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Set-relations and optimality conditions in set-valued maps
Nonlinear Analysis 63 (2005) 1167 – 1179 www.elsevier.com/locate/na
Set-relations and optimality conditions in set-valued maps María Alonso∗ , Luis Rodríguez-Marín Dpto. Matemática Aplicada I. E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Aptdo. 60149, 28080 Madrid, Spain Received 18 December 2003; accepted 2 June 2005
Abstract Problems in set-valued optimization can be solved via set-optimization. In this paper properties of set-optimization are investigated. Conditions for existence of solutions are established. Directional derivatives are studied for set-valued mappings. Necessary and sufficient conditions in the existence of solutions are showed with directional derivatives. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Set-valued optimization; Set optimization with set-valued maps; Directional derivative; Optimality conditions.
1. Introduction and notation In many occasions during the resolution of a problem, a subset among all which belong to a certain family has to be elected. Consequently, relations between subsets and not between the elements of the union are considered. Although a team has the best player of the league championship, it does not mean that it is the best team. In order to introduce new classes of sets for a generalization of the concept of convexity, in [5,8] six relations between sets are defined. These sets are subsets of a linear space ordered by a convex cone. Later in [6,7], properties of optimization are studied. In this way an interesting line of investigation is started. Among these relations and in accordance with the last works, we focus on two of them which are designated as upper and lower relations. Implications between both of ∗ Corresponding author. Tel.: +34 1 398 7990; fax: +34 1 398 6012.
E-mail address: [email protected] (M. Alonso). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.06.002
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them can be of interest and they comprise a certain generalization of the order induced by the cone. In this sense they can constitute more natural criteria to solve a problem. We consider real normed vector spaces X, Y, Z, a convex pointed cone K ⊂ Y which defines on Y a partial order, and a multifunction F from a subset M ⊂ X to Y such that F (x) = ∅ for all x ∈ M. We designate by F (M) the subset ∪x∈M F (x). Researches in set-valued optimization have concentrated on the problems with and without constraints: minimize
F (x), x ∈ M
(1)
minimize
F (x) G(x) ∩ (−C) = ∅, x ∈ M,
(2)
(see for example [4,9,10]) where G is a multifunction from X to Y and C is a convex pointed cone in Z. The optimality concept used is the generalization of the notion established by Pareto which we recall next. Definition 1. A pair (x0 , y0 ) with x0 ∈ M, y0 ∈ F (x0 ) is called a minimizer of F in M if it verifies ({y0 } − K) ∩ F (M) = {y0 }. Then x0 is called a minimum of F and y0 a minimal (Edgeworth Pareto point) of F (M). To avoid confusion we will designate by K-min F the set of minimum of F in the upper sense, by K-minl F (M) the set of minimals of F (M) and by K-minzr F the set of minimizers of F in M. In this work we will study problem (1) with the optimality concepts determined by the following relations. They are defined between the subsets of the linear space Y, partially ordered by the convex pointed cone K. Definition 2 (Kuroiwa [8]). Let A, B be two subsets of Y. A l B if for all b ∈ B there exists a ∈ A such that a b. “ l ” is called lower relation. The above definition is equivalent to B ⊂ A + K and generalizes the order induced by K in Y, in the sense: a b if b ∈ a + K. The concept of minimal element of a family is also a natural generalization of the one corresponding to an order relation. Let S be a family of subsets of Y; an element A ∈ S is a minimal for the relation l (lower minimal or l-minimal) if for each B ∈ S, such that B l A, it satisfies A l B. Similarly A is an l-maximal if for each B ∈ S, such that A l B, B l A. The set of lminimals of S is designated by l-minl S and the set of l-maximals by l-maxl S. In this way problem (1) can be written in the form l-min F (x),
x ∈ M.
(3)
In this case, an l-minimizer of the problem is a pair (x0 , F (x0 )) such that x0 ∈ M and F (x0 ) is a lower minimal of the family of images of F, i.e. the family F = {F (x)|x ∈ M}.
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Then x0 is called a minimum of F and the respective sets are designed by l-minzr F, l-minl F and l-min F. Similarly the concept of upper relation “ u ” and the other considerations made for the lower case are established. Definition 3 (Kuroiwa [8]). Let A, B be two subsets of Y. A u B if for all a ∈ A there exists b ∈ B such that a b. “ u ” is called upper relation. This definition is also a generalization of the order induced by K, in this case in the sense a b if a ∈ b − K. The concepts of u-minimal and u-maximal subsets are defined in a way similar to the lower relation. The set of u-minimals of S is designated by u-minl S and the set of u-maximals by u-maxl S. Problem (1) turns into u-min F (x),
x ∈ M.
(4)
The subsets of minimizers, minimals and minimum are symbolized by u-minzr F, uminl F and u-min F, respectively. It is easy to prove that if −S = {−A | A ∈ S} it verifies −(l-minl S) = u-maxl (−S), −(u-minl S) = l-maxl (−S), and the maximal problems are reduced to (3) and (4). On the other hand, the problems are equivalent in the sense that each of them has the same solutions as the other with a suitable definition of the multifunction. The next proposition has a simple proof. For the subset A we will designate by AC its complementary. Proposition 4. 1. u-min F = l-min G, where G is a multifunction from M to Y defined by G(x) = (F (x) − K)C . 2. l-min F =u-min L, where L is a multifunction from M toY defined by L(x)=(F (x)+K)C . In Section 2 we study several properties of the set relations defined above. Furthermore, several relations among problems (1), (3) and (4) are analyzed; this means relations among the sets K-min F , l-min F and u-min F . Definitions of K-continuity which we recall here are used in this section and in Section 3. Definition 5 (Luc [9]). 1. F is upper K-continuous at x0 ∈ M if for each neighbourhood V of F (x0 ) in Y, there is a neighbourhood U of x0 in X such that F (x) ⊂ V + C
for all x ∈ U ∩ M.
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2. F is lower K-continuous at x0 ∈ M if for any y ∈ F (x0 ), any neighbourhood V of y in Y, there is a neighbourhood U of x0 in X such that F (x) ∩ (V + C) = ∅
for each x ∈ U ∩ M.
3. F is K-continuous at x0 ∈ M if it is upper and lower K-continuous at that point. In Section 3 we provide sufficient conditions for the existence of a solution to problems (3) and (4). With this aim we introduce the concept of K-regular family. We define as well the concept of K-semicompact multifunction from the definition of K-semicompact subset. Definition 6 (Luc [9]). A subset A ⊂ Y is called K-semicompact if any cover of A of the form {(xi − K)C | xi ∈ A, i ∈ I } admits a finite subcover. Definition 7. The multifunction F is called K-semicompact if F (M) is K-semicompact. Continuous selections of multifunctions have been used in different contexts to establish optimality conditions (see for example [11]). In Section 4 we study these conditions for the relation l . For a point x0 ∈ M we designate by T (x0 , M) the contingent cone to M at x0 (see [3] ) and by S(x0 , M) the cone of feasible directions S(x0 , M) = {v ∈ X : ∃ > 0, x0 + tv ∈ M, ∀ 0 < t < }.
(1.1)
We recall that a multifunction F is pseudoconvex at (x0 , y0 ) ∈ graph(F ) if epigraph(F ) ⊂ (x0 , y0 ) + T ((x0 , y0 ), epigraph(F )).
(1.2)
2. Upper and lower relations The relation l ( u ) determines in S the equivalence relation A∼l B ⇔ A l B
and B l A (A∼u B ⇔ A u B and B u A),
whose classes we will represent by [A]l , ([A]u ). In the quotient set S/∼l (S/∼u ) the order [A]l l [B]l ⇔ A l B ([A]u u [B]u ⇔ A u B) is defined in a natural way. Consequently, if a subset is minimal, all the subsets of its class are minimals. The nature of the elements of S can characterize the classes via its K-minimals (K-maximals). Proposition 8. 1. If [A]l = [B]l , then K-minl A = K-minl B. 2. If [A]u = [B]u , then K-maxl A = K-maxl B. Proof. We will prove 1 (2 is similar). We will suppose for example, that K-minl A = ∅. (If K-minl A = K-minl B = ∅, the proposition is true.) Let a1 ∈ K-minl A, such that a1 ∈ / K-minl B. Since B l A, there exists b1 ∈ B verifying b1 a1 . And as A l B there
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exists a2 ∈ A such that a2 b1 . In consequence, a2 ∈ b1 − K ⊂ a1 − K − K = a1 − K. / K-minl A and we arrive at a contradiction. To prove In this way, if a2 = a1 , then a1 ∈ that a2 = a1 let us suppose that a2 = a1 . Then a1 = b1 and from a1 ∈ / K-minl B, we / A. Therefore, there exist a3 ∈ A, such that deduce that there exist b2 = b1 = a1 and b2 ∈ a3 b2 b1 = a1 , which contradicts a1 ∈ K-minl A. Definition 9 (Domination property). A subset A ⊂ Y has the K-minimal (K-maximal) property if for all x ∈ A there exists a ∈ K-minl A (a ∈ K-maxl A) such that a x (x a). K-semicompact sets are examples of sets with K-minimal property as is shown in the next proposition. Proposition 10. Let A ⊂ Y be a K-semicompact subset. Then A has the K-minimal property. Proof. Let us suppose that A does not have the K-minimal property. Then there exists y0 ∈ A such that {y ∈ A : y y0 } ∩ K-minl A = ∅.
(2.1)
Thus we can construct a sequence {yn }n∈N , yn+1 yn , such that it does not have any / y + K. Therefore, lower bound in A. Then for all y ∈ A there exists yn such that yn ∈ {(yn − K)c : n ∈ N} is an open cover of A. We can bring out a finite subcover {(ynj − K)c : j = 1, 2, . . . , p}. Let us suppose that yn1 yn2 · · · ynp . In consequence, (yn1 − K)c ⊃ (yn2 − K)c ⊃ · · · ⊃ (ynp − K)c .
(2.2)
Then A⊂
p
(ynj − K)c = (yn1 − K)c
(2.3)
j =1
and in particular yn1 ∈ (yn1 − K)c , which is a contradiction.
Proposition 11. Let S be a family of subsets of Y whose elements have the K-minimal (Kmaximal) property. Let A and B be in S. If K-minl A=K-minl B (K-maxl A=K-maxl B), then [A]l = [B]l ([A]u = [B]u ). Proof. We will prove it for “ l ”. We suppose that it is false. That is, [A]l = [B]l , for l / a + K for all a ∈ A, and in example A B. Then there exists b ∈ B such that b ∈ consequence A∩(b−K)=∅. If b ∈ K-minl B, we arrive at a contradiction. If b ∈ / K-minl B, since B has the K-minimal property, there exists b0 ∈ K-minl B such that b0 b, and b0 − K ⊂ b − K, therefore A ∩ (b0 − K) = ∅.
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The K-minimal property of the family is a necessary condition in the above proposition. For example, we consider in R2 the family of sets where each of them is the union of the circle {x 2 + y 2 , > 0} with the quadrant {x > 0, y > 0} or with the quadrant {x < 0, y < 0}. Let K = R2+ . Each determines two subsets. They have the same K-minimals; however, they do not belong to the same class. We will consider problems (1), (3) and (4). Usually there is no relation between K-min F , l-min F and u-min F as we see in the next example. Nevertheless, in Proposition 13 it is shown that some relations can be established under certain conditions. Example 12. Let F : [−1, 1] −→ 2R be a multifunction defined by {(a, b) | a 2 + b2 = x 2 , b > 0} if x = −1, 0, F (x) = (−1/2, 1) if x = −1, (1/2, 1) if x = 0. 2
It is easily seen that K-min F = ∅, l-min F = 1, u-min F = −1. Proposition 13. Let t be a function from M toY. Let us suppose that F (x) has the K-minimal property for all x ∈ M and that K-minl F (x) = {t (x)}. Then l-min F = K-min F . Proof. Let x0 ∈ l-min F , that is, F (x0 ) ∈ l-minl F. We shall prove that t (x0 ) ∈ Kminl F (M). Or equivalently F (M) ∩ (t (x0 ) − K) = {t (x0 )}. In fact, in other cases there would exist x ∈ M and y ∈ F (x), with y = t (x0 ), such that y ∈ t (x0 ) − K and in consequence x = x0 . From t (x)y we deduce that t (x0 ) ∈ t (x) + K and from the K-minimal property of F (x0 ) it results F (x0 ) ⊂ t (x0 ) + K ⊂ t (x) + K + K = t (x) + K. Therefore F (x) l F (x0 ). Also, as t (x) = t (x0 ), by Proposition 8, [F (x)]l = [F (x0 )]l and F (x0 ) cannot be an l-minimal of F contrary to the hypothesis. Let us prove that K-min F ⊂ l-min F . Let x0 ∈ K-min F . Then t (x0 ) ∈ K-minl F (M). Let us suppose that there exists x ∈ M such that F (x) l F (x0 ) and [F (x)]l = [F (x0 )]l , / l-minl F. Since t (x) = t (x0 ) and t (x) t (x0 ), then t (x0 ) is not a K-minimal that is, F (x0 ) ∈ of F (M) and we arrive at a contradiction. Corollary 14. Let t be a function from M to Y. Let M be a compact set and F an upper K-continuous multifunction with K-semicompact values. If K-minl F (x) = {t (x)}, then l-min F = K-min F = ∅. Proof. Since M is a compact set and F is an upper K-continuous multifunction with Ksemicompact values, then F (M) is K-semicompact and K-min F = ∅. By Proposition 10, F (x) has the K-minimal property and by Proposition 13 we obtain the result. According to Proposition 13 if the hypotheses are verified and there exists some Kminimal of F (M), problem (3) has some solution and these solutions are the K-minimum
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of F. However, although K-minl F (M) = ∅ the conditions that each F (x) has an only Kminimal and the property K-minimal are necessary to satisfy this proposition as it is shown in the following examples. Example 15. 1. Let F : M = [0, 2] × [0, 1/2) −→ 2R be a multifunction such that each (, ), = 0, has as image the circle {(x − (1 − ) cos )2 + (y − (1 − ) sin )2 2 } (this circle is tangent inside to the unit circumference at the point (cos , sin )). And F (, 0)={(0, 0)}. Then F (M) is the unit closed circle. If K=R2+ , F (, ) is K-semicompact for all (, ) ∈ M. Therefore it has the K-minimal property. Furthermore, K-minl F (, ), = 0, is a line formed by a quarter of circumference, so it does not have a unique K-minimal. In consequence, K-min F = ∅ and it is easy to check that l-minl F = ∅. 2 2. Let F : [0, 2]×[0, 1/2) −→ 2R be a multifunction such that F (, )={(x, y) | (x − 2 2 (1 − ) cos ) + (y − (1 − ) sin ) < 2 } ∪ {(cos , sin ), (− + (1 − ) cos , (1 − ) sin )} if (, ) = (, 0). And F (, 0) = {(0, 0)}. Then F (M) is the unit closed circle. If K = R2+ , each F (, ) has a unique K-minimal but it does not have the K-minimal property if = 0. It verifies K-min F = ∅ but l-minl F = ∅. 2
Proposition 16. Let s be a function from M toY. Let us suppose that F (x) has the K-maximal property for all x ∈ M and that K-maxl F (x) = {s(x)}. Then u-min F = K-min H , where H (x) = s(x) + K. Proof. Let x0 ∈ u-min F , so F (x0 ) ∈ u-minl F. Let us first prove that s(x0 ) ∈ Kminl H (M), that is to say H (M) ∩ (s(x0 ) − K) = {s(x0 )}. Let us suppose that it is false. Then there exist x ∈ M and y ∈ s(x) + K with y = s(x0 ), such that y ∈ H (M) ∩ (s(x0 ) − K) and therefore s(x) y s(x0 ). Furthermore, s(x) = s(x0 ) because in other cases y ∈ (s(x0 ) + K) ∩ (s(x0 ) − K) and y = s(x0 ), which is a contradiction. In consequence, for all y ∈ F (x) it verifies y ∈ s(x) − K ⊂ s(x0 ) − K − K = s(x0 ) − K, u
hence F (x) u F (x0 ) with F (x0 ) F (x), in contradiction to x0 ∈ u-min F . Let x0 ∈ K-min H . Let us prove that F (x0 ) ∈ u-minl F. Let us suppose that it is false u and there exists x ∈ M with x = x0 such that F (x) u F (x0 ) and F (x0 ) F (x). Then s(x) = s(x0 ). Thus there exists y0 ∈ F (x0 ) such that s(x) ∈ y0 − K ⊂ s(x0 ) − K and s(x0 ) ∈ s(x) + K, in contradiction to x0 ∈ K-min H . Corollary 17. . Let F (x) = [t (x), s(x)] where t and s are functions from M to Y, such that t (x) s(x) for all x ∈ M. Then 1. l-min F = K-min [t (.) + K]. 2. u-min F = K-min [s(.) + K]. Proof. The proof is a direct consequence of Propositions 13 and 16. Take into account that F verifies the hypothesis and K-min F = K-min [t (.) + K].
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3. Existence of optimal solutions We next present some conditions of existence of solutions to problems (3) and (4). Definition 18. Let S be a family of subsets of Y. The collection {(Ai − K)C | Ai ∈ S, i ∈ I } is a K-family of intersection of S if for each A ∈ S there exists i ∈ I such that A ∩ (Ai − K)C = ∅. Definition 19. The family S is K-regular if for each K-family of intersection of S there exists a finite K-subfamily of intersection. Definition 20. The multifunction F is K-regular if the family F is K-regular. Proposition 21. Let M be a compact set and let F be lower K-continuous on M. If F (x)−K is closed for all x ∈ M, then F is K-regular. Proof. Let {F (xi ) | i ∈ I } be a K-family of intersection of F. That is, for all x ∈ M there exists i ∈ I such that (F (xi ) − K)C ∩ F (x) = ∅. Since F is lower K-continuous, for each x ∈ M and y ∈ (F (xi ) − K)C ∩ F (x), there exists an open neighbourhood of x, Uy,i ⊂ Msuch that F (x ) ∩ [(F (xi ) − K)C + K] = F (x ) ∩ (F (xi ) − K)C = ∅ for all x ∈ Uy,i . Therefore {Uy,i } comprises an open cover of M. Then there exists a finite subcover which we will designate by {Uj }j =1,2,...,n . And for all x ∈ M there exists j ∈ {1, 2, . . . , n} such that x ∈ Uj . In consequence, (F (xj ) − K)C ∩ F (x) = ∅. Proposition 22. . If F is K-regular, then u-min F = ∅. Proof. Let us suppose that it is false. That is to say there is not any u-minimal class in F. Then there exists a totally ordered chain {[F (xi )]u }i ∈ I such that it does not have a u-lower bound in the quotient set F/∼u . Thus for all x ∈ M there exists i ∈ I such that u [F (x)]u [F (xi )]u . Or equivalently for all x ∈ M there exists i ∈ I such that F (x) ⊂ / F (xi ) − K. Then F (x) ∩ (F (xi ) − K)C = ∅. Therefore (F (xi ) − K)C | i ∈ I } is a K-family of intersection of F. Since F is K-regular, there exists a finite K-subfamily of intersection {F (x1 ), F (x2 ), . . . , F (xn )}. However, since {[F (xi )]u }i ∈ I is totally ordered, we can suppose that F (x1 ) u F (x2 ) u · · · u F (xn ).
(3.1)
We will prove that for all i ∈ I it verifies F (x1 ) u F (xi ) and we will get a contradiction. In fact, for all i ∈ I there exists j ∈ {1, 2, . . . , n} such that F (xi ) ∩ (F (xj ) − K)C = ∅,
(3.2)
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and since the chain is totally ordered one of the two next contents may be satisfied: F (xi ) ⊂ (F (xj ) − K),
(3.3)
F (xj ) ⊂ (F (xi ) − K).
(3.4)
If (3.3) is verified, (3.2) is contradicted. Then (3.4) is satisfied. But taking (3.1) into account, it follows that F (x1 ) u F (xi ). Remark 23. We recall that if there is not any K-family of intersection of F, necessarily u-min F = ∅. Because if {(F (x) − K)C | x ∈ M} is not such a family, then there exists an element a ∈ M such that F (a) ∩ (F (x) − K)c = ∅. That is to say F (a) ⊂ F (x) − K, for all x ∈ M, and then F (a) is a u-minimal of F. Corollary 24. Let M be compact and F lower K-continuous on M. IfF (x) − K is closed for all x ∈ M, then u-min F = ∅ Proof. It is a direct consequence of Propositions 22 and 21.
Definition 25. Let x ∈ M. The set Fx ={F (x ) ∈ F | F (x ) u F (x)} is called a u-section of F in F (x). Theorem 26. u-min F = ∅ if and only if F has a K-regular u-section. Proof. Let x ∈ u-min F . If Fx = [F (x)]u , then x ∈ / u-min F and we get a contradiction. If Fx = [F (x)]u then there does not exist any K-family of intersection of Fx because (F (x ) − K)c = (F (x) − K)c for all x ∈ M with F (x ) ∈ [F (x)]u . Reciprocally, let us suppose that Fx is K-regular. Let H = {x ∈ M | F (x ) u F (x)}. Since the restriction FH of F to H is K-regular, by Proposition 22, u-min FH = ∅. On the other hand, it is obvious that u-min FH ⊂ u-min F because if x0 ∈ u-min FH and x0 ∈ / u-min F , there exists z ∈ M\H such that F (z) ⊂ F (x0 ) − K ⊂ F (x) − K and F (z) ∈ Fx .
(3.5)
The concept of K-semicompacticity of a set is spread in a natural way to a family S. Thus it allows to establish an existence condition for the l-minimals of S in a way similar to the existence of K-minimals for a K-semicompact set. Definition 27. Let S be a family of subsets of Y and let S = ∪A∈S A. A cover {(xi − K)c : xi ∈ S, i ∈ I } is called a KS -cover if for each A ∈ S there exists i ∈ I such that A ⊂ (xi − K)c . S is called K-semicompact if all KS -cover has a finite KS -subcover. The multifunction F is called KF -semicompact if F is K-semicompact. There is no relation between the K-regularity and the KF -semicompacticity of a multifunction as we can see in the following examples.
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Example 28. 1. Let F : (0, 1] −→ 2R be a multifunction, with F (x) = {(x, y) : x y − x + 2}. F is K-regular; however, it is not KF -semicompact. 2 2. Let F : (0, 1] −→ 2R be a multifunction, with F () = {(x, y) : x 2 + y 2 = }. F is KF semicompact; however, it is not K-regular. We remark that there is not any KF -cover. 2
Proposition 29. Let M be compact. If F is upper K-continuous, then F is KF -semicompact. Proof. Let {(ai − K)c : ai ∈ F (M), i ∈ I } be a KF -cover of F. For each x ∈ M, let i(x) ∈ I , such that F (x) ⊂ (ai(x) − K)c . Let us consider U (x) = {y ∈ M | F (y) ⊂ (ai(x) − K)c }. Since F is upper K-continuous, then U (x) is open. Thus {U (x) | x ∈ M} is an open cover of M and there exists a finite subcover {U (xj ) | j = 1, 2, . . . , n}. Therefore, {(ai(xj ) − K)c : j = 1, 2, . . . , n} is a finite KF -cover because for all x ∈ M there exists j ∈ {1, 2, . . . , n} such that x ∈ U (xj ) and in consequence F (x) ⊂ (ai(xj ) − K)c . Proposition 30. Let F be KF -semicompact, then l-min F = ∅. Proof. Let us prove that every totally ordered l-chain {[F (xi )]l }i∈I of classes of Fl has an l-lower bound. Then by Zorn’s lemma there exists an l-minimal class and l-min F = ∅. Let us suppose that it is false. That is to say, for all x ∈ M there exists i ∈ I such that l / F (x) + K. Therefore for all x ∈ M there exist i ∈ I [F (x)]l [F (xi )]l . Then F (xi ) ⊂ and bi ∈ F (xi ) such that bi ∈ / y + K for all y ∈ F (x). Thus F (x) ∩ (bi − K) = ∅ and F (x) ⊂ (bi −K)c . In this way {(bi −K)c : i ∈ I } is a KF -cover and it has a finite subcover which we will denote by {(bj − K)c : j = 1, 2, . . . , n}. On the other hand, since {[F (xi )]l }i∈I is totally ordered, we can suppose that F (x1 ) l F (x2 ) l · · · l F (xn ).
(3.6)
Furthermore, F (x1 ) ⊂ / (b1 − K)c , then there exists j = 1 such that F (x1 ) ⊂ (bj − K)c . In l / F (x1 ) + K. Therefore F (x1 ) F (xj ), which consequence, F (x1 ) ∩ (bj − K) = ∅ and bj ∈ is a contradiction.
4. Optimality conditions with directional derivatives In this section, M will be a convex set. The cone K will be convex and pointed (nor necessarily closed) with nonempty interior. Proposition 31. F (x0 ) is an l-minimal of F if and only if for each x ∈ M one of the conditions below is verified: 1. F (x) ∈ [F (x0 )]l . 2. There exists ∈ F (x0 ) such that (F (x) − ) ∩ (−K) = ∅.
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Proof. Let us suppose that neither 1 nor 2 is verified. Then there exists x such that F (x) ∈ / [F (x0 )]l . And for all ∈ F (x0 ) there exists y ∈ F (x) such that v = y − ∈ −K. Thus l = y − v and F (x0 ) ⊂ F (x) + K. Therefore F (x) l F (x0 ) and necessarily F (x0 ) F (x) because F (x) ∈ / [F (x0 )]l . In consequence, F (x0 ) is not an l-minimal of F. Reciprocally, let us suppose that F (x0 ) ∈ / l-minl F. Then there exists x ∈ M such that l / [F (x0 )]l and for all ∈ F (x0 ), F (x) l F (x0 ) and F (x0 ) F (x). Consequently, F (x) ∈ there exists y ∈ F (x) such that ∈ y+K, and y− ∈ −K. Therefore (F (x)−)∩(−K) = ∅, which is a contradiction. Definition 32. We call limit set of F at x0 on the direction v ∈ S(x0 , M) to the set YF (x0 ,⎧v) ⎫ ⎬ ⎨ f (x0 + tn un ) − f (x0 ) ; f ∈ CS(F ), un ∈ S(x0 , M) , := z ∈ Y : z = lim ⎭ ⎩ tn tn →0+ un →v
(4.1) where CS(F ) denotes the set of continuous selections of F . Remark 33. If f is a function, we denote by Yf (x0 , v) the limit set of f on the direction v ∈ S(x0 , M) and by Yf (x0 , S(x0 , M)) the union of all these sets. Definition 34. x0 ∈ l-min F is called strict if there exists a neighbourhood U of x0 such l that F (x) F (x0 ) for all x ∈ U ∩ M. Theorem 35. Let x0 be a strict l-minimum of F. IfK-minl F (x0 ) = {y0 }, and F (x0 ) has the K-minimal property, then every continuous selection f of F such that f (x0 ) = y0 verifies Yf (x0 , S(x0 , M)) ∩ (−int K) = ∅.
(4.2)
Proof. Let f ∈ CS(F ) such that f (x0 ) = y0 . Let U be a neighbourhood of x0 such that l F (x) F (x0 ) for all x ∈ U ∩ M. If z ∈ Yf (x0 , S(x0 , M)), then there exist v ∈ S(x0 , M), un → v, tn → 0+ such that z = lim
→0+
tn un →v
f (x0 + tn un ) − f (x0 ) . tn
Therefore there exists n0 ∈ N such that if nn0 we get xn = x0 + tn un ∈ U ∩ M and F (xn ) ∈ / [F (x0 )]l . By Proposition 31, for each xn ∈ U ∩ M, there exists n ∈ F (x0 ) such that (F (xn ) − n ) ∩ (−int K) = ∅. Since K-minl F (x0 ) = {y0 }, then n = f (x0 ) + k, where k ∈ K, and we have (F (xn ) − f (x0 ) − k) ∩ (−int K) = ∅.
(4.3)
Furthermore, f is a continuous selection of F, so (f (xn ) − f (x0 )) ∩ (−int K) = ∅
(4.4)
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and we obtain f (x0 + tn un ) − f (x0 ) ∩ (−int K) = ∅. tn
(4.5)
Finally, as −int K is an open set it follows that z ∈ / − int K.
Remark 36. For a function f from X to R, the above theorem is satisfied for all “intermediate” derivatives or local approximations associated with contingent epiderivatives D↓ f (x0 , v), / D↑ f (x0 , v) (see [1,2]). If f (x0 , v) denotes one of these derivatives, then f (x0 , v) ∈ (−int K) and we have the classic result f (x0 , v)0. Lemma 37. Let F be pseudoconvex at (x0 , y0 ) ∈ graph(F ). Let us suppose that for each pair of sequences tn → 0+ , (hn , kn ) → (h, k), such that (x0 + tn hn , y0 + tn kn ) ∈ epigraph(F ), there exists a continuous selection f of F such that y0 = f (x0 ) and (x0 + tn hn , y0 + tn kn ) ∈ epigraph(f ). Then it is verified F (x) − y0 ⊂ YF (x0 , x − x0 ) + K
(4.6)
for all x ∈ M. Proof. Since F is pseudoconvex at (x0 , y0 ), we get epigraph(F ) ⊂ {(x0 , y0 )} + T (epigraph(F ), (x0 , y0 )).
(4.7)
Therefore for all x ∈ M, y ∈ F (x) (x − x0 , y − y0 ) ∈ T (epigraph(F ), (x0 , y0 )).
(4.8)
Thus there exist sequences tn → 0+ , (hn , kn ) → (x − x0 , y − y0 ) such that for all n ∈ N (x0 + tn hn , y0 + tn kn ) ∈ epigraph(F ).
(4.9)
By hypothesis there exists f ∈ CS(F ) such that y0 = f (x0 ) and (x0 + tn hn , y0 + tn kn ) ∈ epigraph(f ).
(4.10)
Then y0 + tn kn ∈ f (x0 + tn hn ) + K and we have kn ∈
f (x0 + tn hn ) − f (x0 ) + K. tn
(4.11)
By calculating limits when n → ∞ y − y0 ∈ lim
n→∞
f (x0 + tn hn ) − f (x0 ) + K, tn
and in consequence y − y0 ∈ YF (x0 , x − x0 ) + K.
Theorem 38. Let F be pseudoconvex at (x0 , y0 ) ∈ graph(F ). Let us suppose that the hypotheses of Lemma 37 are satisfied. If every v ∈ S(x0 , M) verifies YF (x0 , v) ∩ (−K) = ∅, then x0 is an l-minimum of F.
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Proof. Let us prove that F (x0 ) is an l-minimal of F. Let us suppose that it is false. Then by Proposition 31 there exists x ∈ M such that for each ∈ F (x0 ) there exists ∈ F (x) with ( − ) ∈ −K. From Lemma 37 we get F (x) − y0 ⊂ YF (x0 , x − x0 ) + K,
(4.12)
and in particular y0 − y0 ∈ YF (x0 , x − x0 ) + K.
(4.13)
That is to say, there exists z ∈ YF (x0 , x − x0 ) such that y0 − y0 ∈ z + K. Therefore z ∈ y0 − y0 − K ⊂ −K − K = −K, and in consequence z ∈ YF (x0 , x − x0 ) ∩ (−K), which is a contradiction.
(4.14)
References [1] M. Alonso, L. Rodríguez-Marín, Approximations of nondifferentiable functions and local topological equivalence, J. Math. Anal. Appl. 275 (2002) 771–788. [2] M. Alonso, L. Rodríguez-Marín, On the qualitative approximation of Lipschitzian functions, Nonlinear Anal. 56 (2003) 289–307. [3] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990. [4] J. Jahn, R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Method Op. Res. 46 (1997) 193–211. [5] D. Kuroiwa, Convexity for set-valued maps, Appl. Math. Lett. 9 (2) (1996) 97–101. [6] D. Kuroiwa, On set-valued optimization, Nonlinear Anal. 47 (2001) 1395–1400. [7] D. Kuroiwa, Existence theorems of set optimization with set-valued maps, J. Inform. Optim. Sci. 24 (1) (2003) 73–84. [8] D. Kuroiwa, T. Tanaka, T.X.D. Ha, On cone of convexity of set-valued maps, Nonlinear Anal. 30 (3) (1997) 1487–1496. [9] D.T. Luc, Theory of vector optimization, Lectures Notes in Economics and Mathematical Systems, vol. 319, Springer, New York, 1986. [10] D.T. Luc, Contingent derivatives of set-valued maps and applications to vector optimization, Math. Programming 50 (1991) 99–111. [11] X.Q. Yang, Directional derivatives for set-valued mappings and applications, Math. Methods Oper. Res. 48 (1998) 273–285.
Set-relations and optimality conditions in set-valued maps María Alonso∗ , Luis Rodríguez-Marín Dpto. Matemática Aplicada I. E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Aptdo. 60149, 28080 Madrid, Spain Received 18 December 2003; accepted 2 June 2005
Abstract Problems in set-valued optimization can be solved via set-optimization. In this paper properties of set-optimization are investigated. Conditions for existence of solutions are established. Directional derivatives are studied for set-valued mappings. Necessary and sufficient conditions in the existence of solutions are showed with directional derivatives. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Set-valued optimization; Set optimization with set-valued maps; Directional derivative; Optimality conditions.
1. Introduction and notation In many occasions during the resolution of a problem, a subset among all which belong to a certain family has to be elected. Consequently, relations between subsets and not between the elements of the union are considered. Although a team has the best player of the league championship, it does not mean that it is the best team. In order to introduce new classes of sets for a generalization of the concept of convexity, in [5,8] six relations between sets are defined. These sets are subsets of a linear space ordered by a convex cone. Later in [6,7], properties of optimization are studied. In this way an interesting line of investigation is started. Among these relations and in accordance with the last works, we focus on two of them which are designated as upper and lower relations. Implications between both of ∗ Corresponding author. Tel.: +34 1 398 7990; fax: +34 1 398 6012.
E-mail address: [email protected] (M. Alonso). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.06.002
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them can be of interest and they comprise a certain generalization of the order induced by the cone. In this sense they can constitute more natural criteria to solve a problem. We consider real normed vector spaces X, Y, Z, a convex pointed cone K ⊂ Y which defines on Y a partial order, and a multifunction F from a subset M ⊂ X to Y such that F (x) = ∅ for all x ∈ M. We designate by F (M) the subset ∪x∈M F (x). Researches in set-valued optimization have concentrated on the problems with and without constraints: minimize
F (x), x ∈ M
(1)
minimize
F (x) G(x) ∩ (−C) = ∅, x ∈ M,
(2)
(see for example [4,9,10]) where G is a multifunction from X to Y and C is a convex pointed cone in Z. The optimality concept used is the generalization of the notion established by Pareto which we recall next. Definition 1. A pair (x0 , y0 ) with x0 ∈ M, y0 ∈ F (x0 ) is called a minimizer of F in M if it verifies ({y0 } − K) ∩ F (M) = {y0 }. Then x0 is called a minimum of F and y0 a minimal (Edgeworth Pareto point) of F (M). To avoid confusion we will designate by K-min F the set of minimum of F in the upper sense, by K-minl F (M) the set of minimals of F (M) and by K-minzr F the set of minimizers of F in M. In this work we will study problem (1) with the optimality concepts determined by the following relations. They are defined between the subsets of the linear space Y, partially ordered by the convex pointed cone K. Definition 2 (Kuroiwa [8]). Let A, B be two subsets of Y. A l B if for all b ∈ B there exists a ∈ A such that a b. “ l ” is called lower relation. The above definition is equivalent to B ⊂ A + K and generalizes the order induced by K in Y, in the sense: a b if b ∈ a + K. The concept of minimal element of a family is also a natural generalization of the one corresponding to an order relation. Let S be a family of subsets of Y; an element A ∈ S is a minimal for the relation l (lower minimal or l-minimal) if for each B ∈ S, such that B l A, it satisfies A l B. Similarly A is an l-maximal if for each B ∈ S, such that A l B, B l A. The set of lminimals of S is designated by l-minl S and the set of l-maximals by l-maxl S. In this way problem (1) can be written in the form l-min F (x),
x ∈ M.
(3)
In this case, an l-minimizer of the problem is a pair (x0 , F (x0 )) such that x0 ∈ M and F (x0 ) is a lower minimal of the family of images of F, i.e. the family F = {F (x)|x ∈ M}.
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Then x0 is called a minimum of F and the respective sets are designed by l-minzr F, l-minl F and l-min F. Similarly the concept of upper relation “ u ” and the other considerations made for the lower case are established. Definition 3 (Kuroiwa [8]). Let A, B be two subsets of Y. A u B if for all a ∈ A there exists b ∈ B such that a b. “ u ” is called upper relation. This definition is also a generalization of the order induced by K, in this case in the sense a b if a ∈ b − K. The concepts of u-minimal and u-maximal subsets are defined in a way similar to the lower relation. The set of u-minimals of S is designated by u-minl S and the set of u-maximals by u-maxl S. Problem (1) turns into u-min F (x),
x ∈ M.
(4)
The subsets of minimizers, minimals and minimum are symbolized by u-minzr F, uminl F and u-min F, respectively. It is easy to prove that if −S = {−A | A ∈ S} it verifies −(l-minl S) = u-maxl (−S), −(u-minl S) = l-maxl (−S), and the maximal problems are reduced to (3) and (4). On the other hand, the problems are equivalent in the sense that each of them has the same solutions as the other with a suitable definition of the multifunction. The next proposition has a simple proof. For the subset A we will designate by AC its complementary. Proposition 4. 1. u-min F = l-min G, where G is a multifunction from M to Y defined by G(x) = (F (x) − K)C . 2. l-min F =u-min L, where L is a multifunction from M toY defined by L(x)=(F (x)+K)C . In Section 2 we study several properties of the set relations defined above. Furthermore, several relations among problems (1), (3) and (4) are analyzed; this means relations among the sets K-min F , l-min F and u-min F . Definitions of K-continuity which we recall here are used in this section and in Section 3. Definition 5 (Luc [9]). 1. F is upper K-continuous at x0 ∈ M if for each neighbourhood V of F (x0 ) in Y, there is a neighbourhood U of x0 in X such that F (x) ⊂ V + C
for all x ∈ U ∩ M.
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2. F is lower K-continuous at x0 ∈ M if for any y ∈ F (x0 ), any neighbourhood V of y in Y, there is a neighbourhood U of x0 in X such that F (x) ∩ (V + C) = ∅
for each x ∈ U ∩ M.
3. F is K-continuous at x0 ∈ M if it is upper and lower K-continuous at that point. In Section 3 we provide sufficient conditions for the existence of a solution to problems (3) and (4). With this aim we introduce the concept of K-regular family. We define as well the concept of K-semicompact multifunction from the definition of K-semicompact subset. Definition 6 (Luc [9]). A subset A ⊂ Y is called K-semicompact if any cover of A of the form {(xi − K)C | xi ∈ A, i ∈ I } admits a finite subcover. Definition 7. The multifunction F is called K-semicompact if F (M) is K-semicompact. Continuous selections of multifunctions have been used in different contexts to establish optimality conditions (see for example [11]). In Section 4 we study these conditions for the relation l . For a point x0 ∈ M we designate by T (x0 , M) the contingent cone to M at x0 (see [3] ) and by S(x0 , M) the cone of feasible directions S(x0 , M) = {v ∈ X : ∃ > 0, x0 + tv ∈ M, ∀ 0 < t < }.
(1.1)
We recall that a multifunction F is pseudoconvex at (x0 , y0 ) ∈ graph(F ) if epigraph(F ) ⊂ (x0 , y0 ) + T ((x0 , y0 ), epigraph(F )).
(1.2)
2. Upper and lower relations The relation l ( u ) determines in S the equivalence relation A∼l B ⇔ A l B
and B l A (A∼u B ⇔ A u B and B u A),
whose classes we will represent by [A]l , ([A]u ). In the quotient set S/∼l (S/∼u ) the order [A]l l [B]l ⇔ A l B ([A]u u [B]u ⇔ A u B) is defined in a natural way. Consequently, if a subset is minimal, all the subsets of its class are minimals. The nature of the elements of S can characterize the classes via its K-minimals (K-maximals). Proposition 8. 1. If [A]l = [B]l , then K-minl A = K-minl B. 2. If [A]u = [B]u , then K-maxl A = K-maxl B. Proof. We will prove 1 (2 is similar). We will suppose for example, that K-minl A = ∅. (If K-minl A = K-minl B = ∅, the proposition is true.) Let a1 ∈ K-minl A, such that a1 ∈ / K-minl B. Since B l A, there exists b1 ∈ B verifying b1 a1 . And as A l B there
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exists a2 ∈ A such that a2 b1 . In consequence, a2 ∈ b1 − K ⊂ a1 − K − K = a1 − K. / K-minl A and we arrive at a contradiction. To prove In this way, if a2 = a1 , then a1 ∈ that a2 = a1 let us suppose that a2 = a1 . Then a1 = b1 and from a1 ∈ / K-minl B, we / A. Therefore, there exist a3 ∈ A, such that deduce that there exist b2 = b1 = a1 and b2 ∈ a3 b2 b1 = a1 , which contradicts a1 ∈ K-minl A. Definition 9 (Domination property). A subset A ⊂ Y has the K-minimal (K-maximal) property if for all x ∈ A there exists a ∈ K-minl A (a ∈ K-maxl A) such that a x (x a). K-semicompact sets are examples of sets with K-minimal property as is shown in the next proposition. Proposition 10. Let A ⊂ Y be a K-semicompact subset. Then A has the K-minimal property. Proof. Let us suppose that A does not have the K-minimal property. Then there exists y0 ∈ A such that {y ∈ A : y y0 } ∩ K-minl A = ∅.
(2.1)
Thus we can construct a sequence {yn }n∈N , yn+1 yn , such that it does not have any / y + K. Therefore, lower bound in A. Then for all y ∈ A there exists yn such that yn ∈ {(yn − K)c : n ∈ N} is an open cover of A. We can bring out a finite subcover {(ynj − K)c : j = 1, 2, . . . , p}. Let us suppose that yn1 yn2 · · · ynp . In consequence, (yn1 − K)c ⊃ (yn2 − K)c ⊃ · · · ⊃ (ynp − K)c .
(2.2)
Then A⊂
p
(ynj − K)c = (yn1 − K)c
(2.3)
j =1
and in particular yn1 ∈ (yn1 − K)c , which is a contradiction.
Proposition 11. Let S be a family of subsets of Y whose elements have the K-minimal (Kmaximal) property. Let A and B be in S. If K-minl A=K-minl B (K-maxl A=K-maxl B), then [A]l = [B]l ([A]u = [B]u ). Proof. We will prove it for “ l ”. We suppose that it is false. That is, [A]l = [B]l , for l / a + K for all a ∈ A, and in example A B. Then there exists b ∈ B such that b ∈ consequence A∩(b−K)=∅. If b ∈ K-minl B, we arrive at a contradiction. If b ∈ / K-minl B, since B has the K-minimal property, there exists b0 ∈ K-minl B such that b0 b, and b0 − K ⊂ b − K, therefore A ∩ (b0 − K) = ∅.
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The K-minimal property of the family is a necessary condition in the above proposition. For example, we consider in R2 the family of sets where each of them is the union of the circle {x 2 + y 2 , > 0} with the quadrant {x > 0, y > 0} or with the quadrant {x < 0, y < 0}. Let K = R2+ . Each determines two subsets. They have the same K-minimals; however, they do not belong to the same class. We will consider problems (1), (3) and (4). Usually there is no relation between K-min F , l-min F and u-min F as we see in the next example. Nevertheless, in Proposition 13 it is shown that some relations can be established under certain conditions. Example 12. Let F : [−1, 1] −→ 2R be a multifunction defined by {(a, b) | a 2 + b2 = x 2 , b > 0} if x = −1, 0, F (x) = (−1/2, 1) if x = −1, (1/2, 1) if x = 0. 2
It is easily seen that K-min F = ∅, l-min F = 1, u-min F = −1. Proposition 13. Let t be a function from M toY. Let us suppose that F (x) has the K-minimal property for all x ∈ M and that K-minl F (x) = {t (x)}. Then l-min F = K-min F . Proof. Let x0 ∈ l-min F , that is, F (x0 ) ∈ l-minl F. We shall prove that t (x0 ) ∈ Kminl F (M). Or equivalently F (M) ∩ (t (x0 ) − K) = {t (x0 )}. In fact, in other cases there would exist x ∈ M and y ∈ F (x), with y = t (x0 ), such that y ∈ t (x0 ) − K and in consequence x = x0 . From t (x)y we deduce that t (x0 ) ∈ t (x) + K and from the K-minimal property of F (x0 ) it results F (x0 ) ⊂ t (x0 ) + K ⊂ t (x) + K + K = t (x) + K. Therefore F (x) l F (x0 ). Also, as t (x) = t (x0 ), by Proposition 8, [F (x)]l = [F (x0 )]l and F (x0 ) cannot be an l-minimal of F contrary to the hypothesis. Let us prove that K-min F ⊂ l-min F . Let x0 ∈ K-min F . Then t (x0 ) ∈ K-minl F (M). Let us suppose that there exists x ∈ M such that F (x) l F (x0 ) and [F (x)]l = [F (x0 )]l , / l-minl F. Since t (x) = t (x0 ) and t (x) t (x0 ), then t (x0 ) is not a K-minimal that is, F (x0 ) ∈ of F (M) and we arrive at a contradiction. Corollary 14. Let t be a function from M to Y. Let M be a compact set and F an upper K-continuous multifunction with K-semicompact values. If K-minl F (x) = {t (x)}, then l-min F = K-min F = ∅. Proof. Since M is a compact set and F is an upper K-continuous multifunction with Ksemicompact values, then F (M) is K-semicompact and K-min F = ∅. By Proposition 10, F (x) has the K-minimal property and by Proposition 13 we obtain the result. According to Proposition 13 if the hypotheses are verified and there exists some Kminimal of F (M), problem (3) has some solution and these solutions are the K-minimum
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of F. However, although K-minl F (M) = ∅ the conditions that each F (x) has an only Kminimal and the property K-minimal are necessary to satisfy this proposition as it is shown in the following examples. Example 15. 1. Let F : M = [0, 2] × [0, 1/2) −→ 2R be a multifunction such that each (, ), = 0, has as image the circle {(x − (1 − ) cos )2 + (y − (1 − ) sin )2 2 } (this circle is tangent inside to the unit circumference at the point (cos , sin )). And F (, 0)={(0, 0)}. Then F (M) is the unit closed circle. If K=R2+ , F (, ) is K-semicompact for all (, ) ∈ M. Therefore it has the K-minimal property. Furthermore, K-minl F (, ), = 0, is a line formed by a quarter of circumference, so it does not have a unique K-minimal. In consequence, K-min F = ∅ and it is easy to check that l-minl F = ∅. 2 2. Let F : [0, 2]×[0, 1/2) −→ 2R be a multifunction such that F (, )={(x, y) | (x − 2 2 (1 − ) cos ) + (y − (1 − ) sin ) < 2 } ∪ {(cos , sin ), (− + (1 − ) cos , (1 − ) sin )} if (, ) = (, 0). And F (, 0) = {(0, 0)}. Then F (M) is the unit closed circle. If K = R2+ , each F (, ) has a unique K-minimal but it does not have the K-minimal property if = 0. It verifies K-min F = ∅ but l-minl F = ∅. 2
Proposition 16. Let s be a function from M toY. Let us suppose that F (x) has the K-maximal property for all x ∈ M and that K-maxl F (x) = {s(x)}. Then u-min F = K-min H , where H (x) = s(x) + K. Proof. Let x0 ∈ u-min F , so F (x0 ) ∈ u-minl F. Let us first prove that s(x0 ) ∈ Kminl H (M), that is to say H (M) ∩ (s(x0 ) − K) = {s(x0 )}. Let us suppose that it is false. Then there exist x ∈ M and y ∈ s(x) + K with y = s(x0 ), such that y ∈ H (M) ∩ (s(x0 ) − K) and therefore s(x) y s(x0 ). Furthermore, s(x) = s(x0 ) because in other cases y ∈ (s(x0 ) + K) ∩ (s(x0 ) − K) and y = s(x0 ), which is a contradiction. In consequence, for all y ∈ F (x) it verifies y ∈ s(x) − K ⊂ s(x0 ) − K − K = s(x0 ) − K, u
hence F (x) u F (x0 ) with F (x0 ) F (x), in contradiction to x0 ∈ u-min F . Let x0 ∈ K-min H . Let us prove that F (x0 ) ∈ u-minl F. Let us suppose that it is false u and there exists x ∈ M with x = x0 such that F (x) u F (x0 ) and F (x0 ) F (x). Then s(x) = s(x0 ). Thus there exists y0 ∈ F (x0 ) such that s(x) ∈ y0 − K ⊂ s(x0 ) − K and s(x0 ) ∈ s(x) + K, in contradiction to x0 ∈ K-min H . Corollary 17. . Let F (x) = [t (x), s(x)] where t and s are functions from M to Y, such that t (x) s(x) for all x ∈ M. Then 1. l-min F = K-min [t (.) + K]. 2. u-min F = K-min [s(.) + K]. Proof. The proof is a direct consequence of Propositions 13 and 16. Take into account that F verifies the hypothesis and K-min F = K-min [t (.) + K].
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3. Existence of optimal solutions We next present some conditions of existence of solutions to problems (3) and (4). Definition 18. Let S be a family of subsets of Y. The collection {(Ai − K)C | Ai ∈ S, i ∈ I } is a K-family of intersection of S if for each A ∈ S there exists i ∈ I such that A ∩ (Ai − K)C = ∅. Definition 19. The family S is K-regular if for each K-family of intersection of S there exists a finite K-subfamily of intersection. Definition 20. The multifunction F is K-regular if the family F is K-regular. Proposition 21. Let M be a compact set and let F be lower K-continuous on M. If F (x)−K is closed for all x ∈ M, then F is K-regular. Proof. Let {F (xi ) | i ∈ I } be a K-family of intersection of F. That is, for all x ∈ M there exists i ∈ I such that (F (xi ) − K)C ∩ F (x) = ∅. Since F is lower K-continuous, for each x ∈ M and y ∈ (F (xi ) − K)C ∩ F (x), there exists an open neighbourhood of x, Uy,i ⊂ Msuch that F (x ) ∩ [(F (xi ) − K)C + K] = F (x ) ∩ (F (xi ) − K)C = ∅ for all x ∈ Uy,i . Therefore {Uy,i } comprises an open cover of M. Then there exists a finite subcover which we will designate by {Uj }j =1,2,...,n . And for all x ∈ M there exists j ∈ {1, 2, . . . , n} such that x ∈ Uj . In consequence, (F (xj ) − K)C ∩ F (x) = ∅. Proposition 22. . If F is K-regular, then u-min F = ∅. Proof. Let us suppose that it is false. That is to say there is not any u-minimal class in F. Then there exists a totally ordered chain {[F (xi )]u }i ∈ I such that it does not have a u-lower bound in the quotient set F/∼u . Thus for all x ∈ M there exists i ∈ I such that u [F (x)]u [F (xi )]u . Or equivalently for all x ∈ M there exists i ∈ I such that F (x) ⊂ / F (xi ) − K. Then F (x) ∩ (F (xi ) − K)C = ∅. Therefore (F (xi ) − K)C | i ∈ I } is a K-family of intersection of F. Since F is K-regular, there exists a finite K-subfamily of intersection {F (x1 ), F (x2 ), . . . , F (xn )}. However, since {[F (xi )]u }i ∈ I is totally ordered, we can suppose that F (x1 ) u F (x2 ) u · · · u F (xn ).
(3.1)
We will prove that for all i ∈ I it verifies F (x1 ) u F (xi ) and we will get a contradiction. In fact, for all i ∈ I there exists j ∈ {1, 2, . . . , n} such that F (xi ) ∩ (F (xj ) − K)C = ∅,
(3.2)
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and since the chain is totally ordered one of the two next contents may be satisfied: F (xi ) ⊂ (F (xj ) − K),
(3.3)
F (xj ) ⊂ (F (xi ) − K).
(3.4)
If (3.3) is verified, (3.2) is contradicted. Then (3.4) is satisfied. But taking (3.1) into account, it follows that F (x1 ) u F (xi ). Remark 23. We recall that if there is not any K-family of intersection of F, necessarily u-min F = ∅. Because if {(F (x) − K)C | x ∈ M} is not such a family, then there exists an element a ∈ M such that F (a) ∩ (F (x) − K)c = ∅. That is to say F (a) ⊂ F (x) − K, for all x ∈ M, and then F (a) is a u-minimal of F. Corollary 24. Let M be compact and F lower K-continuous on M. IfF (x) − K is closed for all x ∈ M, then u-min F = ∅ Proof. It is a direct consequence of Propositions 22 and 21.
Definition 25. Let x ∈ M. The set Fx ={F (x ) ∈ F | F (x ) u F (x)} is called a u-section of F in F (x). Theorem 26. u-min F = ∅ if and only if F has a K-regular u-section. Proof. Let x ∈ u-min F . If Fx = [F (x)]u , then x ∈ / u-min F and we get a contradiction. If Fx = [F (x)]u then there does not exist any K-family of intersection of Fx because (F (x ) − K)c = (F (x) − K)c for all x ∈ M with F (x ) ∈ [F (x)]u . Reciprocally, let us suppose that Fx is K-regular. Let H = {x ∈ M | F (x ) u F (x)}. Since the restriction FH of F to H is K-regular, by Proposition 22, u-min FH = ∅. On the other hand, it is obvious that u-min FH ⊂ u-min F because if x0 ∈ u-min FH and x0 ∈ / u-min F , there exists z ∈ M\H such that F (z) ⊂ F (x0 ) − K ⊂ F (x) − K and F (z) ∈ Fx .
(3.5)
The concept of K-semicompacticity of a set is spread in a natural way to a family S. Thus it allows to establish an existence condition for the l-minimals of S in a way similar to the existence of K-minimals for a K-semicompact set. Definition 27. Let S be a family of subsets of Y and let S = ∪A∈S A. A cover {(xi − K)c : xi ∈ S, i ∈ I } is called a KS -cover if for each A ∈ S there exists i ∈ I such that A ⊂ (xi − K)c . S is called K-semicompact if all KS -cover has a finite KS -subcover. The multifunction F is called KF -semicompact if F is K-semicompact. There is no relation between the K-regularity and the KF -semicompacticity of a multifunction as we can see in the following examples.
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Example 28. 1. Let F : (0, 1] −→ 2R be a multifunction, with F (x) = {(x, y) : x y − x + 2}. F is K-regular; however, it is not KF -semicompact. 2 2. Let F : (0, 1] −→ 2R be a multifunction, with F () = {(x, y) : x 2 + y 2 = }. F is KF semicompact; however, it is not K-regular. We remark that there is not any KF -cover. 2
Proposition 29. Let M be compact. If F is upper K-continuous, then F is KF -semicompact. Proof. Let {(ai − K)c : ai ∈ F (M), i ∈ I } be a KF -cover of F. For each x ∈ M, let i(x) ∈ I , such that F (x) ⊂ (ai(x) − K)c . Let us consider U (x) = {y ∈ M | F (y) ⊂ (ai(x) − K)c }. Since F is upper K-continuous, then U (x) is open. Thus {U (x) | x ∈ M} is an open cover of M and there exists a finite subcover {U (xj ) | j = 1, 2, . . . , n}. Therefore, {(ai(xj ) − K)c : j = 1, 2, . . . , n} is a finite KF -cover because for all x ∈ M there exists j ∈ {1, 2, . . . , n} such that x ∈ U (xj ) and in consequence F (x) ⊂ (ai(xj ) − K)c . Proposition 30. Let F be KF -semicompact, then l-min F = ∅. Proof. Let us prove that every totally ordered l-chain {[F (xi )]l }i∈I of classes of Fl has an l-lower bound. Then by Zorn’s lemma there exists an l-minimal class and l-min F = ∅. Let us suppose that it is false. That is to say, for all x ∈ M there exists i ∈ I such that l / F (x) + K. Therefore for all x ∈ M there exist i ∈ I [F (x)]l [F (xi )]l . Then F (xi ) ⊂ and bi ∈ F (xi ) such that bi ∈ / y + K for all y ∈ F (x). Thus F (x) ∩ (bi − K) = ∅ and F (x) ⊂ (bi −K)c . In this way {(bi −K)c : i ∈ I } is a KF -cover and it has a finite subcover which we will denote by {(bj − K)c : j = 1, 2, . . . , n}. On the other hand, since {[F (xi )]l }i∈I is totally ordered, we can suppose that F (x1 ) l F (x2 ) l · · · l F (xn ).
(3.6)
Furthermore, F (x1 ) ⊂ / (b1 − K)c , then there exists j = 1 such that F (x1 ) ⊂ (bj − K)c . In l / F (x1 ) + K. Therefore F (x1 ) F (xj ), which consequence, F (x1 ) ∩ (bj − K) = ∅ and bj ∈ is a contradiction.
4. Optimality conditions with directional derivatives In this section, M will be a convex set. The cone K will be convex and pointed (nor necessarily closed) with nonempty interior. Proposition 31. F (x0 ) is an l-minimal of F if and only if for each x ∈ M one of the conditions below is verified: 1. F (x) ∈ [F (x0 )]l . 2. There exists ∈ F (x0 ) such that (F (x) − ) ∩ (−K) = ∅.
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Proof. Let us suppose that neither 1 nor 2 is verified. Then there exists x such that F (x) ∈ / [F (x0 )]l . And for all ∈ F (x0 ) there exists y ∈ F (x) such that v = y − ∈ −K. Thus l = y − v and F (x0 ) ⊂ F (x) + K. Therefore F (x) l F (x0 ) and necessarily F (x0 ) F (x) because F (x) ∈ / [F (x0 )]l . In consequence, F (x0 ) is not an l-minimal of F. Reciprocally, let us suppose that F (x0 ) ∈ / l-minl F. Then there exists x ∈ M such that l / [F (x0 )]l and for all ∈ F (x0 ), F (x) l F (x0 ) and F (x0 ) F (x). Consequently, F (x) ∈ there exists y ∈ F (x) such that ∈ y+K, and y− ∈ −K. Therefore (F (x)−)∩(−K) = ∅, which is a contradiction. Definition 32. We call limit set of F at x0 on the direction v ∈ S(x0 , M) to the set YF (x0 ,⎧v) ⎫ ⎬ ⎨ f (x0 + tn un ) − f (x0 ) ; f ∈ CS(F ), un ∈ S(x0 , M) , := z ∈ Y : z = lim ⎭ ⎩ tn tn →0+ un →v
(4.1) where CS(F ) denotes the set of continuous selections of F . Remark 33. If f is a function, we denote by Yf (x0 , v) the limit set of f on the direction v ∈ S(x0 , M) and by Yf (x0 , S(x0 , M)) the union of all these sets. Definition 34. x0 ∈ l-min F is called strict if there exists a neighbourhood U of x0 such l that F (x) F (x0 ) for all x ∈ U ∩ M. Theorem 35. Let x0 be a strict l-minimum of F. IfK-minl F (x0 ) = {y0 }, and F (x0 ) has the K-minimal property, then every continuous selection f of F such that f (x0 ) = y0 verifies Yf (x0 , S(x0 , M)) ∩ (−int K) = ∅.
(4.2)
Proof. Let f ∈ CS(F ) such that f (x0 ) = y0 . Let U be a neighbourhood of x0 such that l F (x) F (x0 ) for all x ∈ U ∩ M. If z ∈ Yf (x0 , S(x0 , M)), then there exist v ∈ S(x0 , M), un → v, tn → 0+ such that z = lim
→0+
tn un →v
f (x0 + tn un ) − f (x0 ) . tn
Therefore there exists n0 ∈ N such that if nn0 we get xn = x0 + tn un ∈ U ∩ M and F (xn ) ∈ / [F (x0 )]l . By Proposition 31, for each xn ∈ U ∩ M, there exists n ∈ F (x0 ) such that (F (xn ) − n ) ∩ (−int K) = ∅. Since K-minl F (x0 ) = {y0 }, then n = f (x0 ) + k, where k ∈ K, and we have (F (xn ) − f (x0 ) − k) ∩ (−int K) = ∅.
(4.3)
Furthermore, f is a continuous selection of F, so (f (xn ) − f (x0 )) ∩ (−int K) = ∅
(4.4)
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and we obtain f (x0 + tn un ) − f (x0 ) ∩ (−int K) = ∅. tn
(4.5)
Finally, as −int K is an open set it follows that z ∈ / − int K.
Remark 36. For a function f from X to R, the above theorem is satisfied for all “intermediate” derivatives or local approximations associated with contingent epiderivatives D↓ f (x0 , v), / D↑ f (x0 , v) (see [1,2]). If f (x0 , v) denotes one of these derivatives, then f (x0 , v) ∈ (−int K) and we have the classic result f (x0 , v)0. Lemma 37. Let F be pseudoconvex at (x0 , y0 ) ∈ graph(F ). Let us suppose that for each pair of sequences tn → 0+ , (hn , kn ) → (h, k), such that (x0 + tn hn , y0 + tn kn ) ∈ epigraph(F ), there exists a continuous selection f of F such that y0 = f (x0 ) and (x0 + tn hn , y0 + tn kn ) ∈ epigraph(f ). Then it is verified F (x) − y0 ⊂ YF (x0 , x − x0 ) + K
(4.6)
for all x ∈ M. Proof. Since F is pseudoconvex at (x0 , y0 ), we get epigraph(F ) ⊂ {(x0 , y0 )} + T (epigraph(F ), (x0 , y0 )).
(4.7)
Therefore for all x ∈ M, y ∈ F (x) (x − x0 , y − y0 ) ∈ T (epigraph(F ), (x0 , y0 )).
(4.8)
Thus there exist sequences tn → 0+ , (hn , kn ) → (x − x0 , y − y0 ) such that for all n ∈ N (x0 + tn hn , y0 + tn kn ) ∈ epigraph(F ).
(4.9)
By hypothesis there exists f ∈ CS(F ) such that y0 = f (x0 ) and (x0 + tn hn , y0 + tn kn ) ∈ epigraph(f ).
(4.10)
Then y0 + tn kn ∈ f (x0 + tn hn ) + K and we have kn ∈
f (x0 + tn hn ) − f (x0 ) + K. tn
(4.11)
By calculating limits when n → ∞ y − y0 ∈ lim
n→∞
f (x0 + tn hn ) − f (x0 ) + K, tn
and in consequence y − y0 ∈ YF (x0 , x − x0 ) + K.
Theorem 38. Let F be pseudoconvex at (x0 , y0 ) ∈ graph(F ). Let us suppose that the hypotheses of Lemma 37 are satisfied. If every v ∈ S(x0 , M) verifies YF (x0 , v) ∩ (−K) = ∅, then x0 is an l-minimum of F.
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Proof. Let us prove that F (x0 ) is an l-minimal of F. Let us suppose that it is false. Then by Proposition 31 there exists x ∈ M such that for each ∈ F (x0 ) there exists ∈ F (x) with ( − ) ∈ −K. From Lemma 37 we get F (x) − y0 ⊂ YF (x0 , x − x0 ) + K,
(4.12)
and in particular y0 − y0 ∈ YF (x0 , x − x0 ) + K.
(4.13)
That is to say, there exists z ∈ YF (x0 , x − x0 ) such that y0 − y0 ∈ z + K. Therefore z ∈ y0 − y0 − K ⊂ −K − K = −K, and in consequence z ∈ YF (x0 , x − x0 ) ∩ (−K), which is a contradiction.
(4.14)
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