈ Γ such that, for every ai ∈ S, there exists xi(j) ∈ A with ⟨hj , xi(j) ⟩ > ⟨hj , ai ⟩. Let ai(j) ∈ S satisfy ⟨hj , ai(j) ⟩ = maxai ∈S {⟨hj , ai ⟩}; then

⟨hj , xi(j) ⟩ > ⟨hj , ai(j) ⟩ ≥ ⟨hj , ai ⟩ ∀ai ∈ S . 

(ai − K ) ̸=

(ai − K ),



ai ∈S

(ai − K ) = AD .

Theorem 4.12. Let S be a family of subsets of Rn satisfying the following conditions. (i) Every A ∈ S has a K -essential set. (ii) If A, B ∈ S , B ̸∈ [A]l and BD ⊂ AD , then there does not exist a K -essential set {a1 , . . . , ar } of A such that B ⊂ i (ai − K )c . If A0 ∈ l-Min S , then, for every hj ∈ Γ , for all B ∈ Sj (A0 ) with B ̸∈ [A0 ]l .

Proof. Let A0 ∈ l-Min S . Suppose that there exist hj ∈ Γ and B ∈ Sj (A0 ) with B ̸∈ [A0 ]l such that hj ⋆ B ≥ hj ⋆ A0 ; then

ai ∈S

ai ∈S

which contradicts



Corollary 4.11. Let A ⊂ Rn be such that A has a K -essential set. Then hj ⋆ A ≤ 0 for every hj ∈ Γ if and only if 0 ≤l A.

hj ⋆ B < hj ⋆ A0

Applying Lemma 4.7, we have

(xi(j) − K )



Consequently, taking into account that −K ⊂ AD is equivalent to 0 ≤l A, if B = {0}, by Theorem 4.9 we have the following characterization.

xi ∈S

and we conclude the proof.

contradicts that BD ⊂ AD .

Corollary 4.10. Let A, B ⊂ Rn be such that they have K -essential sets. Then hj ⋆ A = hj ⋆ B for every hj ∈ Γ if and only if AD = BD .

xi ∈S



(10) D

hk ⋆ B ≥ hk ⋆ A0



Theorem 4.9. Let A, B ⊂ Rn be such that they have K -essential sets. Then hj ⋆ A ≤ hj ⋆ B for every hj ∈ Γ if and only if BD ⊂ AD . Proof. Suppose that {a1 , . . . , ar } and {b1 , . . . , bs } are K -essential families of A and B, respectively.   (⇒). Since BD = i (bi − K ) and AD = i (ai − K ), if z ∈ BD , we have to show that, for every i ∈ {1, . . . , r },

∀hk ∈ Γ .

According to Theorem 4.9, we deduce that BD ⊂ (A0 )D . Let {a1 , . . . , ar } be a K -essential family of A0 . By (ii), we conclude the proof if B ⊂ (a1 − K )c ∪ · · · ∪ (ar − K )c

(11)

ai − z ∈ K ,

holds. Indeed, suppose that (11) does not hold. Then there exits b ∈ B such that b ∈ [(a1 − K )c ∪ · · · ∪ (ar − K )c ]c ; that is, b ∈ (a1 − K ) ∩ · · · ∩ (ar − K ). Thus,

or, equivalently,

b ≤ ai

⟨hj , ai − z ⟩ ≤ 0 ∀hj ∈ Γ .

On the other hand, since (A0 )D = i (ai − K ), we obtain A0 ⊂ b + K . Thus, B ≤l A0 , which is a contradiction, since A0 ∈ l-Min S and B ̸∈ [A0 ]l . 



From Lemma 4.8 and z ∈ BD , for every hj ∈ Γ there exist bi(j) ∈ {b1 , . . . , bs } and ej ∈ K such that z = bi(j) − ej and hj ⋆ B = ⟨hj , bi(j) ⟩.

(9)

Therefore, for every i ∈ {1, . . . , r },

⟨hj , ai − z ⟩ = ⟨hj , ai − bi(j) + ej ⟩ = ⟨hj , ai ⟩ − ⟨hj , bi(j) ⟩ + ⟨hj , ej ⟩. Since hj ⋆ A ≥ hj ⋆ ai and hj ⋆ A ≤ hj ⋆ B, taking into account (9), we obtain

⟨hj , ai − z ⟩ ≤ ⟨hj , ej ⟩. On the other hand, since ej ∈ K and hj ∈ K − ,

⟨hj , ai − z ⟩ ≤ 0, and we conclude.

∀i ∈ {1, . . . , r }.

Note that assumption (ii) could be omitted by considering set relations stronger than ≤l ; for example, see those given in [7]. Concluding remarks It is worth pointing out that Yu in [10] developed a method to compute all efficient points of a set. Therefore, taking into account Theorems 3.4 and 4.12, we could find similar mathematical programs to locate minimal sets of a family S . Roughly speaking, we would obtain a method to find solutions of problem (P) of set type (working on {F (x): x ∈ M }). On the other hand, it would be interesting to carry out further studies on the K -essential set. For instance, following the line studied in [2], we could establish relationships between a K essential set of A and its image under a transformation that preserves a given ordering.

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E. Hernández, L. Rodríguez-Marín / Operations Research Letters 39 (2011) 466–470

Acknowledgments Part of the research of the first author was supported by Ministerio de Ciencia e Innovación (Spain) Project MTM200909493. The authors want to express their gratitude to the anonymous referee for valuable suggestions and remarks. References [1] M. Alonso, L. Rodríguez-Marín, Set-relations and optimality conditions in setvalued maps, Nonlinear Anal. 63 (2005) 1167–1179. [2] A. Cambini, D.T. Luc, L. Martein, Order-preserving transformations and applications, J. Optim. Theory Appl. 118 (2003) 275–293. [3] E. Hernández, L. Rodríguez-Marín, Existence theorems for set optimization problems, Nonlinear Anal. 67 (2007) 1726–1736.

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