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Improvement sets and vector optimization
European Journal of Operational Research 223 (2012) 304–311
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Continuous Optimization
Improvement sets and vector optimization q C. Gutiérrez a,⇑, B. Jiménez b, V. Novo b a b
Departamento de Matemática Aplicada, E.T.S. de Ingenieros de Telecomunicación, Universidad de Valladolid, Paseo de Belén 15, Campus Miguel Delibes, 47011 Valladolid, Spain Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 15 December 2011 Accepted 31 May 2012 Available online 13 June 2012 Keywords: Improvement set Minimal point Vector optimization e-Efficiency Scalarization
a b s t r a c t In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize wellknown results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems. 2012 Elsevier B.V. All rights reserved.
1. Introduction A current research line related with vector optimization problems is to develop concepts and settings in order to unify the more important solution notions of these problems, like the so-called efficiency, proper efficiency, weak efficiency, strict efficiency, strong efficiency, and e-efficiency (see [1,4,6,12–15,22]). Probably, the first attempt to unify classical solution concepts in vector optimization problems via a more general solution notion that collapses all of them was due to Mordukhovich (see [22]) through the notion of generalized order optimality of vector optimization problems. This notion works when the objective space is normed, it was motivated by the well-known set extremality concept due to Kruger and Mordukhovich (see [18]), extends the concepts of Slater minimizer, efficient solution and weak efficient solution (see [27]) and it has been used to obtain necessary optimality conditions of vector optimization problems (see [22]). Recently, in the framework of a normed space or a Hausdorff locally convex topological linear space Y, several authors have introduced and studied concepts of nondominated point of a set based on binary relations defined as usual, but from a general ordering set H Y:
y1 ; y2 2 Y;
y1 6H y2 () y2 y1 2 H:
ð1Þ
q This research was partially supported by the Ministerio de Ciencia e Innovación (Spain) under project MTM2009-09493. ⇑ Corresponding author. E-mail addresses: [email protected] (C. Gutiérrez), [email protected] (B. Jiménez), [email protected] (V. Novo).
0377-2217/$ - see front matter 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.05.050
This ordering set could not be convex neither a cone and so the relation 6H could not be transitive. Analogously, it could happen that 0 R H and then 6H could not be reflexive. Both facts are interesting from a practical point of view, since preference relations that are not necessarily a preorder are usual in Economics (see [19,24] and the references therein). Sometimes, the closedness of the ordering set is not assumed either, and so certain important preference relations like the lexicographical order can be dealt with. Moreover, some ordering sets could not be solid, i.e., the topological interior of H could be empty, in order to deal with objective spaces as ‘p and Lp, 1 6 p < 1, where the natural ordering is given by a non-solid convex cone. To compensate these generalizations, other mild conditions are assumed on the ordering set. For example, in [16,24] the authors considered nondominated point notions via conic ordering sets, and in [12–15] we introduced and studied a concept of nondominated point based on a coradiant ordering set (aH H, for all a > 1), that generalizes the more important exact and approximate concepts of nondominated point in linear spaces. To our knowledge, the more general approaches in this line are due to Bao and Mordukhovich [1] and Flores–Bazán and Hernández [6], since they consider the usual notion of nondominated point without imposing any condition on the ordering set H. These notions reduce to the concepts of ideal, efficient, weak efficient, relative efficient, quasi relative efficient and Henig proper efficient point among others. However, in order to obtain optimality conditions in vector optimization problems, some conditions on the ordering set are assumed. To be exact, Flores–Bazán and Hernández suppose that zero is in the boundary of H and there exists a point q 2 Yn{0} such
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that (cl H and int H denote the closure and the interior of H, respectively)
cl H þ ð0; 1Þq int H;
ð2Þ
which implies that the ordering set H is solid (see [6]), and Bao and Mordukhovich assume the so-called local closedness property (see [1, Definition 3.3]), which is compatible with non-solid ordering sets. A similar contribution has recently been introduced by Chicco et al. in [4]. Specifically, by considering the finite dimensional setting Y ¼ Rp and via an improvement set E (i.e., 0 R E and E þ Rpþ ¼ E, where Rpþ is the non-negative orthant), these authors have defined the notion of E-optimal point (nondominated with respect to the binary relation (1) given by H = E), that collapses the concepts of Pareto minimality and weak Pareto minimality, and that is useful to deal with approximate minimal points too. This paper focus on this notion. Essentially, in a non-necessarily finite dimensional ordered linear space, an improvement set E is a free disposal set, i.e., its cone expansion E + K is itself, where K is the ordering cone. This kind of sets was introduced by Debreu in [5] and they have frequently been used in mathematical economics and optimization. So, one can find in the literature several previous concepts very close to the notion of improvement set. For example, in the setting of ordered linear spaces where the ordering relation (1) is defined by a convex cone H, the concepts of downward set and upward set (see [20,26]) are the same as the notion of improvement set, but relaxing the condition 0 R E. To be precise, an improvement set E is an upward set such that 0 R E. On the other hand, downward sets can be considered as a certain analog of normal subsets of the non-negative orthant (see [20,26]). Moreover, let us observe that in mathematical economics and game theory, the upward sets are also called free disposal sets and the convex downward sets are known as comprehensive sets. This work has two objectives. The first one is to study the notion of improvement set by Chicco et al. [4] in the setting of an arbitrary quasi ordered linear space. In particular, we are interested in characterizing these sets in terms of sublevel sets of scalar functions. The second one is to state necessary and sufficient conditions for E-optimal solutions of vector optimization problems via scalarization. In the whole paper it is implicit that a lot of ideas and techniques of the works [12–15] can be used to obtain results about E-optimal points and E-optimal solutions. This work is structured as follows. In Section 2, some notations are fixed, several well-known concepts and mathematical tools are recalled and the notion of improvement set by Chicco et al. [4] is extended to a linear space ordered via a convex cone as usual. This notion is showed to be very close to the concepts of free disposal set, downward set, upward set and coradiant set and several general classes of improvement sets are introduced. Moreover, the relation between an improvement set and the ordering cone K is clarified. In Section 3, the improvement sets are characterized. First, by the separation theorem, closed and convex improvement sets are showed to be such that the positive polar cone of their recession cone is included into the positive polar cone of the order cone. Second, nonconvex closed (resp. open) improvement sets are characterized through sublevel sets of decreasing lower semicontinuous (resp. continuous) functionals. The main mathematical tool for proving these last results is the well-known nonconvex separation functional by Tammer and Weidner [8,10]. In Section 4, the notion of nondominated solution of a vector optimization problem with respect to an ordering set given by an improvement set is defined. It is showed that this notion collapses the more popular exact and approximate solution concepts of a
vector optimization problem. Moreover, it is characterized through approximate solutions of associated scalar optimization problems without considering any convexity assumption. These scalarization results are obtained in two steps. First, by considering generic scalar functionals whose sublevel set at zero is the improvement set. Second, by applying these generic characterizations to some specific scalarization functionals. In Section 5, other characterizations for this kind of nondominated solutions via linear scalarization processes are obtained in K-convexlike vector optimization problems, i.e., in problems where the objective function is closely K-convexlike and the improvement set is convex. As a consequence, some necessary and sufficient approximate optimality conditions via linear scalarizations obtained by ourselves in [12] for K-convexlike Pareto optimization problems are improved and extended to vector optimization problems. Finally, in Section 6, some conclusions are presented which summarize this work. 2. Notations and preliminaries Let Y be a Hausdorff locally convex topological linear space and K Y be a convex cone (we consider that 0 2 K), which is assumed to be proper, i.e., {0} – K – Y. In the sequel, we suppose that Y is ordered through the following quasi order:
y1 ; y2 2 Y;
y1 6K y2 () y2 y1 2 K:
Let R :¼ R [ f1g. Given a functional u : Y ! R, we denote
epi u ¼ fðy; rÞ 2 Y R : uðyÞ 6 rg: Moreover, u is said to be 6K-increasing (resp. 6K-decreasing) on a nonempty set M Y if u(y) 6 u(y + d) (resp. u(y) P u(y + d)), for all y 2 M, d 2 K. We say that u is 6K-increasing (resp. 6K-decreasing) if u is 6K-increasing (resp. 6K-decreasing) on Y. Let ; – M Y be closed and q 2 Yn{0}. In the paper, the following Minkowski gauge nM q : Y ! R will be used:
nM q ðyÞ ¼
þ1
if y R Rq þ M;
infft 2 R : y þ tq 2 Mg otherwise:
We denote by cl M, int M, bd M, Mc, conv M and cone M the closure, the topological interior, the boundary, the complement, the convex hull and the cone generated by a set M Y, respectively, and we say that M is solid if int M – ; and free disposal with respect to K (free disposal for short) if M + K = M (see [21] and the references therein). As usual, the topological dual space of Y is denoted by Y⁄. The positive and strict positive polar cones to M are denoted by M+ and M+s, respectively, i.e.,
M þ ¼ fk 2 Y : kðyÞ P 0; 8 y 2 Mg; M þs ¼ fk 2 Y : kðyÞ > 0; 8 y 2 M; y – 0g: We write Rpþ to refer the non-negative orthant of Rp , Rþ :¼ R1þ and Rþþ ¼ Rþ n f0g. Definition 2.1. A nonempty set E Y is said to be an improvement set for 6K (or an improvement set with respect to K) if 0 R E and E is free disposal. For convenience we will say that E is a 6K-i.s. or simply E is an i.s. if there is no confusion. Remark 2.2. Definition 2.1 extends [4, Definition 2.5] to ordered linear spaces. To be precise, the notion of 6K-improvement set reduces to [4, Definition 2.5] if we consider Y ¼ Rp and K ¼ Rpþ . However, there exist in the literature several previous concepts very close to this notion by Chicco et al., as it is showed in the following examples.
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(a) Let (V, 6) be an ordered (non-necessarily linear) space and ; – E V. Let us recall that E is a downward set (resp. upward set, see for instance [20,26]) if
v 2 V;
y 2 E;
v6y)v 2E
ðresp: y 6 v ) v 2 EÞ:
In the setting of the ordered linear space (Y, 6K), it is clear that a nonempty set E Y is upward if and only if E + K = E (i.e., if E is free disposal), and it is downward if and only if E K = E. So, the following statements are equivalent: (i) E is a 6K-i.s. (ii) 0 R E and E is an upward set. (iii) 0 R E and E is a downward set with respect to the ordering 6K. (b) Consider Y ¼ Rp ; K ¼ Rpþ and a nonempty set E Rpþ . Let us recall that E is normal (see, for instance, [25]) if
v 2 Rpþ ;
y 2 E;
v 6R
p þ
y ) v 2 E:
Clearly, E is normal if and only if ðE Rpþ Þ \ Rpþ ¼ E. Thus, if E Rpþ is a 6Rpþ -improvement set, then E is normal. (c) Recall that a nonempty set E Y is coradiant (see, for instance, [13]) if aE E, for all a > 1. Then, any i.s. E K is coradiant. However, the reciprocal statement is false. For example, if dim Y > 1, K is solid, q 2 Kn{0} and c > 0, then the set E = {aq:a P c} Kn{0} is coradiant, but it is not free disposal. If 0 R E, E å K, then the statements ‘‘E is 6K-i.s.’’ and ‘‘E is coradiant’’ have not any relation (see [4, Remark 2.3]). (d) Given a nonempty set H Y, the well-known conic extension of H with respect to K is the set EK ðHÞ :¼ H þ K. Then EK ðHÞ is an i.s. if and only if H \ (K) = ;. By the previous remark, the next example and Proposition 2.4, we see that the class of 6K-i.s. is very wide.
(f) If E is i.s., then y + E is i.s. for all y 2 Y, whenever y R E. If ; – H Y and H \ (E) = ;, then H + E is an i.s. (observe that S H+E= y2Hy + E). In particular, if E1,E2 are i.s. and E1 \ (E2) = ;, then E1 + E2 is an i.s. (g) If E is i.s. and a > 0, then aE is i.s.; if a < 0, then aE is i.s. for 6K. (h) If E is free disposal and 0 2 E, then (E)c is 6K-i.s., as can be easily checked. In particular, (q K)c is an 6K-i.s. for all q 2 K. Let us observe that (K)c is the biggest 6K-i.s. Let us observe that parts (e) and (h) of Example 2.3 reduce to [4, Proposition 3.1] by considering Y ¼ Rp and K ¼ Rpþ . Proposition 2.4. Suppose that E Y is 6K-i.s. Then (a) cl E is free disposal. (b) If E is solid, then int E is 6K-i.s. Proof. (a) We need to check that cl E + K cl E. Let y 2 cl E, z 2 K and take a net (yi) E such that yi ? y. Then
y þ z ¼ limðyi þ zÞ 2 clðE þ KÞ ¼ cl E i
and so cl E is free disposal. (b) It is enough to prove that int E + K int E, which is clear since int E + K is an open set contained in E + K = E. h The following two propositions clarify the relation between the sets E and K when E is 6K-i.s. Proposition 2.5. Let E Y be a nonempty free disposal closed set. Then the following statements are equivalent: (a) 0 R E. (b) (Kn{0}) \ Ec – ;.
Example 2.3. (a) Let us suppose that K is pointed, i.e., K \ (K) = {0}. If ; – H Y and H \ (Kn{0}) = ;, then H + Kn{0} is an i.s. In particular, Kn{0} is an i.s. (b) If K is solid, then int K is an i.s. (c) If u : Y ! R is 6K-increasing, u(0) = 0, d P 0 and m > 0, then [u > d] :¼ {y 2 Y:u(y) > d}, [u P m] :¼ {y 2 Y:u(y) P m} and [u P 0]n{0} are i.s. whenever they are nonempty. Observe that every k 2 K+n{0} and the Minkowski gauge nK (whenq ever K is closed, solid and q 2 int K), are 6K-increasing functionals that satisfy the condition u(0) = 0 (see, for instance, [10]). More generally, if E is free disposal, u : Y ! R is 6K-increasing on E, u(0) = 0, d P 0 and m > 0, then E \ [u > d], E \ [u P m] and E \ ([u P 0]n{0}) are i.s. whenever they are nonempty. Analogously, if E is free disposal and u : Y ! R is 6K-decreasing on E, then the sublevel set E \ [u 6 a] :¼ {y 2 E:u(y) 6 a} and the strict sublevel set E \ [u < a] :¼ {y 2 E: u(y) < a} are 6K-i.s. for each a 2 R whenever they are nonempty sets, 0 R E \ [u 6 a] and 0 R E \ [u < a]. (d) Consider V ¼ Y R ordered by the cone K Rþ . Then u : Y ! R [ fþ1g is 6K-decreasing if and only if epi u V is upward. Therefore, if u is 6K-decreasing and u(0) > 0, then epi u is 6KRþ -i.s. S (e) If (Ei)i2I is a family of i.s., then i2IEi is an i.s. If (Ei)i2I is a family of free disposal sets and 0 R Ei0 for some i0 2 I (i.e., T some set in the family is an i.s.), then i2IEi is an i.s. whenever it is nonempty.
Furthermore, if (b) holds, then aq R E for all q 2 (Kn{0}) \ Ec and for all a 2 (1,1]. Proof. (a) ) (b) If (Kn{0}) \ Ec = ;, then Kn{0} E and by using that K is a proper cone we deduce that 0 2 cl(Kn{0}) cl E = E, which is a contradiction. (b) ) (a) Let q 2 (Kn{0}) \ Ec. If 0 2 E, then q 2 K K + E = E, which is a contradiction. For the last part, consider q 2 (Kn{0}) \ Ec and suppose that there exists a 2 (1,1] such that aq 2 E. Then, q = aq + (1 a)q 2 E + K = E, which is a contradiction. h The second part of the following result is a direct consequence of the well-known separation property between a point and a closed convex set and so its proof is omitted. We denote by E the cone generated by the convex hull of E, i.e., E ¼ cone conv E.
Proposition 2.6. Suppose that E Y is 6K-i.s. Then (a) Eþ K þ and cl K cl E. (b) Additionally suppose that E is closed and convex. Then there exists k 2 Eþs such that infy2Ek(y) > 0. Proof. Let k 2 Eþ . For a fixed point y0 2 E, as y0 + K E it follows that k(y0) 6 k(z) for all z 2 K, and so k(z) P 0 for all z 2 K, i.e., k 2 K+. Then Eþ K þ and by the bipolar theorem we conclude that cl K cl E. h
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3. Characterization of
M 1 ¼ fu 2 Y : M þ Rþþ u Mg: It is well-known that M1 is a convex cone for any set M (see, for instance, [9,23]). Moreover, if M is convex, then M1 = {u 2 Y:y0 + tu 2 M, " t > 0}, where y0 is an arbitrary point of M, and M is free disposal if and only if K M1. In the sequel we obtain a necessary and sufficient condition from which one can deduce if a nonempty closed convex set E Y is 6K-i.s. Theorem 3.1. Let E Y be nonempty, closed and convex. Then E is 6K-i.s. if and only if 0 R E and (E1)+ K+. Proof. Suppose that E is 6K-i.s. From the definition we have that 0 R E and E is free disposal. Thus, K E1 and so (E1)+ K+. Reciprocally, consider 0 R E, (E1)+ K+ and suppose that E is not 6K-i.e. Then E + K å E and so there exist y0 2 E and z0 2 K such that y0 + z0 R E. By applying a separation theorem we see that there exists k 2 Y⁄n{0} such that
kðy0 Þ þ kðz0 Þ < inffkðyÞ : y 2 Eg 6 kðyÞ þ tkðuÞ;
8 y 2 E; u 2 E1 ; t P 0: In particular, by taking y = y0 we have
kðz0 Þ < tkðuÞ;
8 u 2 E1 ; t P 0;
and it follows that k(z0) < 0 and k 2 (E1)+. This is a contradiction, since (E1)+ K+ and z0 2 K, and the proof is complete. h In [20, Remark 4], every closed downward set M of the ordered space Rp ; 6Rpþ was characterized as the zero sublevel set of the following Minkowski gauge n1M : Rp ! R:
nM 1 ðyÞ ¼ infft 2 R : y t1 2 Mg;
8 y 2 Rp ;
where 1 ¼ ð1; 1; . . . ; 1Þ 2 Rp . On the other hand, in [26, Theorem 4.2] every downward set M of Rp ; 6Rpþ was characterized as a ‘‘plus-radiant set’’ (i.e., p y t1 2 M, for all y 2 M, t > 0) such that nM 1 is 6Rþ -increasing. Ob-
serve that M is plus-radiant if and only if 1 2 M1. Following both approaches we characterize every 6K-i.s. E Y via the zero sublevel set of a 6K-decreasing functional u such that u(0) > 0. The following lemma is necessary for this objective. Lemma 3.2. Let E Y be a closed 6K-i.s. Then for each q 2 E1n{0} the Minkowski gauge nEq is lower semicontinuous and satisfies the following properties: (a) nEq ð0Þ > 0, n o (b) y 2 Y : nEq ðyÞ 6 a ¼ aq þ E, for all a 2 R,
307
Remark 3.3. Part (a) ensures nEq ð0Þ > 0 but it may be nEq ð0Þ ¼ þ1. One has nEq ð0Þ < þ1 if and only if Rq \ E – ;. Moreover, under the assumptions of Lemma 3.2 it follows that
n o y 2 Y : nEq ðyÞ < 0 ¼ Rþþ q þ E:
ð3Þ
Indeed, let y 2 Y be such that nEq ðyÞ < 0. If nEq ðyÞ ¼ 1, then y + tq 2 E for all t 2 Rþþ , and so y 2 Rþþ q þ E. In other case, there exists t < 0 such that y + tq 2 E and y 2 Rþþ q þ E too. Reciprocally, by the definition of nEq it is obvious that nEq ðyÞ < 0 when y 2 Rþþ q þ E, which finishes the proof. Theorem 3.4. Let E Y be a nonempty set. Then E is a closed 6Ki.s. if and only if there exists a lower semicontinuous functional u : Y ! R such that (a) u(0) > 0, (b) E = {y 2 Y:u(y) 6 0}, (c) u is 6K-decreasing.
Proof. If E is a closed 6K-i.s., as K is a proper cone, there exists q 2 Kn{0}. As E is a 6K-i.s., K E1, and so q 2 E1n{0}. By Lemma 3.2, the functional u :¼ nEq is lower semicontinuous and satisfies the desirable properties (a)–(c). Reciprocally, assume that there exists a lower semicontinuous functional u : Y ! R satisfying properties (a)–(c). Then it is clear that E is closed since E is a sublevel set of u. Given y 2 E and z 2 K, by (b) and (c) we have that u(y + z) 6 u(y) 6 0 and by (b) we obtain y + z 2 E. Thus E + K = E. Moreover, by (a) and (b) it is clear that 0 R E and so E is 6K-i.s. h Remark 3.5. Every closed 6K-i.s. is the zero sublevel set of a lower semicontinuous 6K-decreasing functional u such that u(0) > 0, but this functional is not unique. For example, if Y ¼ R, K ¼ Rþ , then the 6K-i.s. E = [1, + 1) is the zero sublevel set of the following decreasing functions: f1(y) = 1 y, f2(y) = e1y 1, f3(y) = 1 y3 (all of them are continuous functions satisfying fi(0) > 0) and f4(y) = 1 E(y), where E(y) is the integer part of y, which is discontinuous and lower semicontinuous. However, only the functions R 3 y ! ay þ a with a > 0 are of the form n½1;þ1Þ for some q q 2 ½1; þ1Þ1 n f0g ¼ Rþþ . Next we characterize open 6K-i.s. by following the same approach as Theorem 3.4. Lemma 3.6. Let E Y be a nonempty set. If E is a solid 6K-i.s., then for each q 2 Yn{0} such that
cl E þ Rþþ q int E; the Minkowski gauge properties:
ð4Þ E ncl q
is continuous and satisfies the following
E (a) ncl q ð0Þ P 0, E (b) fy 2 Y : ncl q ðyÞ < ag ¼ aq þ int E, for all a 2 R, cl E (c) nq is 6K-decreasing.
(c) nEq is 6K-decreasing.
Proof. Suppose that E is a closed 6K-i.s. Then for each q 2 E1n{0} one has E þ Rþ q E. By applying [10, Theorem 2.3.1] to the closed set D = E and k0 = q we deduce that the functional nEq is lower semicontinuous and satisfies properties (b) and (c). On the other hand, as 0 R E, by part (b) (with a = 0) we see that nEq ð0Þ > 0, i.e., part (a) holds too. h
Proof. Suppose that E is a solid 6K-i.s. and take q 2 Yn{0} satisfying (4). By applying [10, Theorem 2.3.1 and Remark 2.3.3] to D = cl E E and k0 = q we deduce that the Minkowski gauge ncl q is continuous and satisfies properties (b) and (c). As 0 R E, by part (b) we see that E ncl h q ð0Þ P 0 and the proof is complete. E Under condition (4), observe that ncl takes the value 1 on q each line parallel to q contained in cl E:
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E ncl q ðzÞ ¼ 1;
8 z 2 y þ Rq;
8 y 2 Y such that y þ Rq cl E:
In connection to condition (4) we make two remarks. Remark 3.7. According to [10, Remark 2.3.3], if (4) is satisfied, then
int cl E ¼ int E:
ð5Þ
Let us prove the following reciprocal statement: if E is free disposal, K is solid and (5) holds, then each q 2 int K satisfies (4). Indeed, by Proposition 2.4, cl E + K cl E, and as cl E + int K is an open set contained in cl E, we have that
cl E þ int K int cl E ¼ int E:
ð6Þ
Note that every solid convex set E satisfies (5) and so statement (6) is true for all convex free disposal set E when K is solid. Property (4) is essential in Lemma 3.6, as it is illustrated in the following example. Example 3.8. Let Y ¼ R2 and E ¼ R ð2; 4Þ [ ð1; þ1Þ ð0; 2Þ (observe that assumption (4) is not satisfied for any q 2 R2 ). Consider q = (1, 0) and K ¼ conefqg ¼ fðy1 ; 0Þ 2 R2 : y1 P 0g. Then E is an open 6K-i.s. and
cl E þ Rþþ q cl E: By Lemma 3.2 we see that the Minkowski gauge nclq E is lower semicontinuous and cl E ¼ fy 2 R2 : nclq E ðyÞ 6 0g. However, by (3) it follows that
n o E y : ncl q ðyÞ < 0 ¼ R ½2; 4 [ ð1; þ1Þ ½0; 2
Theorem 3.9. Consider a solid set E Y that satisfies condition (4) for some q 2 Yn{0}. Then E is an open 6K-i.s. if and only if there exists a continuous functional u : Y ! R such that (a) u(0) P 0, (b) E = {y 2 Y:u(y) < 0}, (c) u is 6K-decreasing. Proof. The necessary condition follows from Lemma 3.6 choosing u ¼ nclq E . The proof of the sufficient condition is analogous to the proof of the sufficient condition in Theorem 3.4. h Let us observe in the previous theorem that assumption (4) is not used to prove the sufficient condition. In other words, if u is continuous, statements (a)–(c) in Theorem 3.9 imply that the nonempty set E is an open 6 K-i.s. 4. E-optimality and nonlinear scalarization Consider the following vector optimization problem:
ð7Þ
where f:X ? Y, X is an arbitrary decision space and the feasible set S X is nonempty. We say that (7) is a Pareto problem if Y ¼ Rp and K ¼ Rpþ . From now on we assume that E Y is 6K-i.s. By translating the E-optimality notion by Chicco et al. (see [4, Definition 3.1]) to problem (7), the following optimality notion is defined. Definition 4.1. A point x0 2 S is said to be an E-optimal (resp. weak E-optimal) solution of problem (7), denoted x0 2 Op(f, S; E) (resp. x0 2 WOp(f, S; E)), if
ðf ðSÞ f ðx0 ÞÞ \ ðEÞ ¼ ; ðresp: ðf ðSÞ f ðx0 ÞÞ \ ðint EÞ ¼ ;Þ:
Remark 4.2. (i) Definition 4.1 collapses the notions of ideal, nondominated, efficient (minimal), weak efficient and Henig proper efficient solution. In the following we enumerate the 6K-i.s. from which one can obtain these notions. (a) Y n ðKÞ 2 IK and Op(f, S; Yn(K)) is the set of ideal solutions of problem (7), since
ðf ðSÞ f ðx0 ÞÞ \ ðY n ðKÞÞ ¼ ; () f ðSÞ f ðx0 Þ þ K: This case reduces to [4, Proposition 3.2(iii)] by considering X ¼ Y ¼ Rp ; K ¼ Rpþ and f(x) = x for all x 2 X (see Example 2.3(h)). (b) K n f0g 2 IK and Op(f, S; Kn{0}) is the set of nondominated solutions of problem (7). (c) K \ ðY n ðKÞÞ 2 IK and Op(f, S;K \ (Yn(K))) is the set of efficient solutions of problem (7) since
ðf ðSÞ f ðx0 ÞÞ \ ðK \ ðY n ðKÞÞÞ ¼ ;
and this set is different to E. The same results are obtained choosing q = (a, 0) with a > 0.
Minff ðxÞ : x 2 Sg;
It is clear that WOp(f, S; E) = Op(f, S; int E). Moreover, if int E = ;, then WOp(f, S; E) = S. Thus, in order to deal with non trivial weak E-optimal solution sets, we assume that E is solid whenever this kind of solutions are considered. By taking different 6K-improvement sets E, the E-optimal solution notion reduces to well-known exact and approximate solution concepts of problem (7), as it is showed in the following remark. Next, the class of 6K-i.s. is denoted by IK .
() ðf ðx0 Þ KÞ \ f ðSÞ f ðx0 Þ þ K: R (d) If K is solid, then int K 2 IK and Op(f, S; K) is the set of weak efficient solutions of problem (7). We denote WMin(f, S) :¼ Op(f, S; int K). (e) If K is pointed and D Y is a proper solid convex cone such that Kn{0} int D, then D \ ðY n ðDÞÞ 2 IK and the elements of the set Op(f, S;D \ (Yn(D))) are Henig proper efficient solutions of problem (7). (ii) The main approximate solution concepts for problem (7) can also be seen as particular cases of Definition 4.1 by taking certain i.s. (see [13] for a similar approach based on coradiant sets instead of improvement sets). Next we show some examples. (a) If K is pointed and q 2 K, then q þ K n f0g 2 IK (see Example 2.3(a)) and Op(f, S; q + Kn{0}) is the set of approximate solutions of problem (7) in the Kutateladze’s sense with respect to q. If K is not pointed, a version of this approximate solution notion can be obtained via the 6K-i.s. E = q + K, where q 2 Kn(K). (b) Consider H Kn(K). Then H þ K 2 IK and Op(f, S; H + K) is the set of approximate solutions of problem (7) in the Németh’s sense with respect to H. If K is pointed and H K, then a version of this approximate solution concept can be obtained by considering the 6K-i.s. E = H + Kn{0} (see Example 2.3(a)). (c) If q 2 K, then K \ ðq KÞc 2 IK (see Example 2.3, parts (e) and (h)) and Op(f, S; K \ (q K)c) is the set of approximate solutions in the White’s sense with respect to q. (d) Let u : Y ! R be 6K-increasing on K such that u(0) = 0 and consider d P 0. Then K \ ½u > d 2 IK whenever is nonempty (see Example 2.3(c)) and Op(f, S;K \ [u > d]) is a new set of approximate solutions of problem (7) whose points satisfy the following property: x0 2 Op(f, S; K \ [u > d]) if and only if x0 2 S and
x 2 S;
f ðxÞ6K f ðx0 Þ ) uðf ðx0 Þ f ðxÞÞ 6 d:
C. Gutiérrez et al. / European Journal of Operational Research 223 (2012) 304–311
In particular, if (Y,p) is a normed space and the norm p is 6Kincreasing on K, then Op(f, S; K \ [p > d]) is the set of approximate solutions in the Tanaka’s sense. Analogously, if u = k 2 D+n{0}, then we obtain the set of approximate solutions in the Helbig’s sense with respect to k and by taking u = max{k1, . . . , kr}, where ki 2 K+n{0} for all i, we obtain the set of generalized Helbig approximate solutions introduced in [13, Section 4.3]. Next we obtain necessary and sufficient E-optimality conditions through scalarization processes. We start studying the nonconvex case (i.e., the set E and the objective function f of problem (7) can be nonconvex), since in this case, the E-optimality conditions are consequence of the characterizations of improvement sets obtained in the previous section through sublevel sets. Given a scalar function g : X ! R, the set of suboptimal (resp. sharp suboptimal) solutions with error e P 0 of the following optimization problem
MinfgðxÞ : x 2 Sg <
is denoted by argminS(g,e) (resp. argminS ðg; eÞ), i.e.,
argminS ðg; eÞ ¼ fx0 2 S : gðx0 Þ e 6 gðxÞ; 8 x 2 Sg <
ðresp: argminS ðg; eÞ ¼ fx0 2 S : gðx0 Þ e < gðxÞ; 8 x 2 S n fx0 ggÞ: We denote argminSg :¼ argminS(g, 0), i.e., the set of exact minima of g on S. Remark 4.3. The following properties are clear:
<
(a) If 0 6 e1 6 e2, then argminS ðg; e1 Þ argminS ðg; e1 Þ argminS ðg; e2 Þ. < (b) If 0 6 e1 < e2, then argminS ðg; e1 Þ argminS ðg; e2 Þ. For each y 2 Y, fy:X ? Y denotes the function fy(x) = y f(x), for all x 2 X. In the next theorem we obtain general necessary and sufficient E-optimality conditions. Theorem 4.4. Let E Y be a 6K-i.s. and u : Y ! R such that u(0) < +1.
(a) Suppose that
E ¼ fy 2 Y : uðyÞ 6 0g:
ð8Þ
Then, the following statements are true. < (i) x0 2 Op(f, S; E) if and only if x0 2 argminS ðu ff ðx0 Þ ; uð0ÞÞ. (ii) If 0 6 e < u(0) and x0 2 argminS ðu ff ðx0 Þ ; eÞ, then x0 2 Op(f, S; E). (b) Consider that
E ¼ fy 2 Y : uðyÞ < 0g:
ð9Þ
Then, x0 2 Op(f, S; E) if and only if x0 2 argminS ðu ff ðx0 Þ ; uð0ÞÞ.
Proof. Let us prove part (a), since the proof of part (b) is similar to the proof of part (a)–(i). (a)–(i) Let x0 2 X. By (8) it is obvious that
ðf ðSÞ f ðx0 ÞÞ \ ðEÞ ¼ ; () uðf ðx0 Þ f ðxÞÞ > 0 8 x 2 S n fx0 g () ðu ff ðx0 Þ ÞðxÞ > ðu ff ðx0 Þ Þðx0 Þ uð0Þ
8 x 2 S n fx0 g and the result follows.
309
(a)–(ii) Let x0 2 argminS ðu ff ðx0 Þ ; eÞ and suppose that 0 6 e < u(0). By Remark 4.3(b) we deduce that x0 2 argmin< S ðu ff ðx0 Þ ; uð0ÞÞ and by part (a) we see that x0 2 Op(f, S; E). h Let us observe that if E is a 6K-i.s. and statement (8) (resp. (9)) is true, then uð0Þ 2 Rþþ [ fþ1g (resp. uð0Þ 2 Rþ [ fþ1g). Moreover, in order to use Theorem 4.4, let us recall that every closed 6 K-i.s., and every open 6K-i.s. satisfying (4) fulfills statements (8) and (9), respectively (see Theorems 3.4 and 3.9). Moreover, statements (8) and (9) can be satisfied by different functionals u (see Remark 3.5). For applying Theorem 4.4, let us introduce the following cone: KE :¼ E1 \ coneE. It is clear that 0 2 KE and KE – {0} if and only if E1 \ E – ;, since E1 is a cone and 0 R E. Next we suppose that E is closed or solid, and KE – {0}. Let us observe that an improvement set could not satisfy any of these assumptions. For example, if Y ¼ R2 and K ¼ fðy1 ; 0Þ 2 R2 : y1 P 0g, the 6K-i.s. E ¼ fðy1 ; 1Þ 2 R2 : y1 > 0g is neither solid nor closed and KE = {(0, 0)}. Now, we provide necessary and sufficient conditions for E-optimality by considering first a closed set E (Theorem 4.5) and second E a solid set E (Theorem 4.6), via nEq and ncl respectively. q Theorem 4.5. Let E Y be a closed 6K-i.s. such that KE – {0} and consider the following statements: (a) x0 2 Op(f, S; E). < (b) x0 2 argminS nEq ff ðx0 Þ ; nEq ð0Þ , for all q 2 KEn{0}. < (c) x0 2 argminS nEq ff ðx0 Þ ; nEq ð0Þ , for some q 2 KEn{0}. (d) 0 6 e < nEq ð0Þ and x0 2 argminS nEq ff ðx0 Þ ; e , for q 2 KEn{0}.
some
Then (a), (b) and (c) are equivalent and (d) implies (a). Proof. Let q 2 KEn{0}. By Lemma 3.2 we see that
fy 2 Y : nEq ðyÞ 6 0g ¼ E and by Remark 3.3 we have that nEq ð0Þ < þ1, since q 2 KE coneE. Then, the result follows by applying Theorem 4.4(a) to u ¼ nEq . h In [6, Theorem 2.5] the authors characterized via scalarization H and the Minkowski gauge n q the nondominated solutions of problem (7) with respect to an ordering set H that satisfies statement (2) and such that 0 2 bd H. Then, let us observe that [6, Theorem 2.5] cannot be applied if H is not solid and it cannot either if H is a closed improvement set. Thus, for the class of closed improvement sets, Theorem 4.5 is more general than [6, Theorem 2.5], since it can be applied for solid and non-solid closed improvement sets. For example, if Y ¼ R2 and K ¼ Rþ fð0;1Þg, then E1 ={(0,1)} + K is a non-solid closed 6K-i.s. and E2 ¼ E1 [ fð0; 2Þg þ R2þ is a solid closed 6K-i.s. such that ð0; 1Þ 2 K Ei , i = 1,2, and so Theorem 4.5 can be applied, but Theorem 2.5 of [6] cannot. Moreover, if one translates E2 in order to satisfy the condition 0 2 bd H, statement (2) is not satisfied either for any q 2 Yn{0}. Let us denote
F E ¼ fq 2 Y n f0g : cl E þ Rþþ q int Eg \ coneðcl EÞ:
Theorem 4.6. Let E Y be a solid 6K-i.s. and suppose that FE – ;. The following statements are equivalent: (a) x0 2 WOp(f, S;E). E cl E (b) x0 2 argminS ncl q ff ðx0 Þ ; nq ð0Þ , for all q 2 FE. E cl E (a) x0 2 argminS ncl q ff ðx0 Þ ; nq ð0Þ , for some q 2 FE.
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Theorem 5.3. Let E be a 6K-i.s. and k 2 E+n{0}. Then
Proof. Let q 2 FE. By Lemma 3.6 we see that
n o E y 2 Y : ncl q ðyÞ < 0 ¼ int E
E and by Remark 3.3 we have that ncl since q ð0Þ < þ1, q 2 FE cone(cl E). Then the result follows by applying TheoE rem 4.4(b) to u ¼ ncl h q .
A particular case of Theorem 4.6 is [11, Theorem 5.2], which can be obtained by choosing q 2 int K and E = eq + Kn{0}, since one has E q 2 FE (as q 2 int K), ncl q ð0Þ ¼ e and WOp(f, S; E) is the set of weak approximate solutions in the Kutateladze’s sense according to Remark 4.2(ii)(a). Similarly, [11, Theorems 4.5 and 5.1(a)] are deduced from Theorem 4.6, and [11, Theorem 5.1(b)] is obtained from Theorem 4.5 (part (d) ) (a)). Let us observe that [6, Theorem 2.5] cannot be applied for an open ordering set H such that 0 R bd H. However, if this ordering set is an 6K-i.s., then Theorem 4.6 can be applied whenever FH – ;. On the other hand, if E is a solid 6K-i.s. such that 0 2 bd E, then FE – ; if and only if statement (2) is satisfied, since
fq 2 Y n f0g : cl E þ Rþþ q int Eg int E and so the assumptions of [6, Theorem 2.5] and Theorem 4.6 are the same.
(a) argminS(kf, rE(k)) WOp(f, S; E), < (b) argminS ðk f ; rE ðkÞÞ Opðf ; S; EÞ.
Proof. (a) Choose x0 2 argminS(kf, rE(k)), then
kðf ðxÞÞ P kðf ðx0 ÞÞ rE ðkÞ 8 x 2 S:
Suppose that x0 R WOp(f, S; E). Then there exist ^ x 2 S and ^e 2 int E such that f ð^xÞ f ðx0 Þ ¼ ^e. By applying (13) to x ¼ ^x it results
kð^eÞ ¼ kðf ð^xÞ f ðx0 ÞÞ P rE ðkÞ; i.e., inf e2E kðeÞ P kð^eÞ, and so inf e2E kðeÞ ¼ kð^eÞ. As ^e 2 int E, this implies that k = 0, a contradiction. (b) Similarly, choose x0 2 argmin< S ðk f ; rE ðkÞÞ, then
kðf ðxÞÞ > kðf ðx0 ÞÞ rE ðkÞ 8 x 2 S n fx0 g:
Recall that a function f:X ? Y is closely K-convexlike on S if cl(f(S) + K) is a convex set (see [7]). For k 2 Y⁄, we denote
rE ðkÞ ¼ inf kðeÞ:
kð^eÞ ¼ kðf ð^xÞ f ðx0 ÞÞ > rE ðkÞ;
+
Let us observe that if k 2 E , then rE(k) P 0.
As a consequence of Theorems 5.1 and 5.3, we obtain the following characterization of weak E-optimal points under convexity assumptions.
WOpðf ; S; EÞ ¼ Theorem 5.1. Assume that f is closely K-convexlike on S, and E is a solid convex 6K-i.s. Then
[
argminS ðk f ; rE ðkÞÞ:
k2Eþ nf0g
ð10Þ
ð11Þ
Indeed, if statement (11) was false, then there exist x 2 S, d 2 K and e 2 int E such that f(x) + d f(x0) = e. So f(x) f(x0) = (e + d) 2 int E, since int E is a 6K-i.s. (see Proposition 2.4(b)), which contradicts (10). From (11), we derive that cl(f(S) + K f(x0)) \ (int E) = ;, since (f(S) + K f(x0)) (int E)c and this last set is closed. As cl(f(S) + K f(x0)) is a convex set since f is closely K-convexlike, by the separation theorem (see, for example, [17, Theorem 3.16]) there exist k 2 Y⁄n{0} and a 2 R such that
kðf ðxÞ þ d f ðx0 ÞÞ P a P kðeÞ 8 x 2 S; d 2 K; e 2 E:
argminS ðk f ; rE ðkÞÞ:
In particular, if K is solid and we choose E = int K, this corollary reduces to the following well-known result (see, for instance, [3, Theorem 5.2]):
[
argminS ðk f Þ;
ð15Þ
k2K þ nf0g
From here, it follows that
ðf ðSÞ þ K f ðx0 ÞÞ \ ðint EÞ ¼ ;:
[ k2Eþ nf0g
WMinðf ; SÞ ¼
Proof. Choose x0 2 WOp(f, S; E), then
ðf ðSÞ f ðx0 ÞÞ \ ðint EÞ ¼ ;:
h
Corollary 5.4. Assume that f is closely K-convexlike on S, and E is a solid convex 6K-i.s. Then
e2E
WOpðf ; S; EÞ
ð14Þ
Suppose that x0 R Op(f, S; E). Then there exist ^x 2 S n fx0 g and ^e 2 E such that f ð^xÞ f ðx0 Þ ¼ ^e. By applying (14) to x ¼ ^ x it follows that
i.e., inf e2E kðeÞ > kð^eÞ, which is a contradiction. 5. E-optimality and linear scalarization
ð13Þ
ð12Þ
Taking x = x0 and d = 0 it results 0 P k(e) for all e 2 E, and so k 2 E+n{0}. From (12) taking d = 0, it follows that k(e) P k(f(x0)) k(f(x)) for every e 2 E, x 2 S, and so rE(k) P k(f(x0)) k(f(x)) for every x 2 S. In consequence, k(f(x)) P k(f(x0)) rE(k) for every x 2 S, i.e., x0 2 argminS(kf, rE(k)). h Remark 5.2. If Y is normed, we can suppose kkk = 1 dividing (12) by kkk.
since E+ = K+ and rE(k) = 0 for all k 2 E+. Next we apply Theorem 5.3 and Corollary 5.4 to obtain a characterization of approximate solutions of K-convexlike vector optimization problems. Let C Yn{0} be a convex and solid coradiant set and let e > 0. Definition 5.5. [12, Definition 2.1] It is said that x0 2 S is a (C, e)efficient solution (resp. weak (C, e)-efficient solution) of problem (7), denoted x0 2 AE(f, S, C, e) (resp. x0 2 WAE(f, S, C, e)), if (f(S) f(x0)) \ (eC) = ; (resp. (f(S) f(x0)) \ (eint C) = ;). Let D Y be a solid convex cone. Recall that f is D-subconvexlike on S if f(S) + int D is a convex set. Theorem 5.6. Suppose that f is (cone C)-subconvexlike on S. Then we have that S (a) WAEðf ; S; C; eÞ ¼ k2C þ nf0g argminS ðk f ; erC ðkÞÞ. < (b) argminS ðk f ; erC ðkÞÞ AEðf ; S; C; eÞ. (c) If rC(k) > 0, then argminS(kf,drC(k)) AE(f, S, C, e), for all d 2 [0, e). Proof. Let us define E :¼ eC. As C is convex, then E is an i.s. with respect to the convex cone D :¼ cone C, since eC + cone C eC by [12, Lemma 3.1(v)]. In consequence, it is clear that
C. Gutiérrez et al. / European Journal of Operational Research 223 (2012) 304–311
AEðf ; S; C; eÞ ¼ Opðf ; S; EÞ and WAEðf ; S; C; eÞ ¼ WOpðf ; S; EÞ:
ð16Þ
It is known that, for a solid convex cone D, the sets of D-subconvexlike and closely D-convexlike functions coincide (see, for instance, [7]). Moreover, it is clear that
Dþ ¼ ðcone CÞþ ¼ C þ ¼ Eþ :
ð17Þ
Then the result follows from (16), Corollary 5.4, (17), Theorem 5.3(b) and Remark 4.3(b). Note that reC(k) = erC(k) for all k 2 Y⁄, e > 0. h As a direct consequence of Theorem 5.6 one can deduce [11, Proposition 4.10 and Theorem 5.6] by choosing C = q + D, since C+ = D+, rC(k) = k(q) for all k 2 C+ and f(S) + int D is a convex set whenever f(S) is convex. On the other hand, Theorem 5.6 improves [12, Theorems 5.2 and 5.4] and extends them to a (non-necessarily Pareto) vector optimization problem. The improvement is based on the errors in the necessary and sufficient conditions. Indeed, let us suppose that Y is normed and denote
dYnC þ ðkÞ ¼ inffkn kk : n 2 Y n C þ g
dC ¼ inffkdk : d 2 Cg;
and dðkÞ ¼ dC dYnC þ ðkÞ, where k 2 C+. In [12, Theorems 5.2 and 5.4], the following necessary and sufficient conditions for weak (C, e)efficient solutions of (cone C)-subconvexlike (Pareto) problems were stated (in (19) it is assumed that 0 R cl C):
WAEðf ; S; C; eÞ
[
argminS ðk f ; edC Þ;
ð18Þ
k2C þ ; kkk¼1
argminS ðk f ; dÞ WAEðf ; S; C; d=dðkÞÞ;
8 d P 0; 8k 2 intðC þð19Þ Þ:
By Remark 5.2, one can suppose that kkk = 1 in the statements of Theorem 5.6. Thus,
rC ðkÞ ¼ inf kðdÞ 6 kkk inf kdk ¼ dC d2C
d2C
and the precision erC(k) of Theorem 5.6(a) is better than the precision e dC of (18). Reciprocally, by [2, Lemma 2.7], we have that
dYnCþ ðkÞ 6 inffkðyÞ : y 2 cone C; kyk ¼ 1g: So, for each d 2 C, as d0 = (1/kdk)d 2 cone C and kd0 k = 1, it follows that 0
kðdÞ ¼ kdkkðd Þ P kdk dYnC þ ðkÞ P dC dYnC þ ðkÞ ¼ dðkÞ: Therefore, rC(k) P d(k). The sufficient condition of Theorem 5.6(a) can be rewritten as follows:
argminS ðk f ; dÞ WAEðf ; S; C; d=rC ðkÞÞ;
8 d > 0;
8 k 2 intðC þ Þ; kkk ¼ 1 and the precision d/rC(k) is better than the precision d/d(k) of (19). 6. Conclusions In this work we have studied nondominated points of a set and nondominated solutions of vector optimization problems where the ordering relation is given by an improvement set in the framework of linear spaces. This kind of minimality has been shown to be very suitable to deal in a unified way with well-known exact and approximate nondominated concepts. Moreover, it works with very general ordering sets (non-necessarily cones neither pointed, convex, solid or closed sets) and so it can be used to deal with problems where the ordering relation is not a quasi order, which frequently appear in Economics, and in linear spaces where the ordering set is not solid, such as the spaces ‘p and Lp, 1 6 p < 1, and their natural ordering cones. On the other hand, through this minimality notion it is possible to improve some well-known results on particular exact and
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approximate nondominated concepts, since only essential assumptions are required to deal with it. For example, in Corollary 5.4 we have obtained that the well-known characterization of weak efficient solution in subconvexlike vector optimization problems via solutions of linear scalarizations (see statement (15)) really works for any nondominated notion defined via a solid convex improvement set. Acknowledgments The authors are grateful to the anonymous referees for their helpful comments and suggestions. References [1] T.Q. Bao, B.S. Mordukhovich, Extended Pareto optimality in multiobjective problems, in: Q.H. Ansari, J.-C. Yao (Eds.), Recent Developments in Vector Optimization, vol. 1, Springer-Verlag, Berlin, 2012, pp. 467–515. [2] S. Bolintinéanu, Vector variational principles; e-efficiency and scalar stationarity, Journal of Convex Analysis 8 (1) (2001) 71–85. [3] W.W. Breckner, G. Kassay, A systematization of convexity concepts for sets and functions, Journal of Convex Analysis 4 (1) (1997) 109–127. [4] M. Chicco, F. Mignanego, L. Pusillo, S. Tijs, Vector optimization problems via improvement sets, Journal of Optimization Theory and Applications 150 (3) (2011) 516–529. [5] G. Debreu, Theory of Value, John Wiley, New York, 1959. [6] F. Flores-Bazán, E. Hernández, Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. Journal of Global Optimization. http://dx.doi.org/10.1007/s10898-011-9822-y. [7] J.B.G. Frenk, G. Kassay, On classes of generalized convex functions, Gordan– Farkas type theorems, and Lagrangian duality, Journal of Optimization Theory and Applications 102 (2) (1999) 315–343. [8] C. Gerth, P. Weidner, Nonconvex separation theorems and some applications in vector optimization, Journal of Optimization Theory and Applications 67 (2) (1990) 297–320. [9] G. Giorgi, A. Guerraggio, J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier, Amsterdam, 2004. [10] A. Göpfert, H. Riahi, C. Tammer, C. Za˘linescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003. [11] C. Gutiérrez, B. Jiménez, V. Novo, On approximate solutions in vector optimization problems via scalarization, Computational Optimization and Applications 35 (3) (2006) 305–324. [12] C. Gutiérrez, B. Jiménez, V. Novo, On approximate efficiency in multiobjective programming, Mathematical Methods of Operations Research 64 (1) (2006) 165–185. [13] C. Gutiérrez, B. Jiménez, V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM Journal on Optimization 17 (3) (2006) 688–710. [14] C. Gutiérrez, B. Jiménez, V. Novo, Optimality conditions via scalarization for a new e-efficiency concept in vector optimization problems, European Journal of Operational Research 201 (1) (2010) 11–22. [15] C. Gutiérrez, B. Jiménez, V. Novo, A generic approach to approximate efficiency and applications to vector optimization with set-valued maps, Journal of Global Optimization 49 (2) (2011) 313–342. [16] T.X.D. Ha, Optimality conditions for several types of efficient solutions of setvalued optimization problems, in: P.M. Pardalos, T.M. Rassias, A.A. Khan (Eds.), Nonlinear Analysis and Variational Problems, Springer Optimization and Its Applications, vol. 35, Springer, New York, 2010, pp. 305–324. [17] J. Jahn, Vector Optimization: Theory, Applications, and Extensions, SpringerVerlag, Berlin, 2004. [18] A.Y. Kruger, B.S. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization problems, Doklady Akademii Nauk BSSR 24 (8) (1980) 684–687. [19] V.L. Makarov, M.J. Levin, A.M. Rubinov, Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms, North-Holland, Amsterdam, 1995. [20] J.-E. Martı´nez Legaz, A.M. Rubinov, I. Singer, Downward sets and their separation and approximation properties, Optimization 23 (2) (2002) 111–137. [21] E. Miglierina, E. Molho, Sectionwise connected sets in vector optimization, Operations Research Letters 37 (4) (2009) 295–298. [22] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin, 2006. [23] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. [24] A.M. Rubinov, R.N. Gasimov, Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation, Journal of Global Optimization 29 (4) (2004) 455–477. [25] A.M. Rubinov, I. Singer, Best approximation by normal and conormal sets, Journal of Approximation Theory 107 (2) (2000) 212–243. [26] A.M. Rubinov, I. Singer, Topical and sub-topical functions, downward sets and abstract convexity, Optimization 50 (5–6) (2001) 307–351. [27] N.V. Tuyen, N.D. Yen, On the concept of generalized order optimality, Nonlinear Analysis 75 (3) (2012) 1592–1601.
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Continuous Optimization
Improvement sets and vector optimization q C. Gutiérrez a,⇑, B. Jiménez b, V. Novo b a b
Departamento de Matemática Aplicada, E.T.S. de Ingenieros de Telecomunicación, Universidad de Valladolid, Paseo de Belén 15, Campus Miguel Delibes, 47011 Valladolid, Spain Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 15 December 2011 Accepted 31 May 2012 Available online 13 June 2012 Keywords: Improvement set Minimal point Vector optimization e-Efficiency Scalarization
a b s t r a c t In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize wellknown results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems. 2012 Elsevier B.V. All rights reserved.
1. Introduction A current research line related with vector optimization problems is to develop concepts and settings in order to unify the more important solution notions of these problems, like the so-called efficiency, proper efficiency, weak efficiency, strict efficiency, strong efficiency, and e-efficiency (see [1,4,6,12–15,22]). Probably, the first attempt to unify classical solution concepts in vector optimization problems via a more general solution notion that collapses all of them was due to Mordukhovich (see [22]) through the notion of generalized order optimality of vector optimization problems. This notion works when the objective space is normed, it was motivated by the well-known set extremality concept due to Kruger and Mordukhovich (see [18]), extends the concepts of Slater minimizer, efficient solution and weak efficient solution (see [27]) and it has been used to obtain necessary optimality conditions of vector optimization problems (see [22]). Recently, in the framework of a normed space or a Hausdorff locally convex topological linear space Y, several authors have introduced and studied concepts of nondominated point of a set based on binary relations defined as usual, but from a general ordering set H Y:
y1 ; y2 2 Y;
y1 6H y2 () y2 y1 2 H:
ð1Þ
q This research was partially supported by the Ministerio de Ciencia e Innovación (Spain) under project MTM2009-09493. ⇑ Corresponding author. E-mail addresses: [email protected] (C. Gutiérrez), [email protected] (B. Jiménez), [email protected] (V. Novo).
0377-2217/$ - see front matter 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.05.050
This ordering set could not be convex neither a cone and so the relation 6H could not be transitive. Analogously, it could happen that 0 R H and then 6H could not be reflexive. Both facts are interesting from a practical point of view, since preference relations that are not necessarily a preorder are usual in Economics (see [19,24] and the references therein). Sometimes, the closedness of the ordering set is not assumed either, and so certain important preference relations like the lexicographical order can be dealt with. Moreover, some ordering sets could not be solid, i.e., the topological interior of H could be empty, in order to deal with objective spaces as ‘p and Lp, 1 6 p < 1, where the natural ordering is given by a non-solid convex cone. To compensate these generalizations, other mild conditions are assumed on the ordering set. For example, in [16,24] the authors considered nondominated point notions via conic ordering sets, and in [12–15] we introduced and studied a concept of nondominated point based on a coradiant ordering set (aH H, for all a > 1), that generalizes the more important exact and approximate concepts of nondominated point in linear spaces. To our knowledge, the more general approaches in this line are due to Bao and Mordukhovich [1] and Flores–Bazán and Hernández [6], since they consider the usual notion of nondominated point without imposing any condition on the ordering set H. These notions reduce to the concepts of ideal, efficient, weak efficient, relative efficient, quasi relative efficient and Henig proper efficient point among others. However, in order to obtain optimality conditions in vector optimization problems, some conditions on the ordering set are assumed. To be exact, Flores–Bazán and Hernández suppose that zero is in the boundary of H and there exists a point q 2 Yn{0} such
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that (cl H and int H denote the closure and the interior of H, respectively)
cl H þ ð0; 1Þq int H;
ð2Þ
which implies that the ordering set H is solid (see [6]), and Bao and Mordukhovich assume the so-called local closedness property (see [1, Definition 3.3]), which is compatible with non-solid ordering sets. A similar contribution has recently been introduced by Chicco et al. in [4]. Specifically, by considering the finite dimensional setting Y ¼ Rp and via an improvement set E (i.e., 0 R E and E þ Rpþ ¼ E, where Rpþ is the non-negative orthant), these authors have defined the notion of E-optimal point (nondominated with respect to the binary relation (1) given by H = E), that collapses the concepts of Pareto minimality and weak Pareto minimality, and that is useful to deal with approximate minimal points too. This paper focus on this notion. Essentially, in a non-necessarily finite dimensional ordered linear space, an improvement set E is a free disposal set, i.e., its cone expansion E + K is itself, where K is the ordering cone. This kind of sets was introduced by Debreu in [5] and they have frequently been used in mathematical economics and optimization. So, one can find in the literature several previous concepts very close to the notion of improvement set. For example, in the setting of ordered linear spaces where the ordering relation (1) is defined by a convex cone H, the concepts of downward set and upward set (see [20,26]) are the same as the notion of improvement set, but relaxing the condition 0 R E. To be precise, an improvement set E is an upward set such that 0 R E. On the other hand, downward sets can be considered as a certain analog of normal subsets of the non-negative orthant (see [20,26]). Moreover, let us observe that in mathematical economics and game theory, the upward sets are also called free disposal sets and the convex downward sets are known as comprehensive sets. This work has two objectives. The first one is to study the notion of improvement set by Chicco et al. [4] in the setting of an arbitrary quasi ordered linear space. In particular, we are interested in characterizing these sets in terms of sublevel sets of scalar functions. The second one is to state necessary and sufficient conditions for E-optimal solutions of vector optimization problems via scalarization. In the whole paper it is implicit that a lot of ideas and techniques of the works [12–15] can be used to obtain results about E-optimal points and E-optimal solutions. This work is structured as follows. In Section 2, some notations are fixed, several well-known concepts and mathematical tools are recalled and the notion of improvement set by Chicco et al. [4] is extended to a linear space ordered via a convex cone as usual. This notion is showed to be very close to the concepts of free disposal set, downward set, upward set and coradiant set and several general classes of improvement sets are introduced. Moreover, the relation between an improvement set and the ordering cone K is clarified. In Section 3, the improvement sets are characterized. First, by the separation theorem, closed and convex improvement sets are showed to be such that the positive polar cone of their recession cone is included into the positive polar cone of the order cone. Second, nonconvex closed (resp. open) improvement sets are characterized through sublevel sets of decreasing lower semicontinuous (resp. continuous) functionals. The main mathematical tool for proving these last results is the well-known nonconvex separation functional by Tammer and Weidner [8,10]. In Section 4, the notion of nondominated solution of a vector optimization problem with respect to an ordering set given by an improvement set is defined. It is showed that this notion collapses the more popular exact and approximate solution concepts of a
vector optimization problem. Moreover, it is characterized through approximate solutions of associated scalar optimization problems without considering any convexity assumption. These scalarization results are obtained in two steps. First, by considering generic scalar functionals whose sublevel set at zero is the improvement set. Second, by applying these generic characterizations to some specific scalarization functionals. In Section 5, other characterizations for this kind of nondominated solutions via linear scalarization processes are obtained in K-convexlike vector optimization problems, i.e., in problems where the objective function is closely K-convexlike and the improvement set is convex. As a consequence, some necessary and sufficient approximate optimality conditions via linear scalarizations obtained by ourselves in [12] for K-convexlike Pareto optimization problems are improved and extended to vector optimization problems. Finally, in Section 6, some conclusions are presented which summarize this work. 2. Notations and preliminaries Let Y be a Hausdorff locally convex topological linear space and K Y be a convex cone (we consider that 0 2 K), which is assumed to be proper, i.e., {0} – K – Y. In the sequel, we suppose that Y is ordered through the following quasi order:
y1 ; y2 2 Y;
y1 6K y2 () y2 y1 2 K:
Let R :¼ R [ f1g. Given a functional u : Y ! R, we denote
epi u ¼ fðy; rÞ 2 Y R : uðyÞ 6 rg: Moreover, u is said to be 6K-increasing (resp. 6K-decreasing) on a nonempty set M Y if u(y) 6 u(y + d) (resp. u(y) P u(y + d)), for all y 2 M, d 2 K. We say that u is 6K-increasing (resp. 6K-decreasing) if u is 6K-increasing (resp. 6K-decreasing) on Y. Let ; – M Y be closed and q 2 Yn{0}. In the paper, the following Minkowski gauge nM q : Y ! R will be used:
nM q ðyÞ ¼
þ1
if y R Rq þ M;
infft 2 R : y þ tq 2 Mg otherwise:
We denote by cl M, int M, bd M, Mc, conv M and cone M the closure, the topological interior, the boundary, the complement, the convex hull and the cone generated by a set M Y, respectively, and we say that M is solid if int M – ; and free disposal with respect to K (free disposal for short) if M + K = M (see [21] and the references therein). As usual, the topological dual space of Y is denoted by Y⁄. The positive and strict positive polar cones to M are denoted by M+ and M+s, respectively, i.e.,
M þ ¼ fk 2 Y : kðyÞ P 0; 8 y 2 Mg; M þs ¼ fk 2 Y : kðyÞ > 0; 8 y 2 M; y – 0g: We write Rpþ to refer the non-negative orthant of Rp , Rþ :¼ R1þ and Rþþ ¼ Rþ n f0g. Definition 2.1. A nonempty set E Y is said to be an improvement set for 6K (or an improvement set with respect to K) if 0 R E and E is free disposal. For convenience we will say that E is a 6K-i.s. or simply E is an i.s. if there is no confusion. Remark 2.2. Definition 2.1 extends [4, Definition 2.5] to ordered linear spaces. To be precise, the notion of 6K-improvement set reduces to [4, Definition 2.5] if we consider Y ¼ Rp and K ¼ Rpþ . However, there exist in the literature several previous concepts very close to this notion by Chicco et al., as it is showed in the following examples.
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(a) Let (V, 6) be an ordered (non-necessarily linear) space and ; – E V. Let us recall that E is a downward set (resp. upward set, see for instance [20,26]) if
v 2 V;
y 2 E;
v6y)v 2E
ðresp: y 6 v ) v 2 EÞ:
In the setting of the ordered linear space (Y, 6K), it is clear that a nonempty set E Y is upward if and only if E + K = E (i.e., if E is free disposal), and it is downward if and only if E K = E. So, the following statements are equivalent: (i) E is a 6K-i.s. (ii) 0 R E and E is an upward set. (iii) 0 R E and E is a downward set with respect to the ordering 6K. (b) Consider Y ¼ Rp ; K ¼ Rpþ and a nonempty set E Rpþ . Let us recall that E is normal (see, for instance, [25]) if
v 2 Rpþ ;
y 2 E;
v 6R
p þ
y ) v 2 E:
Clearly, E is normal if and only if ðE Rpþ Þ \ Rpþ ¼ E. Thus, if E Rpþ is a 6Rpþ -improvement set, then E is normal. (c) Recall that a nonempty set E Y is coradiant (see, for instance, [13]) if aE E, for all a > 1. Then, any i.s. E K is coradiant. However, the reciprocal statement is false. For example, if dim Y > 1, K is solid, q 2 Kn{0} and c > 0, then the set E = {aq:a P c} Kn{0} is coradiant, but it is not free disposal. If 0 R E, E å K, then the statements ‘‘E is 6K-i.s.’’ and ‘‘E is coradiant’’ have not any relation (see [4, Remark 2.3]). (d) Given a nonempty set H Y, the well-known conic extension of H with respect to K is the set EK ðHÞ :¼ H þ K. Then EK ðHÞ is an i.s. if and only if H \ (K) = ;. By the previous remark, the next example and Proposition 2.4, we see that the class of 6K-i.s. is very wide.
(f) If E is i.s., then y + E is i.s. for all y 2 Y, whenever y R E. If ; – H Y and H \ (E) = ;, then H + E is an i.s. (observe that S H+E= y2Hy + E). In particular, if E1,E2 are i.s. and E1 \ (E2) = ;, then E1 + E2 is an i.s. (g) If E is i.s. and a > 0, then aE is i.s.; if a < 0, then aE is i.s. for 6K. (h) If E is free disposal and 0 2 E, then (E)c is 6K-i.s., as can be easily checked. In particular, (q K)c is an 6K-i.s. for all q 2 K. Let us observe that (K)c is the biggest 6K-i.s. Let us observe that parts (e) and (h) of Example 2.3 reduce to [4, Proposition 3.1] by considering Y ¼ Rp and K ¼ Rpþ . Proposition 2.4. Suppose that E Y is 6K-i.s. Then (a) cl E is free disposal. (b) If E is solid, then int E is 6K-i.s. Proof. (a) We need to check that cl E + K cl E. Let y 2 cl E, z 2 K and take a net (yi) E such that yi ? y. Then
y þ z ¼ limðyi þ zÞ 2 clðE þ KÞ ¼ cl E i
and so cl E is free disposal. (b) It is enough to prove that int E + K int E, which is clear since int E + K is an open set contained in E + K = E. h The following two propositions clarify the relation between the sets E and K when E is 6K-i.s. Proposition 2.5. Let E Y be a nonempty free disposal closed set. Then the following statements are equivalent: (a) 0 R E. (b) (Kn{0}) \ Ec – ;.
Example 2.3. (a) Let us suppose that K is pointed, i.e., K \ (K) = {0}. If ; – H Y and H \ (Kn{0}) = ;, then H + Kn{0} is an i.s. In particular, Kn{0} is an i.s. (b) If K is solid, then int K is an i.s. (c) If u : Y ! R is 6K-increasing, u(0) = 0, d P 0 and m > 0, then [u > d] :¼ {y 2 Y:u(y) > d}, [u P m] :¼ {y 2 Y:u(y) P m} and [u P 0]n{0} are i.s. whenever they are nonempty. Observe that every k 2 K+n{0} and the Minkowski gauge nK (whenq ever K is closed, solid and q 2 int K), are 6K-increasing functionals that satisfy the condition u(0) = 0 (see, for instance, [10]). More generally, if E is free disposal, u : Y ! R is 6K-increasing on E, u(0) = 0, d P 0 and m > 0, then E \ [u > d], E \ [u P m] and E \ ([u P 0]n{0}) are i.s. whenever they are nonempty. Analogously, if E is free disposal and u : Y ! R is 6K-decreasing on E, then the sublevel set E \ [u 6 a] :¼ {y 2 E:u(y) 6 a} and the strict sublevel set E \ [u < a] :¼ {y 2 E: u(y) < a} are 6K-i.s. for each a 2 R whenever they are nonempty sets, 0 R E \ [u 6 a] and 0 R E \ [u < a]. (d) Consider V ¼ Y R ordered by the cone K Rþ . Then u : Y ! R [ fþ1g is 6K-decreasing if and only if epi u V is upward. Therefore, if u is 6K-decreasing and u(0) > 0, then epi u is 6KRþ -i.s. S (e) If (Ei)i2I is a family of i.s., then i2IEi is an i.s. If (Ei)i2I is a family of free disposal sets and 0 R Ei0 for some i0 2 I (i.e., T some set in the family is an i.s.), then i2IEi is an i.s. whenever it is nonempty.
Furthermore, if (b) holds, then aq R E for all q 2 (Kn{0}) \ Ec and for all a 2 (1,1]. Proof. (a) ) (b) If (Kn{0}) \ Ec = ;, then Kn{0} E and by using that K is a proper cone we deduce that 0 2 cl(Kn{0}) cl E = E, which is a contradiction. (b) ) (a) Let q 2 (Kn{0}) \ Ec. If 0 2 E, then q 2 K K + E = E, which is a contradiction. For the last part, consider q 2 (Kn{0}) \ Ec and suppose that there exists a 2 (1,1] such that aq 2 E. Then, q = aq + (1 a)q 2 E + K = E, which is a contradiction. h The second part of the following result is a direct consequence of the well-known separation property between a point and a closed convex set and so its proof is omitted. We denote by E the cone generated by the convex hull of E, i.e., E ¼ cone conv E.
Proposition 2.6. Suppose that E Y is 6K-i.s. Then (a) Eþ K þ and cl K cl E. (b) Additionally suppose that E is closed and convex. Then there exists k 2 Eþs such that infy2Ek(y) > 0. Proof. Let k 2 Eþ . For a fixed point y0 2 E, as y0 + K E it follows that k(y0) 6 k(z) for all z 2 K, and so k(z) P 0 for all z 2 K, i.e., k 2 K+. Then Eþ K þ and by the bipolar theorem we conclude that cl K cl E. h
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3. Characterization of
M 1 ¼ fu 2 Y : M þ Rþþ u Mg: It is well-known that M1 is a convex cone for any set M (see, for instance, [9,23]). Moreover, if M is convex, then M1 = {u 2 Y:y0 + tu 2 M, " t > 0}, where y0 is an arbitrary point of M, and M is free disposal if and only if K M1. In the sequel we obtain a necessary and sufficient condition from which one can deduce if a nonempty closed convex set E Y is 6K-i.s. Theorem 3.1. Let E Y be nonempty, closed and convex. Then E is 6K-i.s. if and only if 0 R E and (E1)+ K+. Proof. Suppose that E is 6K-i.s. From the definition we have that 0 R E and E is free disposal. Thus, K E1 and so (E1)+ K+. Reciprocally, consider 0 R E, (E1)+ K+ and suppose that E is not 6K-i.e. Then E + K å E and so there exist y0 2 E and z0 2 K such that y0 + z0 R E. By applying a separation theorem we see that there exists k 2 Y⁄n{0} such that
kðy0 Þ þ kðz0 Þ < inffkðyÞ : y 2 Eg 6 kðyÞ þ tkðuÞ;
8 y 2 E; u 2 E1 ; t P 0: In particular, by taking y = y0 we have
kðz0 Þ < tkðuÞ;
8 u 2 E1 ; t P 0;
and it follows that k(z0) < 0 and k 2 (E1)+. This is a contradiction, since (E1)+ K+ and z0 2 K, and the proof is complete. h In [20, Remark 4], every closed downward set M of the ordered space Rp ; 6Rpþ was characterized as the zero sublevel set of the following Minkowski gauge n1M : Rp ! R:
nM 1 ðyÞ ¼ infft 2 R : y t1 2 Mg;
8 y 2 Rp ;
where 1 ¼ ð1; 1; . . . ; 1Þ 2 Rp . On the other hand, in [26, Theorem 4.2] every downward set M of Rp ; 6Rpþ was characterized as a ‘‘plus-radiant set’’ (i.e., p y t1 2 M, for all y 2 M, t > 0) such that nM 1 is 6Rþ -increasing. Ob-
serve that M is plus-radiant if and only if 1 2 M1. Following both approaches we characterize every 6K-i.s. E Y via the zero sublevel set of a 6K-decreasing functional u such that u(0) > 0. The following lemma is necessary for this objective. Lemma 3.2. Let E Y be a closed 6K-i.s. Then for each q 2 E1n{0} the Minkowski gauge nEq is lower semicontinuous and satisfies the following properties: (a) nEq ð0Þ > 0, n o (b) y 2 Y : nEq ðyÞ 6 a ¼ aq þ E, for all a 2 R,
307
Remark 3.3. Part (a) ensures nEq ð0Þ > 0 but it may be nEq ð0Þ ¼ þ1. One has nEq ð0Þ < þ1 if and only if Rq \ E – ;. Moreover, under the assumptions of Lemma 3.2 it follows that
n o y 2 Y : nEq ðyÞ < 0 ¼ Rþþ q þ E:
ð3Þ
Indeed, let y 2 Y be such that nEq ðyÞ < 0. If nEq ðyÞ ¼ 1, then y + tq 2 E for all t 2 Rþþ , and so y 2 Rþþ q þ E. In other case, there exists t < 0 such that y + tq 2 E and y 2 Rþþ q þ E too. Reciprocally, by the definition of nEq it is obvious that nEq ðyÞ < 0 when y 2 Rþþ q þ E, which finishes the proof. Theorem 3.4. Let E Y be a nonempty set. Then E is a closed 6Ki.s. if and only if there exists a lower semicontinuous functional u : Y ! R such that (a) u(0) > 0, (b) E = {y 2 Y:u(y) 6 0}, (c) u is 6K-decreasing.
Proof. If E is a closed 6K-i.s., as K is a proper cone, there exists q 2 Kn{0}. As E is a 6K-i.s., K E1, and so q 2 E1n{0}. By Lemma 3.2, the functional u :¼ nEq is lower semicontinuous and satisfies the desirable properties (a)–(c). Reciprocally, assume that there exists a lower semicontinuous functional u : Y ! R satisfying properties (a)–(c). Then it is clear that E is closed since E is a sublevel set of u. Given y 2 E and z 2 K, by (b) and (c) we have that u(y + z) 6 u(y) 6 0 and by (b) we obtain y + z 2 E. Thus E + K = E. Moreover, by (a) and (b) it is clear that 0 R E and so E is 6K-i.s. h Remark 3.5. Every closed 6K-i.s. is the zero sublevel set of a lower semicontinuous 6K-decreasing functional u such that u(0) > 0, but this functional is not unique. For example, if Y ¼ R, K ¼ Rþ , then the 6K-i.s. E = [1, + 1) is the zero sublevel set of the following decreasing functions: f1(y) = 1 y, f2(y) = e1y 1, f3(y) = 1 y3 (all of them are continuous functions satisfying fi(0) > 0) and f4(y) = 1 E(y), where E(y) is the integer part of y, which is discontinuous and lower semicontinuous. However, only the functions R 3 y ! ay þ a with a > 0 are of the form n½1;þ1Þ for some q q 2 ½1; þ1Þ1 n f0g ¼ Rþþ . Next we characterize open 6K-i.s. by following the same approach as Theorem 3.4. Lemma 3.6. Let E Y be a nonempty set. If E is a solid 6K-i.s., then for each q 2 Yn{0} such that
cl E þ Rþþ q int E; the Minkowski gauge properties:
ð4Þ E ncl q
is continuous and satisfies the following
E (a) ncl q ð0Þ P 0, E (b) fy 2 Y : ncl q ðyÞ < ag ¼ aq þ int E, for all a 2 R, cl E (c) nq is 6K-decreasing.
(c) nEq is 6K-decreasing.
Proof. Suppose that E is a closed 6K-i.s. Then for each q 2 E1n{0} one has E þ Rþ q E. By applying [10, Theorem 2.3.1] to the closed set D = E and k0 = q we deduce that the functional nEq is lower semicontinuous and satisfies properties (b) and (c). On the other hand, as 0 R E, by part (b) (with a = 0) we see that nEq ð0Þ > 0, i.e., part (a) holds too. h
Proof. Suppose that E is a solid 6K-i.s. and take q 2 Yn{0} satisfying (4). By applying [10, Theorem 2.3.1 and Remark 2.3.3] to D = cl E E and k0 = q we deduce that the Minkowski gauge ncl q is continuous and satisfies properties (b) and (c). As 0 R E, by part (b) we see that E ncl h q ð0Þ P 0 and the proof is complete. E Under condition (4), observe that ncl takes the value 1 on q each line parallel to q contained in cl E:
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E ncl q ðzÞ ¼ 1;
8 z 2 y þ Rq;
8 y 2 Y such that y þ Rq cl E:
In connection to condition (4) we make two remarks. Remark 3.7. According to [10, Remark 2.3.3], if (4) is satisfied, then
int cl E ¼ int E:
ð5Þ
Let us prove the following reciprocal statement: if E is free disposal, K is solid and (5) holds, then each q 2 int K satisfies (4). Indeed, by Proposition 2.4, cl E + K cl E, and as cl E + int K is an open set contained in cl E, we have that
cl E þ int K int cl E ¼ int E:
ð6Þ
Note that every solid convex set E satisfies (5) and so statement (6) is true for all convex free disposal set E when K is solid. Property (4) is essential in Lemma 3.6, as it is illustrated in the following example. Example 3.8. Let Y ¼ R2 and E ¼ R ð2; 4Þ [ ð1; þ1Þ ð0; 2Þ (observe that assumption (4) is not satisfied for any q 2 R2 ). Consider q = (1, 0) and K ¼ conefqg ¼ fðy1 ; 0Þ 2 R2 : y1 P 0g. Then E is an open 6K-i.s. and
cl E þ Rþþ q cl E: By Lemma 3.2 we see that the Minkowski gauge nclq E is lower semicontinuous and cl E ¼ fy 2 R2 : nclq E ðyÞ 6 0g. However, by (3) it follows that
n o E y : ncl q ðyÞ < 0 ¼ R ½2; 4 [ ð1; þ1Þ ½0; 2
Theorem 3.9. Consider a solid set E Y that satisfies condition (4) for some q 2 Yn{0}. Then E is an open 6K-i.s. if and only if there exists a continuous functional u : Y ! R such that (a) u(0) P 0, (b) E = {y 2 Y:u(y) < 0}, (c) u is 6K-decreasing. Proof. The necessary condition follows from Lemma 3.6 choosing u ¼ nclq E . The proof of the sufficient condition is analogous to the proof of the sufficient condition in Theorem 3.4. h Let us observe in the previous theorem that assumption (4) is not used to prove the sufficient condition. In other words, if u is continuous, statements (a)–(c) in Theorem 3.9 imply that the nonempty set E is an open 6 K-i.s. 4. E-optimality and nonlinear scalarization Consider the following vector optimization problem:
ð7Þ
where f:X ? Y, X is an arbitrary decision space and the feasible set S X is nonempty. We say that (7) is a Pareto problem if Y ¼ Rp and K ¼ Rpþ . From now on we assume that E Y is 6K-i.s. By translating the E-optimality notion by Chicco et al. (see [4, Definition 3.1]) to problem (7), the following optimality notion is defined. Definition 4.1. A point x0 2 S is said to be an E-optimal (resp. weak E-optimal) solution of problem (7), denoted x0 2 Op(f, S; E) (resp. x0 2 WOp(f, S; E)), if
ðf ðSÞ f ðx0 ÞÞ \ ðEÞ ¼ ; ðresp: ðf ðSÞ f ðx0 ÞÞ \ ðint EÞ ¼ ;Þ:
Remark 4.2. (i) Definition 4.1 collapses the notions of ideal, nondominated, efficient (minimal), weak efficient and Henig proper efficient solution. In the following we enumerate the 6K-i.s. from which one can obtain these notions. (a) Y n ðKÞ 2 IK and Op(f, S; Yn(K)) is the set of ideal solutions of problem (7), since
ðf ðSÞ f ðx0 ÞÞ \ ðY n ðKÞÞ ¼ ; () f ðSÞ f ðx0 Þ þ K: This case reduces to [4, Proposition 3.2(iii)] by considering X ¼ Y ¼ Rp ; K ¼ Rpþ and f(x) = x for all x 2 X (see Example 2.3(h)). (b) K n f0g 2 IK and Op(f, S; Kn{0}) is the set of nondominated solutions of problem (7). (c) K \ ðY n ðKÞÞ 2 IK and Op(f, S;K \ (Yn(K))) is the set of efficient solutions of problem (7) since
ðf ðSÞ f ðx0 ÞÞ \ ðK \ ðY n ðKÞÞÞ ¼ ;
and this set is different to E. The same results are obtained choosing q = (a, 0) with a > 0.
Minff ðxÞ : x 2 Sg;
It is clear that WOp(f, S; E) = Op(f, S; int E). Moreover, if int E = ;, then WOp(f, S; E) = S. Thus, in order to deal with non trivial weak E-optimal solution sets, we assume that E is solid whenever this kind of solutions are considered. By taking different 6K-improvement sets E, the E-optimal solution notion reduces to well-known exact and approximate solution concepts of problem (7), as it is showed in the following remark. Next, the class of 6K-i.s. is denoted by IK .
() ðf ðx0 Þ KÞ \ f ðSÞ f ðx0 Þ þ K: R (d) If K is solid, then int K 2 IK and Op(f, S; K) is the set of weak efficient solutions of problem (7). We denote WMin(f, S) :¼ Op(f, S; int K). (e) If K is pointed and D Y is a proper solid convex cone such that Kn{0} int D, then D \ ðY n ðDÞÞ 2 IK and the elements of the set Op(f, S;D \ (Yn(D))) are Henig proper efficient solutions of problem (7). (ii) The main approximate solution concepts for problem (7) can also be seen as particular cases of Definition 4.1 by taking certain i.s. (see [13] for a similar approach based on coradiant sets instead of improvement sets). Next we show some examples. (a) If K is pointed and q 2 K, then q þ K n f0g 2 IK (see Example 2.3(a)) and Op(f, S; q + Kn{0}) is the set of approximate solutions of problem (7) in the Kutateladze’s sense with respect to q. If K is not pointed, a version of this approximate solution notion can be obtained via the 6K-i.s. E = q + K, where q 2 Kn(K). (b) Consider H Kn(K). Then H þ K 2 IK and Op(f, S; H + K) is the set of approximate solutions of problem (7) in the Németh’s sense with respect to H. If K is pointed and H K, then a version of this approximate solution concept can be obtained by considering the 6K-i.s. E = H + Kn{0} (see Example 2.3(a)). (c) If q 2 K, then K \ ðq KÞc 2 IK (see Example 2.3, parts (e) and (h)) and Op(f, S; K \ (q K)c) is the set of approximate solutions in the White’s sense with respect to q. (d) Let u : Y ! R be 6K-increasing on K such that u(0) = 0 and consider d P 0. Then K \ ½u > d 2 IK whenever is nonempty (see Example 2.3(c)) and Op(f, S;K \ [u > d]) is a new set of approximate solutions of problem (7) whose points satisfy the following property: x0 2 Op(f, S; K \ [u > d]) if and only if x0 2 S and
x 2 S;
f ðxÞ6K f ðx0 Þ ) uðf ðx0 Þ f ðxÞÞ 6 d:
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In particular, if (Y,p) is a normed space and the norm p is 6Kincreasing on K, then Op(f, S; K \ [p > d]) is the set of approximate solutions in the Tanaka’s sense. Analogously, if u = k 2 D+n{0}, then we obtain the set of approximate solutions in the Helbig’s sense with respect to k and by taking u = max{k1, . . . , kr}, where ki 2 K+n{0} for all i, we obtain the set of generalized Helbig approximate solutions introduced in [13, Section 4.3]. Next we obtain necessary and sufficient E-optimality conditions through scalarization processes. We start studying the nonconvex case (i.e., the set E and the objective function f of problem (7) can be nonconvex), since in this case, the E-optimality conditions are consequence of the characterizations of improvement sets obtained in the previous section through sublevel sets. Given a scalar function g : X ! R, the set of suboptimal (resp. sharp suboptimal) solutions with error e P 0 of the following optimization problem
MinfgðxÞ : x 2 Sg <
is denoted by argminS(g,e) (resp. argminS ðg; eÞ), i.e.,
argminS ðg; eÞ ¼ fx0 2 S : gðx0 Þ e 6 gðxÞ; 8 x 2 Sg <
ðresp: argminS ðg; eÞ ¼ fx0 2 S : gðx0 Þ e < gðxÞ; 8 x 2 S n fx0 ggÞ: We denote argminSg :¼ argminS(g, 0), i.e., the set of exact minima of g on S. Remark 4.3. The following properties are clear:
<
(a) If 0 6 e1 6 e2, then argminS ðg; e1 Þ argminS ðg; e1 Þ argminS ðg; e2 Þ. < (b) If 0 6 e1 < e2, then argminS ðg; e1 Þ argminS ðg; e2 Þ. For each y 2 Y, fy:X ? Y denotes the function fy(x) = y f(x), for all x 2 X. In the next theorem we obtain general necessary and sufficient E-optimality conditions. Theorem 4.4. Let E Y be a 6K-i.s. and u : Y ! R such that u(0) < +1.
(a) Suppose that
E ¼ fy 2 Y : uðyÞ 6 0g:
ð8Þ
Then, the following statements are true. < (i) x0 2 Op(f, S; E) if and only if x0 2 argminS ðu ff ðx0 Þ ; uð0ÞÞ. (ii) If 0 6 e < u(0) and x0 2 argminS ðu ff ðx0 Þ ; eÞ, then x0 2 Op(f, S; E). (b) Consider that
E ¼ fy 2 Y : uðyÞ < 0g:
ð9Þ
Then, x0 2 Op(f, S; E) if and only if x0 2 argminS ðu ff ðx0 Þ ; uð0ÞÞ.
Proof. Let us prove part (a), since the proof of part (b) is similar to the proof of part (a)–(i). (a)–(i) Let x0 2 X. By (8) it is obvious that
ðf ðSÞ f ðx0 ÞÞ \ ðEÞ ¼ ; () uðf ðx0 Þ f ðxÞÞ > 0 8 x 2 S n fx0 g () ðu ff ðx0 Þ ÞðxÞ > ðu ff ðx0 Þ Þðx0 Þ uð0Þ
8 x 2 S n fx0 g and the result follows.
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(a)–(ii) Let x0 2 argminS ðu ff ðx0 Þ ; eÞ and suppose that 0 6 e < u(0). By Remark 4.3(b) we deduce that x0 2 argmin< S ðu ff ðx0 Þ ; uð0ÞÞ and by part (a) we see that x0 2 Op(f, S; E). h Let us observe that if E is a 6K-i.s. and statement (8) (resp. (9)) is true, then uð0Þ 2 Rþþ [ fþ1g (resp. uð0Þ 2 Rþ [ fþ1g). Moreover, in order to use Theorem 4.4, let us recall that every closed 6 K-i.s., and every open 6K-i.s. satisfying (4) fulfills statements (8) and (9), respectively (see Theorems 3.4 and 3.9). Moreover, statements (8) and (9) can be satisfied by different functionals u (see Remark 3.5). For applying Theorem 4.4, let us introduce the following cone: KE :¼ E1 \ coneE. It is clear that 0 2 KE and KE – {0} if and only if E1 \ E – ;, since E1 is a cone and 0 R E. Next we suppose that E is closed or solid, and KE – {0}. Let us observe that an improvement set could not satisfy any of these assumptions. For example, if Y ¼ R2 and K ¼ fðy1 ; 0Þ 2 R2 : y1 P 0g, the 6K-i.s. E ¼ fðy1 ; 1Þ 2 R2 : y1 > 0g is neither solid nor closed and KE = {(0, 0)}. Now, we provide necessary and sufficient conditions for E-optimality by considering first a closed set E (Theorem 4.5) and second E a solid set E (Theorem 4.6), via nEq and ncl respectively. q Theorem 4.5. Let E Y be a closed 6K-i.s. such that KE – {0} and consider the following statements: (a) x0 2 Op(f, S; E). < (b) x0 2 argminS nEq ff ðx0 Þ ; nEq ð0Þ , for all q 2 KEn{0}. < (c) x0 2 argminS nEq ff ðx0 Þ ; nEq ð0Þ , for some q 2 KEn{0}. (d) 0 6 e < nEq ð0Þ and x0 2 argminS nEq ff ðx0 Þ ; e , for q 2 KEn{0}.
some
Then (a), (b) and (c) are equivalent and (d) implies (a). Proof. Let q 2 KEn{0}. By Lemma 3.2 we see that
fy 2 Y : nEq ðyÞ 6 0g ¼ E and by Remark 3.3 we have that nEq ð0Þ < þ1, since q 2 KE coneE. Then, the result follows by applying Theorem 4.4(a) to u ¼ nEq . h In [6, Theorem 2.5] the authors characterized via scalarization H and the Minkowski gauge n q the nondominated solutions of problem (7) with respect to an ordering set H that satisfies statement (2) and such that 0 2 bd H. Then, let us observe that [6, Theorem 2.5] cannot be applied if H is not solid and it cannot either if H is a closed improvement set. Thus, for the class of closed improvement sets, Theorem 4.5 is more general than [6, Theorem 2.5], since it can be applied for solid and non-solid closed improvement sets. For example, if Y ¼ R2 and K ¼ Rþ fð0;1Þg, then E1 ={(0,1)} + K is a non-solid closed 6K-i.s. and E2 ¼ E1 [ fð0; 2Þg þ R2þ is a solid closed 6K-i.s. such that ð0; 1Þ 2 K Ei , i = 1,2, and so Theorem 4.5 can be applied, but Theorem 2.5 of [6] cannot. Moreover, if one translates E2 in order to satisfy the condition 0 2 bd H, statement (2) is not satisfied either for any q 2 Yn{0}. Let us denote
F E ¼ fq 2 Y n f0g : cl E þ Rþþ q int Eg \ coneðcl EÞ:
Theorem 4.6. Let E Y be a solid 6K-i.s. and suppose that FE – ;. The following statements are equivalent: (a) x0 2 WOp(f, S;E). E cl E (b) x0 2 argminS ncl q ff ðx0 Þ ; nq ð0Þ , for all q 2 FE. E cl E (a) x0 2 argminS ncl q ff ðx0 Þ ; nq ð0Þ , for some q 2 FE.
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Theorem 5.3. Let E be a 6K-i.s. and k 2 E+n{0}. Then
Proof. Let q 2 FE. By Lemma 3.6 we see that
n o E y 2 Y : ncl q ðyÞ < 0 ¼ int E
E and by Remark 3.3 we have that ncl since q ð0Þ < þ1, q 2 FE cone(cl E). Then the result follows by applying TheoE rem 4.4(b) to u ¼ ncl h q .
A particular case of Theorem 4.6 is [11, Theorem 5.2], which can be obtained by choosing q 2 int K and E = eq + Kn{0}, since one has E q 2 FE (as q 2 int K), ncl q ð0Þ ¼ e and WOp(f, S; E) is the set of weak approximate solutions in the Kutateladze’s sense according to Remark 4.2(ii)(a). Similarly, [11, Theorems 4.5 and 5.1(a)] are deduced from Theorem 4.6, and [11, Theorem 5.1(b)] is obtained from Theorem 4.5 (part (d) ) (a)). Let us observe that [6, Theorem 2.5] cannot be applied for an open ordering set H such that 0 R bd H. However, if this ordering set is an 6K-i.s., then Theorem 4.6 can be applied whenever FH – ;. On the other hand, if E is a solid 6K-i.s. such that 0 2 bd E, then FE – ; if and only if statement (2) is satisfied, since
fq 2 Y n f0g : cl E þ Rþþ q int Eg int E and so the assumptions of [6, Theorem 2.5] and Theorem 4.6 are the same.
(a) argminS(kf, rE(k)) WOp(f, S; E), < (b) argminS ðk f ; rE ðkÞÞ Opðf ; S; EÞ.
Proof. (a) Choose x0 2 argminS(kf, rE(k)), then
kðf ðxÞÞ P kðf ðx0 ÞÞ rE ðkÞ 8 x 2 S:
Suppose that x0 R WOp(f, S; E). Then there exist ^ x 2 S and ^e 2 int E such that f ð^xÞ f ðx0 Þ ¼ ^e. By applying (13) to x ¼ ^x it results
kð^eÞ ¼ kðf ð^xÞ f ðx0 ÞÞ P rE ðkÞ; i.e., inf e2E kðeÞ P kð^eÞ, and so inf e2E kðeÞ ¼ kð^eÞ. As ^e 2 int E, this implies that k = 0, a contradiction. (b) Similarly, choose x0 2 argmin< S ðk f ; rE ðkÞÞ, then
kðf ðxÞÞ > kðf ðx0 ÞÞ rE ðkÞ 8 x 2 S n fx0 g:
Recall that a function f:X ? Y is closely K-convexlike on S if cl(f(S) + K) is a convex set (see [7]). For k 2 Y⁄, we denote
rE ðkÞ ¼ inf kðeÞ:
kð^eÞ ¼ kðf ð^xÞ f ðx0 ÞÞ > rE ðkÞ;
+
Let us observe that if k 2 E , then rE(k) P 0.
As a consequence of Theorems 5.1 and 5.3, we obtain the following characterization of weak E-optimal points under convexity assumptions.
WOpðf ; S; EÞ ¼ Theorem 5.1. Assume that f is closely K-convexlike on S, and E is a solid convex 6K-i.s. Then
[
argminS ðk f ; rE ðkÞÞ:
k2Eþ nf0g
ð10Þ
ð11Þ
Indeed, if statement (11) was false, then there exist x 2 S, d 2 K and e 2 int E such that f(x) + d f(x0) = e. So f(x) f(x0) = (e + d) 2 int E, since int E is a 6K-i.s. (see Proposition 2.4(b)), which contradicts (10). From (11), we derive that cl(f(S) + K f(x0)) \ (int E) = ;, since (f(S) + K f(x0)) (int E)c and this last set is closed. As cl(f(S) + K f(x0)) is a convex set since f is closely K-convexlike, by the separation theorem (see, for example, [17, Theorem 3.16]) there exist k 2 Y⁄n{0} and a 2 R such that
kðf ðxÞ þ d f ðx0 ÞÞ P a P kðeÞ 8 x 2 S; d 2 K; e 2 E:
argminS ðk f ; rE ðkÞÞ:
In particular, if K is solid and we choose E = int K, this corollary reduces to the following well-known result (see, for instance, [3, Theorem 5.2]):
[
argminS ðk f Þ;
ð15Þ
k2K þ nf0g
From here, it follows that
ðf ðSÞ þ K f ðx0 ÞÞ \ ðint EÞ ¼ ;:
[ k2Eþ nf0g
WMinðf ; SÞ ¼
Proof. Choose x0 2 WOp(f, S; E), then
ðf ðSÞ f ðx0 ÞÞ \ ðint EÞ ¼ ;:
h
Corollary 5.4. Assume that f is closely K-convexlike on S, and E is a solid convex 6K-i.s. Then
e2E
WOpðf ; S; EÞ
ð14Þ
Suppose that x0 R Op(f, S; E). Then there exist ^x 2 S n fx0 g and ^e 2 E such that f ð^xÞ f ðx0 Þ ¼ ^e. By applying (14) to x ¼ ^ x it follows that
i.e., inf e2E kðeÞ > kð^eÞ, which is a contradiction. 5. E-optimality and linear scalarization
ð13Þ
ð12Þ
Taking x = x0 and d = 0 it results 0 P k(e) for all e 2 E, and so k 2 E+n{0}. From (12) taking d = 0, it follows that k(e) P k(f(x0)) k(f(x)) for every e 2 E, x 2 S, and so rE(k) P k(f(x0)) k(f(x)) for every x 2 S. In consequence, k(f(x)) P k(f(x0)) rE(k) for every x 2 S, i.e., x0 2 argminS(kf, rE(k)). h Remark 5.2. If Y is normed, we can suppose kkk = 1 dividing (12) by kkk.
since E+ = K+ and rE(k) = 0 for all k 2 E+. Next we apply Theorem 5.3 and Corollary 5.4 to obtain a characterization of approximate solutions of K-convexlike vector optimization problems. Let C Yn{0} be a convex and solid coradiant set and let e > 0. Definition 5.5. [12, Definition 2.1] It is said that x0 2 S is a (C, e)efficient solution (resp. weak (C, e)-efficient solution) of problem (7), denoted x0 2 AE(f, S, C, e) (resp. x0 2 WAE(f, S, C, e)), if (f(S) f(x0)) \ (eC) = ; (resp. (f(S) f(x0)) \ (eint C) = ;). Let D Y be a solid convex cone. Recall that f is D-subconvexlike on S if f(S) + int D is a convex set. Theorem 5.6. Suppose that f is (cone C)-subconvexlike on S. Then we have that S (a) WAEðf ; S; C; eÞ ¼ k2C þ nf0g argminS ðk f ; erC ðkÞÞ. < (b) argminS ðk f ; erC ðkÞÞ AEðf ; S; C; eÞ. (c) If rC(k) > 0, then argminS(kf,drC(k)) AE(f, S, C, e), for all d 2 [0, e). Proof. Let us define E :¼ eC. As C is convex, then E is an i.s. with respect to the convex cone D :¼ cone C, since eC + cone C eC by [12, Lemma 3.1(v)]. In consequence, it is clear that
C. Gutiérrez et al. / European Journal of Operational Research 223 (2012) 304–311
AEðf ; S; C; eÞ ¼ Opðf ; S; EÞ and WAEðf ; S; C; eÞ ¼ WOpðf ; S; EÞ:
ð16Þ
It is known that, for a solid convex cone D, the sets of D-subconvexlike and closely D-convexlike functions coincide (see, for instance, [7]). Moreover, it is clear that
Dþ ¼ ðcone CÞþ ¼ C þ ¼ Eþ :
ð17Þ
Then the result follows from (16), Corollary 5.4, (17), Theorem 5.3(b) and Remark 4.3(b). Note that reC(k) = erC(k) for all k 2 Y⁄, e > 0. h As a direct consequence of Theorem 5.6 one can deduce [11, Proposition 4.10 and Theorem 5.6] by choosing C = q + D, since C+ = D+, rC(k) = k(q) for all k 2 C+ and f(S) + int D is a convex set whenever f(S) is convex. On the other hand, Theorem 5.6 improves [12, Theorems 5.2 and 5.4] and extends them to a (non-necessarily Pareto) vector optimization problem. The improvement is based on the errors in the necessary and sufficient conditions. Indeed, let us suppose that Y is normed and denote
dYnC þ ðkÞ ¼ inffkn kk : n 2 Y n C þ g
dC ¼ inffkdk : d 2 Cg;
and dðkÞ ¼ dC dYnC þ ðkÞ, where k 2 C+. In [12, Theorems 5.2 and 5.4], the following necessary and sufficient conditions for weak (C, e)efficient solutions of (cone C)-subconvexlike (Pareto) problems were stated (in (19) it is assumed that 0 R cl C):
WAEðf ; S; C; eÞ
[
argminS ðk f ; edC Þ;
ð18Þ
k2C þ ; kkk¼1
argminS ðk f ; dÞ WAEðf ; S; C; d=dðkÞÞ;
8 d P 0; 8k 2 intðC þð19Þ Þ:
By Remark 5.2, one can suppose that kkk = 1 in the statements of Theorem 5.6. Thus,
rC ðkÞ ¼ inf kðdÞ 6 kkk inf kdk ¼ dC d2C
d2C
and the precision erC(k) of Theorem 5.6(a) is better than the precision e dC of (18). Reciprocally, by [2, Lemma 2.7], we have that
dYnCþ ðkÞ 6 inffkðyÞ : y 2 cone C; kyk ¼ 1g: So, for each d 2 C, as d0 = (1/kdk)d 2 cone C and kd0 k = 1, it follows that 0
kðdÞ ¼ kdkkðd Þ P kdk dYnC þ ðkÞ P dC dYnC þ ðkÞ ¼ dðkÞ: Therefore, rC(k) P d(k). The sufficient condition of Theorem 5.6(a) can be rewritten as follows:
argminS ðk f ; dÞ WAEðf ; S; C; d=rC ðkÞÞ;
8 d > 0;
8 k 2 intðC þ Þ; kkk ¼ 1 and the precision d/rC(k) is better than the precision d/d(k) of (19). 6. Conclusions In this work we have studied nondominated points of a set and nondominated solutions of vector optimization problems where the ordering relation is given by an improvement set in the framework of linear spaces. This kind of minimality has been shown to be very suitable to deal in a unified way with well-known exact and approximate nondominated concepts. Moreover, it works with very general ordering sets (non-necessarily cones neither pointed, convex, solid or closed sets) and so it can be used to deal with problems where the ordering relation is not a quasi order, which frequently appear in Economics, and in linear spaces where the ordering set is not solid, such as the spaces ‘p and Lp, 1 6 p < 1, and their natural ordering cones. On the other hand, through this minimality notion it is possible to improve some well-known results on particular exact and
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approximate nondominated concepts, since only essential assumptions are required to deal with it. For example, in Corollary 5.4 we have obtained that the well-known characterization of weak efficient solution in subconvexlike vector optimization problems via solutions of linear scalarizations (see statement (15)) really works for any nondominated notion defined via a solid convex improvement set. Acknowledgments The authors are grateful to the anonymous referees for their helpful comments and suggestions. References [1] T.Q. Bao, B.S. Mordukhovich, Extended Pareto optimality in multiobjective problems, in: Q.H. Ansari, J.-C. Yao (Eds.), Recent Developments in Vector Optimization, vol. 1, Springer-Verlag, Berlin, 2012, pp. 467–515. [2] S. Bolintinéanu, Vector variational principles; e-efficiency and scalar stationarity, Journal of Convex Analysis 8 (1) (2001) 71–85. [3] W.W. Breckner, G. Kassay, A systematization of convexity concepts for sets and functions, Journal of Convex Analysis 4 (1) (1997) 109–127. [4] M. Chicco, F. Mignanego, L. Pusillo, S. Tijs, Vector optimization problems via improvement sets, Journal of Optimization Theory and Applications 150 (3) (2011) 516–529. [5] G. Debreu, Theory of Value, John Wiley, New York, 1959. [6] F. Flores-Bazán, E. Hernández, Optimality conditions for a unified vector optimization problem with not necessarily preordering relations. Journal of Global Optimization. http://dx.doi.org/10.1007/s10898-011-9822-y. [7] J.B.G. Frenk, G. Kassay, On classes of generalized convex functions, Gordan– Farkas type theorems, and Lagrangian duality, Journal of Optimization Theory and Applications 102 (2) (1999) 315–343. [8] C. Gerth, P. Weidner, Nonconvex separation theorems and some applications in vector optimization, Journal of Optimization Theory and Applications 67 (2) (1990) 297–320. [9] G. Giorgi, A. Guerraggio, J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier, Amsterdam, 2004. [10] A. Göpfert, H. Riahi, C. Tammer, C. Za˘linescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003. [11] C. Gutiérrez, B. Jiménez, V. Novo, On approximate solutions in vector optimization problems via scalarization, Computational Optimization and Applications 35 (3) (2006) 305–324. [12] C. Gutiérrez, B. Jiménez, V. Novo, On approximate efficiency in multiobjective programming, Mathematical Methods of Operations Research 64 (1) (2006) 165–185. [13] C. Gutiérrez, B. Jiménez, V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM Journal on Optimization 17 (3) (2006) 688–710. [14] C. Gutiérrez, B. Jiménez, V. Novo, Optimality conditions via scalarization for a new e-efficiency concept in vector optimization problems, European Journal of Operational Research 201 (1) (2010) 11–22. [15] C. Gutiérrez, B. Jiménez, V. Novo, A generic approach to approximate efficiency and applications to vector optimization with set-valued maps, Journal of Global Optimization 49 (2) (2011) 313–342. [16] T.X.D. Ha, Optimality conditions for several types of efficient solutions of setvalued optimization problems, in: P.M. Pardalos, T.M. Rassias, A.A. Khan (Eds.), Nonlinear Analysis and Variational Problems, Springer Optimization and Its Applications, vol. 35, Springer, New York, 2010, pp. 305–324. [17] J. Jahn, Vector Optimization: Theory, Applications, and Extensions, SpringerVerlag, Berlin, 2004. [18] A.Y. Kruger, B.S. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization problems, Doklady Akademii Nauk BSSR 24 (8) (1980) 684–687. [19] V.L. Makarov, M.J. Levin, A.M. Rubinov, Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms, North-Holland, Amsterdam, 1995. [20] J.-E. Martı´nez Legaz, A.M. Rubinov, I. Singer, Downward sets and their separation and approximation properties, Optimization 23 (2) (2002) 111–137. [21] E. Miglierina, E. Molho, Sectionwise connected sets in vector optimization, Operations Research Letters 37 (4) (2009) 295–298. [22] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin, 2006. [23] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. [24] A.M. Rubinov, R.N. Gasimov, Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation, Journal of Global Optimization 29 (4) (2004) 455–477. [25] A.M. Rubinov, I. Singer, Best approximation by normal and conormal sets, Journal of Approximation Theory 107 (2) (2000) 212–243. [26] A.M. Rubinov, I. Singer, Topical and sub-topical functions, downward sets and abstract convexity, Optimization 50 (5–6) (2001) 307–351. [27] N.V. Tuyen, N.D. Yen, On the concept of generalized order optimality, Nonlinear Analysis 75 (3) (2012) 1592–1601.