Optimality conditions via scalarization for a new ε -efficiency concept in vector optimization problems

European Journal of Operational Research 201 (2010) 11–22

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Continuous Optimization

Optimality conditions via scalarization for a new e-efficiency concept in vector optimization problems q C. Gutiérrez a, B. Jiménez b, V. Novo b,* a

´as de la Información y las Telecomunicaciones, Departamento de Matemática Aplicada, E.T.S.I. Informática, Universidad de Valladolid, Edificio de Tecnologı Co Cementerio s/n, Campus Miguel Delibes, 47011 Valladolid, Spain b Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, c/ Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 6 October 2007 Accepted 5 February 2009 Available online 13 February 2009 Keywords: Vector optimization e-efficiency Scalarization Necessary and sufficient optimality conditions Approximate optimality

a b s t r a c t In this work, necessary and sufficient conditions for approximate solutions of vector optimization problems are obtained via scalarization, i.e., by considering approximate solutions of associated scalar optimization problems. These conditions are proved through a new e-efficiency concept and two very general assumptions on the scalarization that extend the usual order representing and monotonicity properties. Moreover, neither solidness hypothesis on the order cone nor monotonicity property on the scalarization are assumed. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction In the literature there exist a lot of results that relate e-efficient solutions of vector optimization problems with approximate solutions of scalar optimization problems obtained by scalarization processes (see, for example, [2,3,5–12] and the references therein). These results have been used to prove multiplier rules for e-efficient solutions (see, for example, [2,6–8] and the references therein) and Ekeland variational principles in vector optimization problems (see the recent survey by Chen et al. in [1, Chapter 4]). Some of them use specific scalarizations, like those obtained in [2,5–9,11]. The reader can see [3, Section 3] for a brief survey of these results in the framework of a multiobjective optimization problem with the Pareto order and by considering a well-known e-efficiency notion due to Kutateladze [13]. In [10,12], we have extended to the approximate efficiency two classical approaches very used in proving, via scalarization, necessary and sufficient conditions for efficient (exact) solutions of vector optimization problems. To be precise, in these papers we obtain necessary conditions for e-efficient solutions through approximate solutions of scalarization processes that satisfy an approximate order representing property. This property can be used when the order cone is solid and extends the usual order representing property (see [14,16]). Analogously, we prove sufficient conditions by means of approximate solutions of scalarizations that verify some kind of monotonicity property. These results work for e-efficient solutions in the senses of Kutateladze [13] and Tanaka [15] and generalize the previous ones based on specific scalarizations. In this work we introduce a new e-efficiency concept that extends several e-efficiency notions proposed in the literature, and we provide new and more general necessary and sufficient conditions for e-efficient solutions, which are obtained via scalarization. For this aim, we consider two properties more general than those described above since they are not based on any solidness and monotonicity assumption. Moreover, both properties jointly show a way to find scalarization processes from which one can obtain characterizations of e-efficient solutions via approximate solutions of these scalar optimization problems.

q This research was partially supported by the Ministerio de Educación y Ciencia (Spain) under projects MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD200600032 (Consolider-Ingenio 2010), and by the Consejerı´a de Educación de la Junta de Castilla y León (Spain), project VA027B06. * Corresponding author. Tel.: +34 91 3 98 64 36; fax: +34 91 3 98 60 12. E-mail addresses: [email protected] (C. Gutiérrez), [email protected] (B. Jiménez), [email protected] (V. Novo).

0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.02.007

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C. Gutiérrez et al. / European Journal of Operational Research 201 (2010) 11–22

The paper is structured as follows: in Section 2 the framework of this work and a new concept of approximate efficiency, called Q ðeÞefficiency, are introduced. After that, in Sections 3 and 4, necessary and sufficient conditions for Q ðeÞ-efficient points are obtained via approximate solutions of associated scalar optimization problems. Finally, in Section 5 we use these conditions in order to propose specific scalarization processes from which one can characterize Q ðeÞ-efficient solutions through scalarization. 2. A new e-efficiency notion Let Y be a Hausdorff locally convex space and let Y  be its topological dual space. Let D  Y be a nontrivial ðD – f0gÞ pointed ðD \ ðDÞ ¼ f0gÞ convex cone and assume that Y is partially ordered by the relation

y1 ; y2 2 Y;

y1 6 y2 () y1  y2 2 D:

In this paper we consider the vector optimization problem

Minff ðxÞ : x 2 Sg;

ð1Þ

where f : X ! Y, X is a decision space and S  X; S – ;. For each set A  Y we denote the interior, the closure, the complement and the cone generated of A by intðAÞ; clðAÞ; Ac and coneðAÞ, respectively. If Y is normed, A – ; and y 2 Y then we denote

dðy; AÞ ¼ inffkz  yk : z 2 Ag: We say that A is solid if intðAÞ – ;, and co-radiant if aA  A; 8a > 1. Let

Dþ ¼ fk 2 Y  : kðdÞ P 0; 8d 2 Dg; Dþs ¼ fk 2 Y  : kðdÞ > 0; 8d 2 D n f0gg: By Rpþ we denote the nonnegative orthant of Rp . Moreover, Rþ :¼ R1þ and Rþþ :¼ Rþ n f0g. Given two sets E; H and a set-valued map F : EH we denote

DomðFÞ ¼ fu 2 E : FðuÞ – ;g;

ImðFÞ ¼

[

FðuÞ:

u2E

We say that F is proper if DomðFÞ ¼ E. To prove our results we work in the objective space Y. Then, for each nonempty set M  Y we say that y 2 M is an efficient (resp. weak efficient) point of M, denoted y 2 EðMÞ (resp. y 2 WEðMÞ), if ðM  yÞ \ ðD n f0gÞ ¼ ; (resp. ðM  yÞ \ ðintðDÞÞ ¼ ;). In order to get nontrivial weak efficient points we assume that D is solid whenever the weak efficiency set is considered. To deal with approximate solutions of (1) we introduce the following new concept of approximate efficient point. Let Q : Rþ D be a proper set-valued map and define Q D : Rþ D by Q D ðeÞ ¼ Q ðeÞ þ D n f0g, 8e 2 Rþ . Let us observe that 0 R ImðQ D Þ. Definition 2.1. Let M  Y be a nonempty set and

ðM  yÞ \ ðQ D ðeÞÞ ¼ ;:

e 2 Rþ . y 2 M is a Q ðeÞ-efficient point of M, denoted y 2 EðM; Q ðeÞÞ, if ð2Þ

Take note that EðM; Q ðeÞÞ ¼ EðM þ D; Q ðeÞÞ \ M. Remark 2.1. (a) Let us recall the e-efficiency notion introduced by ourselves in [9,11]. Let C  D n f0g be a nonempty co-radiant set and M  Y; M – ;. We say that y 2 M is a ðC; eÞ-efficient point of M if ðM  yÞ \ ðQ ðeÞÞ ¼ ;, where

Q ðeÞ ¼



eC

if

coneðCÞ n f0g if

e 2 Rþþ ; e ¼ 0:

ð3Þ

If C þ D n f0g ¼ C then the ðC; eÞ-efficiency notion can be seen as a Q ðeÞ-efficiency concept since Q D ðeÞ ¼ Q ðeÞ; 8e 2 Rþ . Let us observe that conditions C  D n f0g and C þ D n f0g ¼ C imply that C is a co-radiant set. Reciprocally, if for all e 2 Rþþ we have Q D ðeÞ ¼ eQ D ð1Þ and Q D ð0Þ ¼ coneðQ D ð1ÞÞ n f0g then the Q ðeÞ-efficiency notion is the ðC; eÞefficiency concept given by the co-radiant set C ¼ Q D ð1Þ. Therefore, roughly speaking, it is clear that Definition 2.1 is more general than the ðC; eÞ-efficiency concept because one can consider set-valued maps Q such that Q D ðeÞ – eQ D ð1Þ; 8e 2 Rþþ . (b) Analogously, if Q ðeÞ ¼ D n f0g, 8e 2 Rþ then the Q ðeÞ-efficiency set reduces to the (exact) efficiency set for each e 2 Rþ and M  Y. If D is solid then one can consider statement (2) with respect to the set-valued map Q intðDÞ[f0g ðeÞ. In this case we say that y 2 M is a weak Q ðeÞ-efficient point of M if ðM  yÞ \ ðQ intðDÞ[f0g ðeÞÞ ¼ ;. The set of weak Q ðeÞ-efficient points of M is denoted by WEðM; Q ðeÞÞ. The notions of efficient, weak efficient and Q ðeÞ-efficient point are translated to problem (1) as follows. A feasible point x 2 S is an efficient (resp. weak efficient) solution of (1) if f ðxÞ 2 Eðf ðSÞÞ (resp. f ðxÞ 2 WEðf ðSÞÞ). The set of efficient (resp. weak efficient) solutions of (1) is denoted by Eðf ; SÞ (resp. WEðf ; SÞ). Analogously, a feasible point x 2 S is a Q ðeÞ-efficient (resp. weak Q ðeÞ-efficient) solution of (1), denoted by x 2 Eðf ; S; Q ðeÞÞ (resp. x 2 WEðf ; S; Q ðeÞÞ) if f ðxÞ 2 Eðf ðSÞ; Q ðeÞÞ (resp. f ðxÞ 2 WEðf ðSÞ; Q ðeÞÞ). Let us observe that

[

Eðf ; SÞ ¼

f 1 ðyÞ \ S;

y2Eðf ðSÞÞ

WEðf ; SÞ ¼

[

f 1 ðyÞ \ S;

y2WEðf ðSÞÞ

[

Eðf ; S; Q ðeÞÞ ¼

f 1 ðyÞ \ S;

y2Eðf ðSÞ;QðeÞÞ

WEðf ; S; Q ðeÞÞ ¼

[

y2WEðf ðSÞ;Q ðeÞÞ

f 1 ðyÞ \ S:

ð4Þ

C. Gutiérrez et al. / European Journal of Operational Research 201 (2010) 11–22

13

Next, in order to illustrate the usefulness of Definition 2.1, we show a particular Q ðeÞ-efficiency concept that extends and improves (in a certain sense) the well-known e-efficiency notion due to Kutateladze [13]. Example 2.1. Let q : Rþ ! D and define Q ðeÞ ¼ fqðeÞg; 8e 2 Rþ . If d 2 D n f0g and qðeÞ ¼ ed; 8e 2 Rþ then this Q ðeÞ-efficiency concept reduces to Kutateladze’s e-efficiency notion. Kutateladze’s notion is not useful at all to approximate the efficiency set. This drawback is shown in the following simple problem (1). Let X ¼ Y ¼ R2 ; D ¼ S ¼ R2þ ; f ðxÞ ¼ x; 8x 2 R2 ; d ¼ ðd1 ; d2 Þ 2 R2þ n f0g and qðeÞ ¼ ed; 8e 2 Rþ . It is easy to check that Eðf ; SÞ ¼ fð0; 0Þg; WEðf ; SÞ ¼ fðx1 ; x2 Þ 2 R2þ : x1 x2 ¼ 0g,

8 if d 2 intðR2þ Þ; > < WEðf ; SÞ 2 Eðf ; S; Q ðeÞÞ ¼ fðx1 ; x2 Þ 2 Rþ : x2 ¼ 0g if d1 ¼ 0; > : e2Rþþ fðx1 ; x2 Þ 2 R2þ : x1 ¼ 0g if d2 ¼ 0: \

T and so e2Rþþ Eðf ; S; Q ðeÞÞ – Eðf ; SÞ. However, it is possible to overcome this drawback by considering a not linear single valued map Q. Indeed, let e1 ¼ ð1; 0Þ; e2 ¼ ð0; 1Þ and

8 > < eðe1 þ e2 Þ if qðeÞ ¼ ee1 if > : ee2 if

e – 1=k; 8k 2 N; e ¼ 1=ð2k þ 1Þ; k 2 N [ f0g; e ¼ 1=2k; k 2 N:

In this case it is easy to check that

\

Eðf ; S; Q ðeÞÞ ¼ Eðf ; SÞ:

ð5Þ

e2Rþþ

With respect to the general problem (1), if C  D n f0g is a co-radiant set such that coneðCÞ ¼ D, then statement (5) is satisfied for the setvalued map Q given in (3) (see [11, Theorem 3.4(iii)]). For example, if Dþs – ; and k 2 Dþs then one can consider

C ¼ fd 2 D : kðdÞ > 1g: In the literature, a lot of resolution methods for problem (1) are based on scalarization processes, i.e., they consider solutions of certain associated scalar optimization problems

Minfðu  f ÞðxÞ : x 2 Sg;

ð6Þ

where u : Y ! R [ f1g. We denote DomðuÞ :¼ fy 2 Y : juðyÞj < þ1g and we suppose that u is proper, i.e., DomðuÞ – ;. For each e P 0 we denote

ASðu  f ; S; eÞ :¼ fx 2 S : uðf ðxÞÞ  e 6 uðf ðzÞÞ; 8z 2 Sg: Observe that the statement x 2 ASðu  f ; S; 0Þ means that x is an exact solution of problem (6). 3. Necessary conditions Let M  Y be a nonempty set and consider a proper set-valued map Q. One can obtain information on the set EðM; Q ðeÞÞ via approximate solutions of some scalar optimization problems

MinfuðyÞ : y 2 Mg: To be exact, let

ð7Þ

e 2 Rþ and denote by ASðu; M; eÞ the set of approximate solutions with precision e of problem (7), i.e.,

ASðu; M; eÞ :¼ fy 2 M : uðyÞ  e 6 uðzÞ; 8z 2 Mg: Then

EðM; Q ðeÞÞ 

[

ASðu; M; eÞ;

u2F

where F is a class of functionals that satisfy suitable properties (see, for example, [3,10] and the references therein). Definition 3.1. We say that u satisfies the approximate order representing property (AORP for short) with respect to Q at 0 2 DomðuÞ and

fy 2 Y : uðyÞ < 0g  Q D ðeÞ:

e 2 Rþ if ð8Þ

Remark 3.1. (a) Let us observe that (8) implies uð0Þ P 0, because otherwise 0 2 Q D ðeÞ that is a contradiction. As 0 2 DomðuÞ we derive that uð0Þ 2 Rþ .(b) For each y 2 Y we define the functional uy : Y ! R [ f1g by uy ðzÞ :¼ uðz  yÞ; 8z 2 Y. Then

fz 2 Y : uðzÞ < 0g  Q D ðeÞ () fz 2 Y : uy ðzÞ < 0g  y  Q D ðeÞ: Next we compare AORP with other similar order representing properties proposed in the literature. The following proposition is necessary. Proposition 3.1. Assume that D is solid and let C  D n f0g be a nonempty open set such that C þ D n f0g ¼ C. If 0 2 DomðuÞ; e 2 Rþþ (resp. e ¼ 0) and

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C. Gutiérrez et al. / European Journal of Operational Research 201 (2010) 11–22

fy 2 Y : uðyÞ < 0g  eC (resp. fy 2 Y : uðyÞ < 0g  intðDÞ), then u satisfies AORP with respect to the set-valued map (3) at

e (resp. at zero).

Proof The result follows by Remark 2.1(a), since Q ðeÞ ¼ Q D ðeÞ ¼ eC; 8e 2 Rþþ and

Q D ð0Þ ¼ coneðCÞ n f0g ¼ intðDÞ: Let us check this last equality. As C is an open set contained in D n f0g, the set coneðCÞ n f0g ¼ [a>0 aC is an open set contained in D n f0g, and so coneðCÞ n f0g  intðDÞ. On the other hand, it is easy to prove that intðDÞ  D þ coneðCÞ n f0g and as C þ D n f0g ¼ C we have

intðDÞ  D þ coneðCÞ n f0g ¼ coneðCÞ n f0g: Then intðDÞ  intðconeðCÞ n f0gÞ ¼ coneðCÞ n f0g and the proof is completed. h The following example is a consequence of Remark 3.1(b) and Proposition 3.1. It illustrates that AORP is more general than other wellknown order representing properties. In particular, let us observe that AORP can be used when D is not solid and for different efficiency and e-efficiency concepts. Example 3.1. Suppose that D is solid. (a) In [14, property R2] and [16, statement (30)] the authors consider AORP with respect to the set-valued map (3) given by the set C ¼ intðDÞ. In this case, the set-valued maps Q and Q D are constant: Q D ðeÞ ¼ Q ðeÞ ¼ intðDÞ; 8e 2 Rþ (see Remark 2.1(a)). (b) In [12, statement (6)] AORP was considered with respect to the set-valued map (3) given by the set C ¼ intðDÞ \ Bc , where B denotes the closed unit ball defined by a norm p : Y ! R. If p is D-monotone on intðDÞ (i.e., pðq þ dÞ P pðqÞ; 8q 2 intðDÞ; 8d 2 D) then C  D n f0g is a nonempty open set such that C þ D n f0g ¼ C. (c) In [10, statement (11)], we use AORP with respect to the set-valued map (3) given by the set C ¼ q þ intðDÞ, where q 2 D n f0g. Theorem 3.1. Suppose that u satisfies AORP with respect to Q at

e 2 Rþ . If y 2 EðM; Q ðeÞÞ then y 2 ASðuy ; M; uð0ÞÞ.

Proof. By the hypothesis we have that ðM  yÞ \ ðQ D ðeÞÞ ¼ ; and so

uy ðzÞ ¼ uðz  yÞ P 0; 8z 2 M;

ð9Þ

since u verifies (8). By Remark 3.1(a), uð0Þ 2 Rþ and by (9) we deduce

uy ðyÞ  uð0Þ ¼ 0 6 uy ðzÞ; 8z 2 M; which completes the proof. h It is easy to check that Theorem 3.1 extends [10, Lemma 4.2], [12, Proposition 3.2], [14, Proposition 5.2] and [16, Theorem 10 (partially)]. Let us observe, as opposed to these results, that Theorem 3.1 does not require any solidness assumption on the order cone D and can be applied on different efficiency and e-efficiency concepts. In order to use Theorem 3.1, some explicit scalarization processes satisfying AORP are necessary. This problem has been solved in the literature for other order representing properties by using gauge functionals (see, for instance, [4, Theorem 2.1] and [5, Theorem 2.3.1]). Here we propose a similar but in a certain sense more general result, since we obtain a representation like (8) without assuming any solidness assumption on D. Let us recall that a nonempty set A  Y is star-shaped if there exists q 2 A such that

aq þ ð1  aÞy 2 A; 8y 2 A; 8a 2 ð0; 1Þ:

ð10Þ

We denote by kernðAÞ the set of all points q 2 A satisfying (10). Theorem 3.2. Let A  Y be a nonempty star-shaped co-radiant set and consider q 2 kernðAÞ; q – 0. The following statements hold: (a) aq þ A  A; 8a P 0. (b) For each y 2 Y, the set

CA;q ðyÞ :¼ ft 2 R : y 2 tq  Ag is empty or an upper unbounded interval. (c) The functional uA;q : Y ! R [ f1g given by

uA;q ðyÞ :¼



if CA;q ðyÞ ¼ ;;

þ1

infft 2 R : t 2 CA;q ðyÞg if CA;q ðyÞ – ;;

satisfies

fy 2 Y : uA;q ðyÞ < 0g ¼ Rþþ q  A:

ð11Þ

(d) If 0 R A then 0 2 DomðuA;q Þ. (e) If A is a convex cone then uA;q is sublinear. Proof Part (a) is clear because A is a star-shaped co-radiant set and

aq þ y ¼ ð1 þ aÞ



a

1þa

qþ

 1 y ; 1þa

8y 2 A; 8a P 0:

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C. Gutiérrez et al. / European Journal of Operational Research 201 (2010) 11–22

Part (b). Let y 2 Y; s 2 Rþþ and suppose that t 2 CA;q ðyÞ. By part (a) it follows that

y 2 tq  A ¼ ðt þ sÞq  ðsq þ AÞ  ðt þ sÞq  A and so t þ s 2 CA;q ðyÞ. From here it is obvious that CA;q ðyÞ is empty or an upper unbounded interval. Part (c). Let y 2 Y be such that uA;q ðyÞ < 0. If uA;q ðyÞ ¼ 1 then CA;q ðyÞ ¼ R and y 2 tq  A; 8t 2 Rþþ . In the other case, there exists t < 0 such that t 2 CA;q ðyÞ and so y 2 tq  A.Reciprocally, it is obvious that uA;q ðyÞ < 0; 8y 2 Rþþ q  A and so this part is completed. Part (d). Suppose that uA;q ð0Þ < 0. Then, by part (c), 0 ¼ aq  y for some a > 0 and y 2 A. But, by part (a), 0 ¼ aq þ y 2 A, a contradiction. So uA;q ð0Þ P 0. Moreover, 1 2 CA;q ð0Þ and so uA;q ð0Þ < 1, which finishes the proof. Part (e) follows from [5, Theorem 2.3.1(a)]. h Remark 3.2 (a) It is clear that

DomðuA;q Þ ¼ ðRq  AÞ n

\

ðtq  AÞ:

t2R

(b) Let us observe that (11) and Theorem 3.2(a) imply that

fy 2 Y : uA;q ðyÞ < 0g  A: Under the hypotheses of Theorem 3.2, if additionally A is open then Rþþ q  A ¼ A and so

fy 2 Y : uA;q ðyÞ < 0g ¼ A: Example 3.2. Suppose that (1) is a Pareto problem, i.e., Y ¼ Rp and D ¼ Rpþ . If A ¼ D; q ¼ ðq1 ; q2 ; . . . ; qp Þ 2 Rpþ n f0g and we denote IðqÞ :¼ fi 2 f1; 2; . . . ; pg : qi ¼ 0g it is easy to check that

uA;q ðyÞ :¼



if yi > 0 for some i 2 IðqÞ;

þ1

maxfyi =qi : i R IðqÞg if yi 6 0; 8i 2 IðqÞ:

Next, we apply Theorem 3.1 to the vector optimization problem (1). Given a point x 2 X and a mapping h : Y ! R [ f1g we denote by h  ðf  f ðxÞÞ the mapping ðh  ðf  f ðxÞÞÞðzÞ ¼ hðf ðzÞ  f ðxÞÞ for all z 2 X. Corollary 3.1. Suppose that u satisfies AORP with respect to Q at

e 2 Rþ . If x 2 Eðf ; S; Q ðeÞÞ then x 2 ASðu  ðf  f ðxÞÞ; S; uð0ÞÞ.

Proof. For each x 2 Eðf ; S; Q ðeÞÞ it follows that f ðxÞ 2 Eðf ðSÞ; Q ðeÞÞ and by Theorem 3.1 we see that f ðxÞ 2 ASðuf ðxÞ ; f ðSÞ; uð0ÞÞ and so

uf ðxÞ ðf ðxÞÞ  uð0Þ 6 uf ðxÞ ðf ðzÞÞ; 8z 2 S: As uf ðxÞ ðf ðzÞÞ ¼ uðf ðzÞ  f ðxÞÞ ¼ ðu  ðf  f ðxÞÞÞðzÞ, 8z 2 S, the proof is completed. h Corollary 3.1 extends [10, Theorem 4.5], [11, Theorem 5.3], and [12, Theorem 3.1], by Example 3.1, Theorem 3.2 and Remark 3.2. The following result is very useful in order to apply a penalization reasoning to the scalarization process. Let W be a normed space and consider a closed convex cone P  W and a mapping g : X ! W. We denote S1 ¼ fl 2 W  : klk ¼ 1g and

Sg ¼ fx 2 X : gðxÞ 2 Pg: Define the functional w : W ! R [ fþ1g by wðwÞ ¼ supflðwÞ : l 2 P þ \ S1 g. Theorem 3.3. Let u be satisfying AORP with respect to Q at x 2 ASð/x ; N; uð0ÞÞ where /x : X ! R [ fþ1g is defined by

/x ðzÞ :¼ maxfðu  ðf  f ðxÞÞÞðzÞ; wðgðzÞÞg;

e 2 Rþ and suppose that S ¼ N \ Sg , where N  X; N – ;. If x 2 Eðf ; S; Q ðeÞÞ then

8z 2 X:

Proof. By Corollary 3.1 we have that

ðu  ðf  f ðxÞÞÞðzÞ P 0;

8z 2 S:

ð12Þ

By Remark 3.1(a) we see that uð0Þ 2 Rþ . Let z 2 N. If z 2 Sg then z 2 S and from (12) we deduce that

/x ðzÞ P ðu  ðf  f ðxÞÞÞðzÞ P 0: If z R Sg then gðzÞ R P and by the separation theorem (see, for example, [5, Theorem 2.2.8]) there exists

lðgðzÞÞ > lðwÞ; 8w 2 P:

ð13Þ þ

As P is a cone it follows that l 2 P and we can suppose that lðgðzÞÞ > 0. Then wðgðzÞÞ > 0 and so

/x ðzÞ P wðgðzÞÞ > 0: Since x 2 Sg it follows that wðgðxÞÞ 6 0. Therefore,

/x ðxÞ ¼ maxfuð0Þ; wðgðxÞÞg ¼ uð0Þ and the proof is completed. h

l 2 W  ; l – 0, such that

1

þ

l 2 S \ P since l – 0. Moreover, by taking w ¼ 0 in (13) we deduce that

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C. Gutiérrez et al. / European Journal of Operational Research 201 (2010) 11–22

1 r Remark 3.3. If W ¼ Rmþr , P ¼ Rm þ  f0r g (where 0r ¼ ð0; 0; . . . ; 0Þ 2 R ) and S is given by the norm k k1 then

wðwÞ ¼ maxfw1 ; w2 ; . . . ; wm ; jwmþ1 j; jwmþ2 j; . . . ; jwmþr jg;

8w 2 W:

ð14Þ

By using [11, Theorem 6.1] and (14) the reader can check that Theorem 3.3 generalizes the necessary conditions of [7, Lemma 3.1(i)] and [8, Lemma 3.1(i)].

4. Sufficient conditions To obtain sufficient conditions for approximate efficient points of M we consider the following property, which is new to our knowledge. Definition 4.1. We say that u : Y ! R [ f1g satisfies the approximate (strong) inclusion property (AIP and ASIP for short, respectively) with respect to Q if uð0Þ 2 Rþ and

 Q D ðuð0ÞÞ  fy 2 Y : uðyÞ 6 0g; ðQ D ðuð0ÞÞ  fy 2 Y : uðyÞ < 0gÞ:

Using scalarization processes based on these properties we obtain sufficient conditions for Q ðeÞ-efficient points of M. Theorem 4.1. Suppose that u satisfies AIP with respect to Q and uð0Þ 2 Rþþ . Consider 0 6 d < uð0Þ. If y 2 ASðuy ; M; dÞ then y 2 EðM; Q ðuð0ÞÞÞ. Proof. If y R EðM; Q ðuð0ÞÞÞ then there exists z 2 M such that z  y 2 Q D ðuð0ÞÞ. As u satisfies AIP with respect to Q it follows that

uy ðzÞ ¼ uðz  yÞ 6 0:

ð15Þ

From the hypotheses it is clear that

uy ðzÞ P uy ðyÞ  d ¼ uð0Þ  d > 0; which contradicts to (15) and the proof is completed. h The following result is obtained in a similar way. Let us recall that y 2 ASðu; M; dÞ is sharp if

uðzÞ > uðyÞ  d; 8z 2 M n fyg:

Theorem 4.2. Suppose that u satisfies ASIP (resp. AIP) with respect to Q. If 0 6 d 6 uð0Þ and y 2 ASðuy ; M; dÞ (resp. y 2 ASðuy ; M; dÞ is sharp) then y 2 EðM; Q ðuð0ÞÞÞ. Next, we apply Theorems 4.1 and 4.2 to the vector optimization problem (1). Theorem 4.3. Consider 0 6 d 6 uð0Þ and x 2 ASðu  ðf  f ðxÞÞ; S; dÞ. Then x 2 Eðf ; S; Q ðuð0ÞÞÞ if one of the following conditions is true: (a) uð0Þ 2 Rþþ ; u satisfies AIP with respect to Q and d < uð0Þ, (b) u satisfies AIP with respect to Q and x is sharp, (c) u satisfies ASIP with respect to Q.

Proof. We only prove part (b). As x 2 ASðu  ðf  f ðxÞÞ; S; dÞ is sharp we have that f ðxÞ 2 ASðuf ðxÞ ; f ðSÞ; dÞ is sharp and by Theorem 4.2 we deduce that f ðxÞ 2 Eðf ðSÞ; Q ðuð0ÞÞÞ. Then the result follows by (4). h In the literature, sufficient conditions like the previous ones are obtained by means of monotone functionals. In Proposition 4.1 we show that these functionals satisfy AIP or ASIP and so a lot of these conditions based on monotone functionals can be proved by Theorem 4.3. Definition 4.2. Let A  Y; A – ;. We say that u : Y ! R [ f1g is D-monotone (resp. strong D-monotone) on A if uðy  dÞ 6 uðyÞ (resp.

uðy  dÞ < uðyÞ), 8y 2 A; 8d 2 D; d – 0. The following result is immediate and its proof is omitted. Proposition 4.1. Consider u : Y ! R [ f1g such that uð0Þ 2 Rþ . If u is D-monotone (resp. strong D-monotone) on Q ðuð0ÞÞ and uðqÞ 6 0, 8q 2 Q ðuð0ÞÞ then u satisfies AIP (resp. ASIP) with respect to Q. Remark 4.1. Let us observe that AIP and ASIP do not assume any monotonicity property on the functional u. However, for each y 2 Y,

 Q D ðuð0ÞÞ  fz 2 Y : uðzÞ 6 0g () y  Q D ðuð0ÞÞ  fz 2 Y : uy ðzÞ 6 0g;  Q D ðuð0ÞÞ  fz 2 Y : uðzÞ < 0g () y  Q D ðuð0ÞÞ  fz 2 Y : uy ðzÞ < 0g and so, if Q D ðuð0ÞÞ [ f0g is a convex cone then both statements can be seen as two very general pointwise monotonicity properties since uð0Þ 2 Rþ . To be precise, if H ¼ Q D ðuð0ÞÞ [ f0g is a convex cone then AIP (resp. ASIP) implies the H-monotonicity (resp. strong H-monotonicity) of uy on fyg, for all y 2 Y.

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Example 4.1. Let us consider problem (1) with the following data: X ¼ Y ¼ R2 ; S ¼ fðx; yÞ 2 R2 : x 6 0; y ¼ x2 g; f ðx; yÞ ¼ ðx; yÞ for all ðx; yÞ 2 R2 and D ¼ R2þ . Let Q : Rþ R2þ and u : R2 ! R be the mappings Q ðeÞ ¼ intðR2þ Þ for all e P 0 and uðx; yÞ ¼ xy for all ðx; yÞ 2 R2 , respectively. Next we prove that WEðf ; SÞ ¼ S. Indeed, scalarization u is strong R2þ -monotone on intðR2þ Þ (observe that is not R2þ -monotone on R2 ) and uðqÞ 6 0 for all q 2 R2þ . Then, by Proposition 4.1 we deduce that u satisfies property ASIP with respect to Q. Moreover, uð0; 0Þ ¼ 0 and for all ðx; yÞ; ðu; v Þ 2 S we have

ðu  ðf  f ðx; yÞÞÞðu; v Þ ¼ ðu  xÞ2 ðu þ xÞ P 0:

ð16Þ

Therefore, ðx; yÞ 2 ASðu  ðf  f ðx; yÞÞ; S; 0Þ for all ðx; yÞ 2 S and by Theorem 4.3(c) we conclude that WEðf ; SÞ ¼ Eðf ; S; Q ð0ÞÞ ¼ S. Theorem 4.4, Corollary 4.1 and Remark 4.2 illustrate that Theorem 4.3 includes different sufficient conditions for e-efficient solutions of vector optimization problems obtained in the literature by approximate solutions of monotone scalarization processes. We say that the set-valued map Q is nonincreasing if Q ðe1 Þ  Q ðe2 Þ; 8e1 ; e2 2 Rþ ; e1 > e2 . Theorem 4.4. Suppose that Q is nonincreasing, u is finite valued, uð0Þ 2 Rþ and let x 2 ASðu  f ; S; uð0ÞÞ. If u is D-monotone (resp. strong Dmonotone) on f ðxÞ  Q ðuð0ÞÞ and

e :¼ uðf ðxÞÞ  sup fuðf ðxÞ  qÞg > uð0Þ

ð17Þ

q2Qðuð0ÞÞ

(resp.

e P uð0Þ) then x 2 Eðf ; S; Q ðeÞÞ.

Proof. Let us consider that u is D-monotone on f ðxÞ  Q ðuð0ÞÞ and (17) is satisfied. Define the functional U : Y ! R by

UðyÞ ¼ uðy þ f ðxÞÞ  sup fuðf ðxÞ  qÞg;

8y 2 Y:

q2Q ðuð0ÞÞ

From (17) it is clear that Uð0Þ ¼ e > uð0Þ. On the other hand,

ðU  ðf  f ðxÞÞÞðzÞ ¼ Uðf ðzÞ  f ðxÞÞ ¼ uðf ðzÞÞ  sup fuðf ðxÞ  qÞg;

8z 2 X

q2Qðuð0ÞÞ

and as x 2 ASðu  f ; S; uð0ÞÞ it follows that

ðU  ðf  f ðxÞÞÞðzÞ P uðf ðxÞÞ  sup fuðf ðxÞ  qÞg  uð0Þ ¼ ðU  ðf  f ðxÞÞÞðxÞ  uð0Þ;

8z 2 S:

q2Qðuð0ÞÞ

Therefore, x 2 ASðU  ðf  f ðxÞÞ; S; uð0ÞÞ. As Q is nonincreasing it follows that Q ðUð0ÞÞ  Q ðuð0ÞÞ and so, 8d1 2 Q ðUð0ÞÞ; 8d2 2 D n f0g,

Uðd1  d2 Þ ¼ uðd1  d2 þ f ðxÞÞ  sup fuðf ðxÞ  qÞg 6 uðd1 þ f ðxÞÞ  sup fuðf ðxÞ  qÞg ¼ Uðd1 Þ q2Q ðuð0ÞÞ

q2Qðuð0ÞÞ

since u is D-monotone on f ðxÞ  Q ðuð0ÞÞ. Therefore U is D-monotone on Q ðUð0ÞÞ. Moreover,

UðdÞ ¼ uðd þ f ðxÞÞ  sup fuðf ðxÞ  qÞg 6 0;

8d 2 Q ðUð0ÞÞ;

q2Qðuð0ÞÞ

since Q ðUð0ÞÞ  Q ðuð0ÞÞ. Then, by Proposition 4.1 we deduce that U satisfies AIP with respect to Q and the result follows by applying Theorem 4.3(a) to the scalarization U and d ¼ uð0Þ. This finishes the proof since the reasoning is analogous if u is strong D-monotone on f ðxÞ  Q ðuð0ÞÞ and e P uð0Þ. h Let us observe that Theorem 4.4 cannot be applied if 0 2 Q ðuð0ÞÞ and u is D-monotone (but not strong D-monotone) on f ðxÞ  Q ðuð0ÞÞ, since in this case e ¼ 0. Corollary 4.1. Consider q 2 D n f0g and the set-valued map

Q ðeÞ ¼ eq þ D;

8e 2 Rþ :

Suppose that u is finite valued, uð0Þ 2 Rþ and let x 2 ASðu  f ; S; uð0ÞÞ. If u is D-monotone (resp. strong D-monotone) on f ðxÞ  uð0Þq  D and

e :¼ uðf ðxÞÞ  uðf ðxÞ  uð0ÞqÞ > uð0Þ (resp.

e P uð0Þ) then x 2 Eðf ; S; Q ðeÞÞ.

Proof. This result is a direct consequence of Theorem 4.4, since Q is nonincreasing and supd2Q ðuð0ÞÞ fuðf ðxÞ  dÞg ¼ uðf ðxÞ  uð0ÞqÞ. h Remark 4.2. (a) Let us observe that Theorem 4.4 is based on monotonicity properties of the scalarization mapping u that depend on the point f ðxÞ. However, Theorem 4.3 is true under monotonicity assumptions of the scalarization mapping u on zero (see Remark 4.1). In this sense Theorem 4.3 is more general than Theorem 4.4. For instance, Example 4.1 has been solved by applying Theorem 4.3 to the scalarization mapping uðx; yÞ ¼ xy, which is not R2þ -monotone on f ðu; v Þ  R2þ for each ðu; v Þ 2 S. Thus Theorem 4.4 cannot be applied by considering this scalarization mapping. (b) Corollary 4.1 is similar to [10, Theorem 3.4], although the assumptions of the former are stronger than those of the latter. However, both results can be used if u 2 Dþ ; u 2 Dþs or u is the gauge functional uD;q given in Theorem 3.2(c). Moreover, let us observe that Corollary 4.1 is a consequence of Theorem 4.4, which can be used on different e-efficiency notions. Next we show that Theorem 4.3 generalizes well-known sufficient conditions on e-efficient solutions of (1) based on approximate solutions of specific scalarization processes, like the weighted-sum method, the constrained-objective scalarization, etc. (see [3, Section 3]). The following theorem is necessary.

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Theorem 4.5. Let C  D n f0g be a nonempty set such that C þ D n f0g ¼ C, let Q be the set-valued map defined by (3) and let e 2 Rþþ . Consider x 2 X and a proper positively homogeneous functional u : Y ! R [ f1g such that

uðyÞ  uðf ðxÞÞ 6 uðy  f ðxÞÞ; 8y 2 Y:

ð18Þ

(a) Assume that C  fy 2 Y : uðyÞ < 1g. If x 2 ASðu  f ; S; eÞ then x 2 Eðf ; S; Q ðeÞÞ. (b) Assume that C  fy 2 Y : uðyÞ 6 1g. If x 2 ASðu  f ; S; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ. Proof. We only prove part (a) since the proof of part (b) is analogous. Let us define the functional U : Y ! R [ f1g by U ¼ u þ e. It is clear that Uð0Þ ¼ e 2 Rþþ and

UðUð0ÞdÞ ¼ Uð0ÞuðdÞ þ e ¼ euðdÞ þ e < 0;

8d 2 C:

As Q D ðUð0ÞÞ ¼ Uð0ÞC we see that U satisfies ASIP with respect to Q. Moreover, ðU  ðf  f ðxÞÞÞðxÞ ¼ e and from (18) we deduce that

ðU  ðf  f ðxÞÞÞðzÞ ¼ uðf ðzÞ  f ðxÞÞ þ e P uðf ðzÞÞ  uðf ðxÞÞ þ e P 0;

8z 2 S;

since x 2 ASðu  f ; S; eÞ. Therefore, x 2 ASðU  ðf  f ðxÞÞ; S; eÞ and the result follows by Theorem 4.3(c). h Remark 4.3 (a) Condition (18) is satisfied if u is subadditive. (b) By Theorem 4.5 we prove sufficient conditions for approximate solutions of vector optimization problems via scalarization processes that do not satisfy any monotonicity hypotheses. (c) Theorem 4.5 can be extended to e ¼ 0 if statement C  fy 2 Y : uðyÞ < 1g (resp. C  fy 2 Y : uðyÞ 6 1g) are changed into coneðCÞ n f0g  fy 2 Y : uðyÞ < 0g ðconeðCÞ n f0g  fy 2 Y : uðyÞ 6 0gÞ. To illustrate the power of Theorem 4.5, we obtain several relations between approximate solutions of well-known scalarization processes and e-efficient solutions of (1) in the sense of Kutateladze (see Remark 2.1). Corollary 4.2. Suppose that q 2 D n f0g and

e 2 Rþþ . Let Q be the set-valued map Q ðeÞ ¼ eq þ D n f0g; 8e 2 Rþþ .

(a) (Weighted-sum scalarization) Consider k 2 Dþ n f0g, the functional u : Y ! R,

uðyÞ ¼ kðyÞ; 8y 2 Y;

ð19Þ

and suppose that kðqÞ P 1. (i) If x 2 ASðu  f ; S; eÞ then x 2 WEðf ; S; Q ðeÞÞ. (ii) If x 2 ASðu  f ; S; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ. (iii) If k 2 Dþs and x 2 ASðu  f ; S; eÞ then x 2 Eðf ; S; Q ðeÞÞ. (b) (Constrained-objective scalarization) Let k1 ; k2 ; . . . ; kp 2 Dþ n f0g and consider the functional u : Y ! R defined by uðyÞ ¼ k1 ðyÞ; 8y 2 Y and suppose that k1 ðqÞ P 1. Let ðv 2 ; v 3 ; . . . ; v p Þ 2 Rp1 and

K ¼ fz 2 X : ki ðf ðzÞÞ 6 v i ; i ¼ 2; 3; . . . ; pg: (i) (ii)

If x 2 ASðu  f ; S \ K; eÞ then x 2 WEðf ; S; Q ðeÞÞ. If x 2 ASðu  f ; S \ K; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ.

(c) (Guddat scalarization) Consider the functional u : Y ! R defined by (19), x 2 S,

K ¼ fz 2 X : f ðzÞ 6 f ðxÞg and suppose that kðqÞ P 1. (i) If x 2 ASðu  f ; S \ K; eÞ then x 2 WEðf ; S; Q ðeÞÞ. (ii) If x 2 ASðu  f ; S \ K; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ. (iii) If k 2 Dþs and x 2 ASðu  f ; S \ K; eÞ then x 2 Eðf ; S; Q ðeÞÞ. (d) (Tchebycheff-norm scalarization) Let T : Y ! Y be a continuous linear operator such that TðintðDÞÞ  intðDÞ; k1 ; k2 ; . . . ; kp 2 Dþ n f0g; r 2 Y and the functional u : Y ! R defined by

uðyÞ ¼ maxfki ðTðy  rÞÞ : i ¼ 1; 2; . . . ; pg; 8y 2 Y: Suppose that minfki ðTðqÞÞ : i ¼ 1; 2; . . . ; pg P 1. (i) If x 2 ASðu  f ; S; eÞ then x 2 WEðf ; S; Q ðeÞÞ. (ii) If x 2 ASðu  f ; S; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ.

Proof. In order to apply Theorem 4.5 consider C ¼ q þ D n f0g. We know that C  D n f0g and C þ D n f0g ¼ C. Moreover, it is clear that Eðf ; S; Q ðeÞÞ is equal to WEðf ; S; Q ðeÞÞ if we take intðDÞ [ f0g instead of D in the definition of C. Part (a)(i). For each d 2 intðDÞ it follows that

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uðq  dÞ < uðqÞ 6 1 and part (a)(i) follows by Theorem 4.5(a). Part (a)(ii) is analogous to part (a)(i), but now d 2 D n f0g; uðq  dÞ 6 1 and we apply Theorem 4.5(b). Part (a)(iii). For each d 2 D n f0g we see that

uðq  dÞ < uðqÞ 6 1; since u 2 Dþs and this part is completed by Theorem 4.5(a). Part (b). It is obvious that this scalarization function is a particular case of the previous one. Moreover, as Q D ðeÞ  D it follows that

x 2 Eðf ; S \ K; Q ðeÞÞ ) x 2 Eðf ; S; Q ðeÞÞ and similarly,

x 2 WEðf ; S \ K; Q ðeÞÞ ) x 2 WEðf ; S; QðeÞÞ: Then, part (b) is a direct consequence of part (a). The proof of part (c) is similar to the proof of part (b). Part (d)(i) Let us define fr : X ! Y by fr ðzÞ ¼ f ðzÞ  r; 8z 2 X and U : Y ! R by

8y 2 Y:

UðyÞ ¼ maxfki ðTðyÞÞ : i ¼ 1; 2; . . . ; pg;

By hypotheses we have that x 2 ASðU  fr ; S; eÞ and

Uðq  dÞ ¼ maxfki ðTðqÞÞ  ki ðTðdÞÞ : i ¼ 1; 2; . . . ; pg < maxfki ðTðqÞÞ : i ¼ 1; 2; . . . ; pg ¼  minfki ðTðqÞÞ : i ¼ 1; 2; . . . ; pg 8d 2 intðDÞ:

6 1;

It is clear that U is sublinear, and by Remark 4.3(a) U satisfies (18). Then, by Theorem 4.5(a) we deduce that x 2 Eðfr ; S; Q ðeÞÞ if we consider intðDÞ [ f0g instead of D in the definition of C, and so x 2 WEðfr ; S; Q ðeÞÞ. From here the result follows because WEðfr ; S; Q ðeÞÞ ¼ WEðf ; S; Q ðeÞÞ. The proof of Part (d)(ii) is analogous to the previous one. h Next we study the direction method. Fix

v 2 D n f0g; r 2 Y

and consider the problem

Minfa 2 R : r þ av 2 f ðSÞ þ Dg:

ð20Þ

Let us denote the feasible set of (20) by R (observe that R is empty or an upper unbounded interval) and for each

e 2 Rþ ,

ADMðeÞ ¼ fa 2 R : a  e 6 a; 8a 2 Rg; SADMðeÞ ¼ fa 2 R : a  e < a; 8a 2 Rg; [ fx 2 S : f ðxÞ 2 r þ av  Dg; EDMðeÞ ¼ a2ADMðeÞ

SEDMðeÞ ¼

[

fx 2 S : f ðxÞ 2 r þ av  Dg:

a2SADMðeÞ

Theorem 4.6. Consider

e 2 Rþ and fr ¼ f  r. The following statements hold.

(a) If x 2 EDMðeÞ then x 2 ASðuD;v  fr ; S; eÞ. (b) If D is closed and x 2 SEDMðeÞ then x 2 ASðuD;v  fr ; S; eÞ is sharp.

Proof. Part (a). For each a 2 ADMðeÞ we have that

f ðzÞ  r R ða  e  dÞv  D;

8d 2 Rþþ ; 8z 2 S:

ð21Þ

Indeed, if there exist d 2 Rþþ and z 2 S such that f ðzÞ  r 2 ða  e  dÞv  D then a ¼ a  e  d 2 R and a < a  e, which is a contradiction since a 2 ADMðeÞ. By (21) we deduce that

uD;v ðf ðzÞ  rÞ ¼ infft 2 R : f ðzÞ  r 2 tv  Dg P a  e; 8z 2 S:

ð22Þ

On the other hand, if x 2 EDMðeÞ then there exists a 2 ADMðeÞ such that f ðxÞ  r 2 av  D and so uD;v ðf ðxÞ  rÞ 6 a. Therefore,

uD;v ðf ðxÞ  rÞ  e 6 a  e 6 uD;v ðf ðzÞ  rÞ; 8z 2 S and part (a) follows. Part (b). For each a 2 SADMðeÞ we see that

f ðzÞ  r R ða  eÞv  D;

8z 2 S:

ð23Þ

Indeed, if there exists z 2 S such that f ðzÞ  r 2 ða  eÞv  D then a ¼ a  e 2 R, which is a contradiction since a 2 SADMðeÞ. By (23) we deduce that uD;v ðf ðzÞ  rÞ > a  e; 8z 2 S, since D is closed. From here, the proof finishes as in part (a). h Corollary 4.3. Consider q 2 v þ D, the set-valued map Q ðeÞ ¼ eq þ D n f0g and (a) If x 2 EDMðeÞ then x 2 WEðf ; S; Q ðeÞÞ. (b) If D is closed and x 2 SEDMðeÞ then x 2 Eðf ; S; Q ðeÞÞ.

e 2 Rþþ .

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Proof. Part (a). By Theorem 4.6(a) we deduce that x 2 ASðuD;v  fr ; S; eÞ. By Theorem 3.2(e) we see that uD;v is sublinear. Moreover,

q  intðDÞ  fy 2 Y : uD;v ðyÞ < 1g since q 2 v þ D. Indeed, 8d 2 intðDÞ; q  d 2 v  D  intðDÞ ¼ v  intðDÞ and so uD;v ðq  dÞ < 1. Therefore, Theorem 4.5(a) can be applied to C ¼ q þ intðDÞ and we obtain that x 2 WEðfr ; S; Q ðeÞÞ and so x 2 WEðf ; S; Q ðeÞÞ. Part (b) follows similarly and the proof is finished. h Remark 4.4. (a) Corollaries 4.2 and 4.3 reduce to the results of [3, Section 3] when (1) is a Pareto problem and one considers specific funcm tions ki and T. For example, Corollary 4.2(d) reduces to [3, Corollary 3.6] by taking Y ¼ Rm ; D ¼ Rm þ ; p ¼ m; ðw1 ; w2 ; . . . ; wm Þ 2 Rþþ ; ki ðy1 ; y2 ; . . . ; ym Þ ¼ wi yi ; 8 i ¼ 1; 2; . . . ; m and Tðy1 ; y2 ; . . . ; ym Þ ¼ ðy1 ; y2 ; . . . ; ym Þ. In particular, let us observe that all results of [3, Section 3] are consequences of Theorem 4.5. (b) Corollary 4.2(a), Theorem 4.6 and Corollary 4.3 complete the relations given in [10, Corollary 3.6 and Theorem 5.1(b)]. (c) The reader can extend Corollaries 4.2 and 4.3 to e ¼ 0 by considering Remark 4.3(c). The following simple result is very useful in order to apply a penalization reasoning to the scalarization process. We consider the same notations as those used in Theorem 3.3. Theorem 4.7. Let u satisfies AIP with respect to Q. Consider S ¼ N \ Sg and x 2 S. If x 2 ASð/x ; N; dÞ and 0 6 d < uð0Þ then x 2 Eðf ; S; Q ðuð0ÞÞÞ. Proof. As x 2 ASð/x ; N; dÞ we have that

/x ðzÞ P /x ðxÞ  d P uð0Þ  d > 0;

8z 2 N:

ð24Þ

Let us prove that x 2 ASðu  ðf  f ðxÞÞ; S; uð0ÞÞ is sharp. Indeed, for each z 2 S n fxg we have wðgðzÞÞ 6 0 and by (24) we deduce that

ðu  ðf  f ðxÞÞÞðzÞ ¼ /x ðzÞ > 0: Moreover, ðu  ðf  f ðxÞÞÞðxÞ ¼ uð0Þ and so

ðu  ðf  f ðxÞÞÞðzÞ > ðu  fx ÞðxÞ  uð0Þ;

z 2 S n fxg:

Then x 2 ASðu  ðf  f ðxÞÞ; S; uð0ÞÞ is sharp and the conclusion follows from Theorem 4.3(b). h Theorem 4.7 generalizes the sufficient conditions of [7, Lemma 3.1(ii)] and [8, Lemma 3.1(ii)].

5. Characterization of approximate efficiency One can use Corollary 3.1 and Theorem 4.3 to obtain scalarization processes that characterize approximate solutions of (1). Next we show this approach and in Example 5.1 we apply it on a Pareto problem. Fix e 2 Rþ . In order to characterize Q ðeÞ-efficient solutions of (1) via a function u : Y ! R [ f1g by using Corollary 3.1 and the statements in Theorem 4.3(a), (b) (resp. Theorem 4.3(c)), the following conditions must be satisfied:

fy 2 Y : uðyÞ < 0g  Q D ðeÞ;

Q D ðuð0ÞÞ  fy 2 Y : uðyÞ 6 0g

(resp. fy 2 Y : uðyÞ < 0g  Q D ðeÞ; Q D ðuð0ÞÞ  fy 2 Y : uðyÞ < 0g). Indeed, to apply Corollary 3.1 one requires that u satisfies property AORP with respect to Q at e, i.e., statement (8) must be true. Similarly, to apply parts (a) or (b) (resp. part (c)) of Theorem 4.3 it is necessary that u satisfies property AIP (resp. ASIP) with respect to Q, i.e., one needs that Q D ðuð0ÞÞ  fy 2 Y : uðyÞ 6 0g (resp. Q D ðuð0ÞÞ  fy 2 Y : uðyÞ < 0g). If uð0Þ ¼ e then both conditions are

Q D ðeÞ  fy 2 Y : uðyÞ 6 0g;

ðQ D ðeÞÞc  fy 2 Y : uðyÞ P 0g:

ð25Þ

(resp. Q D ðeÞ ¼ fy 2 Y : uðyÞ < 0g). For example, one could define

uðyÞ ¼



e c

if y R Q D ðeÞ; if y 2 Q D ðeÞ;

ð26Þ

where c 2 Rþ (resp. c 2 Rþþ ). Then, the following result follows easily. Corollary 5.1. Consider e; c 2 Rþ , a proper set-valued map Q and the functional u given by (26). (a) (b) (c) (d)

If x 2 Eðf ; S; Q ðeÞÞ then x 2 ASðu  ðf  f ðxÞÞ; S; eÞ. If e 2 Rþþ ; 0 6 d < e and x 2 ASðu  ðf  f ðxÞÞ; S; dÞ then x 2 Eðf ; S; Q ðeÞÞ. If x 2 ASðu  ðf  f ðxÞÞ; S; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ. If c 2 Rþþ then x 2 Eðf ; S; Q ðeÞÞ if and only if x 2 ASðu  ðf  f ðxÞÞ; S; eÞ.

Next we propose a scalarization process based on the functional uA;q studied in Theorem 3.2. Proposition 5.1. Let Q be a proper set-valued map and

e 2 Rþþ such that Q D ðeÞ is a star-shaped co-radiant set. Let q 2 kernðQ D ðeÞÞ satisfying

b :¼ infft 2 R : tq 2 Q D ðeÞg > 0: Then, the functional g :¼ ðe=bÞuQ D ðeÞ;q satisfies the following properties:

ð27Þ

C. Gutiérrez et al. / European Journal of Operational Research 201 (2010) 11–22

21

(a) 0 2 DomðgÞ and gð0Þ ¼ e. (b) fy 2 Y : gðyÞ < 0g  Q D ðeÞ  fy 2 Y : gðyÞ 6 0g. (c) If Q D ðeÞ is open then fy 2 Y : gðyÞ < 0g ¼ Q D ðeÞ. Proof. These statements are consequences of Theorem 3.2 and Remark 3.2(b).

h

Let us observe that hypothesis (27) holds, for example, if 0 R clðQ D ðeÞÞ. Now, we apply Corollary 3.1 and Theorem 4.3 to provide necessary and sufficient conditions for Q ðeÞ-efficient solutions of (1) via the scalarization g. Corollary 5.2. Under the hypotheses of Proposition 5.1 we have the following statements. (a) (b) (c) (d)

If If If If

x 2 Eðf ; S; Q ðeÞÞ then x 2 ASðg  ðf  f ðxÞÞ; S; eÞ. 0 6 d < e and x 2 ASðg  ðf  f ðxÞÞ; S; dÞ then x 2 Eðf ; S; Q ðeÞÞ. x 2 ASðg  ðf  f ðxÞÞ; S; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ. Q D ðeÞ is open then x 2 Eðf ; S; Q ðeÞÞ if and only if x 2 ASðg  ðf  f ðxÞÞ; S; eÞ.

Assume that Y is normed. In this framework a function satisfying (25) is, for instance, the so-called oriented distance:

DQ D ðeÞ ðyÞ ¼ dðy; Q D ðeÞÞ  dðy; Q D ðeÞc Þ;

8y 2 Y:

Suppose that dð0; Q D ðeÞÞ ¼ e. Then DQ D ðeÞ ð0Þ ¼ e and the oriented distance can be used to provide necessary and sufficient conditions for Q ðeÞ-efficient solutions of (1) through Corollary 3.1 and Theorem 4.3. To be precise, properties (a)–(c) of Proposition 5.1 are also true by considering DQ D ðeÞ instead of g and so the following conditions hold. Corollary 5.3. Consider (a) (b) (c) (d)

If If If If

e 2 Rþ , a proper set-valued map Q such that dð0; Q D ðeÞÞ ¼ e and the functional DQ D ðeÞ .

x 2 Eðf ; S; Q ðeÞÞ then x 2 ASðDQ D ðeÞ  ðf  f ðxÞÞ; S; eÞ. 0 6 d < e and x 2 ASðDQ D ðeÞ  ðf  f ðxÞÞ; S; dÞ then x 2 Eðf ; S; Q ðeÞÞ. x 2 ASðDQ D ðeÞ  ðf  f ðxÞÞ; S; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ. Q D ðeÞ is open then x 2 Eðf ; S; Q ðeÞÞ if and only if x 2 ASðDQ D ðeÞ  ðf  f ðxÞÞ; S; eÞ.

Let us observe that Corollary 5.3 can be applied to As ImðQ D Þ  D, the functional

e ¼ 0 whereas e 2 Rþþ in Corollary 5.2.

UQ D ðeÞ :¼ DQ D ðeÞ þ ID ; where ID is the indicator function of the set D, satisfies properties (a)–(c) of Proposition 5.1 and so Corollary 5.3 is also true for the scalarization UQ D ðeÞ . In some problems, it is easier to obtain this functional than the oriented distance DQ D ðeÞ , as it is showed in the following example. Example 5.1. If (1) is a Pareto problem, Rp is normed by the norm k k1 and Q ðeÞ ¼ Rpþ \ eBc , i.e., Tanaka’s e-efficiency notion is considered (see [15] and Example 3.1(b)), then it is easy to check that

8   p > < max maxfy g; e þ P y if y 2 Rpþ ; i i 16i6p UQ D ðeÞ ðyÞ ¼ i¼1 > : þ1 if y R Rpþ : Consider f ¼ ðf1 ; f2 ; . . . ; fp Þ; x 2 S and the mapping hð; xÞ :¼ UQ D ðeÞ  ðf  f ðxÞÞ : X ! Rp , i.e., for each z 2 X; hðz; xÞ ¼ þ1 if f ðzÞ R f ðxÞ  Rpþ and if f ðzÞ 2 f ðxÞ  Rpþ then

(

hðz; xÞ ¼ max maxffi ðzÞ  fi ðxÞg; e þ 16i6p

p X i¼1

fi ðzÞ 

p X

)

fi ðxÞ

i¼1

and the following conditions hold: (a) If x 2 Eðf ; S; Q ðeÞÞ then x 2 ASðhð; xÞ; S; eÞ. (b) If 0 6 d < e and x 2 ASðhð; xÞ; S; dÞ then x 2 Eðf ; S; Q ðeÞÞ. (c) If x 2 ASðhð; xÞ; S; eÞ is sharp then x 2 Eðf ; S; Q ðeÞÞ.

Acknowledgements The authors are grateful to the anonymous referees for their helpful comments and suggestions. References [1] G.Y. Chen, X.X. Huang, X. Yang, Vector optimization, Set-valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, vol. 541, SpringerVerlag, Berlin, 2005. [2] J. Dutta, V. Vetrivel, On approximate minima in vector optimization, Numerical Functional Analysis and Optimization 22 (7&8) (2001) 845–859.

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