Optimality conditions of set-valued optimization problem involving relative algebraic interior in ordered linear spaces

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1 Optimality conditions of set-valued optimization problem involving relative algebraic interior in ordered linear spaces Zhi-Ang Zhou a, Xin-Min Yang b and Jian-Wen Peng c1 a Department of Applied Mathematics, Chongqing University of Technology, Chongqing , P.R. China; b School of Mathematics, Chongqing Normal University, Chongqing , P.R. China; c School of Mathematics, Chongqing Normal University, Chongqing , P.R. China In this paper, firstly, a generalized subconvexlike set-valued map involving the relative algebraic interior is introduced in ordered linear spaces. Secondly, some properties of a generalized subconvexlike set-valued map are investigated. Finally, the optimality conditions of set-valued optimization problem are established. Keywords Relative algebraic interior Generalized cone subconvexlikeness Set-valued map Separation property Optimality condition AMS 2010 Subject Classifications: 90C26, 90C29, 90C30 1 Introduction Recently, many authors have been interested in vector optimization problems with set-valued maps. For example, Rong and Wu [11] gave characterizations of super efficiency for vector 1 Corresponding author. jwpeng6@yahoo.com.cn 1

2 optimization with cone convexlike set-valued maps. Li [2] established an alternative theorem for cone subconvexlike set-valued maps and Kuhn-Tucker conditions for vector optimization problems with cone subconvexlike set-valued maps. Yang et al. [12] established an alternative theorem for the generalized cone subconvexlike set-valued maps. Yang et al. [13] established an alternative theorem for the nearly cone subconvexlike set-valued maps. We note that the results in [2] and [11-13] are established in linear topological spaces. It is well-known that linear spaces are wider than linear topological spaces. Hence, it is natural to consider the following interesting and meaningful problem: How to generalize those results in [2, 12, 13] from linear topological spaces to linear spaces. Li [1] generalized those results in [2] from linear topological spaces to linear spaces. Huang and Li [3] also generalized the results obtained in [1] from cone subconvexlikeness case to generalized cone-subconvexlikeness case. Note that the results in [1] and [3] were established under the condition that the algebraic interior of ordered cone C denoted by cor(c) is nonempty. However, in some optimization problems, it is possible that cor(c) =. In order to overcome this flaw, the authors in [15-16] introduced the notion of relative algebraic interiors in linear spaces. It is worth noting that the algebraic interior of a set C is the subset of the relative algebraic interior of C. Thus, the notion of the relative algebraic interior generalizes that of the algebraic interior. Adán and Novo [5-8] investigated weak or proper efficiency of vector optimization problems with generalized convex set-valued maps involving relative algebraic interior and vector closure of C in linear spaces. Hernández et al. [9] introduced a cone subconvexlike set-valued map and established optimality conditions and Lagrangian multiplier rule involving relative algebraic interior of C in linear spaces. This paper is organized as follows. In section 2, we give some preliminaries. In section 3, the new notion of generalized cone subconvexlike set-valued map involving the relative 2

3 algebraic interior of C is introduced, and some properties of the generalized cone subconvexlike set-valued maps are investigated. In section 4, a separation property is obtained, and the optimality conditions of set-valued optimization problem are established. The results in this paper generalize some known results in some literature. 2 Preliminaries Let X be a nonempty set, and let Y and Z be two ordered linear spaces. Let 0 stand for the zero element of every space. Let K be a nonempty subset in Y. The generated cone of K is defined as conek = {λa a K, λ 0}. A cone K Y is said to be pointed if K ( K) = {0}. A cone K Y is said to be nontrivial if K {0} and K Y. Let Y and Z stand for algebraic dual spaces in Y and Z, respectively. From now on, let C and D be nontrivial pointed convex cones in Y and Z, respectively. The algebraic dual cone C + and strictly algebraic dual cone C +i of C are defined as C + = {y Y y, y 0, y C} and C +i = {y Y y, y > 0, y C \ {0}}, where y, y denotes the value of the linear functional y at the point y. The meanings of D + and D +i are similar. Let K be a nonempty subset in Y. We denote by aff(k), span(k) and L(K) = span(k K) the affine hull, linear hull and generated linear subspace of K, respectively. Definition 2.1 [14] Let K be a nonempty subset in Y. The algebraic interior of K is the set cor(k) = {k K v Y, λ 0 > 0, λ [0, λ 0 ], k + λv K}. 3

4 Definition 2.2 [15,16] Let K be a nonempty subset in Y. The relative algebraic interior of K is the set icr(k) = {k K v L(K), λ 0 > 0, λ [0, λ 0 ], k + λv K}. LEMMA 2.1 Let K be a nonempty subset in Y. Then aff(k) = x + L(K), x K. (1) Proof Firstly, we will show that aff(k) x + L(K), x K. (2) Let y aff(k). Then, there exist k i K, α i R with n α i = 1 such that y = n α i k i. Thus, we obtain y = x + y x = x + n α i k i n α i x = x + n α i (k i x) x + span(k K) = x + L(K). Therefore, (2) holds. Finally, we will show that x + L(K) aff(k), x K. (3) Let z x + L(K). Then, there exist x i, y i K and λ i R(i = 1,, n) such that z = x + n λ i (x i y i ) = x + n λ i x i n λ i y i. (4) Clearly, 1 + n λ i n λ i = 1. (5) 4

5 It follows from (4) and (5) that z aff(k). Therefore, (3) holds. It follows from (2) and (3) that (1) holds. Remark 2.1 It follows from Lemma 2.1 that icr(k) = {k K v aff(k) k, λ 0 > 0, λ [0, λ 0 ], k + λv K}. Remark 2.2 It follows from Lemma 2.1 that if 0 K Y, then icr(k) = {k K v aff(k), λ 0 > 0, λ [0, λ 0 ], k + λv K}. Remark 2.3 If K be a nontrivial pointed cone in Y then 0 / icr(k). In fact, if 0 icr(k), then it follows from Remark 2.2 that aff(k) = K. (6) Since 0 K, it is clear that aff(k) = span(k). (7) By (6) and (7), we have span(k) = K. (8) Since K is nontrivial, by (8), there exists a nonzero k K such that k K, which contradicts that K is pointed. Remark 2.4 It is easy to check that icr(k) is a convex set and icr(k) {0} is a convex cone if K is a convex cone. LEMMA 2.2 Let K be a convex cone in Y. Then K + icr(k) = icr(k). (9) Proof Clearly, (9) holds if icr(k) =. 5

6 Now, let k 1 K and k 2 icr(k). k 2 icr(k) implies that k 2 K, v L(K), λ 0 > 0, λ [0, λ 0 ], we have k 2 + λv K. Since K is a convex cone, we obtain (k 1 + k 2 ) + λv = k 1 + (k 2 + λv) K + K K, which implies k 1 + k 2 icr(k). Therefore, K + icr(k) icr(k). (10) Since 0 K, we have icr(k) K + icr(k). (11) According to (10) and (11), (9) holds. LEMMA 2.3 Let K 1 and K 2 be two nonempty subsets in Y. Then aff(k 1 + K 2 ) = aff(k 1 ) + aff(k 2 ). (12) Proof Clearly, Now, we will show that aff(k 1 + K 2 ) aff(k 1 ) + aff(k 2 ). (13) aff(k 1 ) + aff(k 2 ) aff(k 1 + K 2 ). (14) Let y aff(k 1 ) + aff(k 2 ). Then, there exist k i 1 K 1, k j 2 K 2, λ i 1 R, λ j 2 R(i = 1, m 1 ; j = 1, m 2 ) with m 1 λ i 1 = 1 and m 2 j=1 λ j 2 = 1 such that m 1 m 2 m 1 m 2 y = λ i 1k1 i + λ j 2k j 2 = λ ij (k1 i + k2), j j=1 where λ ij = λ i 1λ j 2(i = 1,, m 1 ; j = 1,, m 2 ). j=1 6

7 It is easy to check that m 1 m 2 j=1 λ ij = 1 and k i 1 +k j 2 K 1 +K 2 (i = 1,, m 1 ; j = 1,, m 2 ). Hence, y aff(k 1 +K 2 ). Thus, (14) holds. By (13) and (14), (12) holds. LEMMA 2.4 Let K be a nonempty subset in Y. Then (a) aff(k) = k + aff(k K), k K. If K is convex and icr(k), then (b) icr(icr(k)) = icr(k); (c) aff(icr(k)) = aff(k). Proof (b) and (c) can be found in [5] and [9], respectively. We will only prove (a). Clearly, it follows from that Lemma 2.3 that aff(k) k aff(k) aff(k) aff(k K), k K, i.e., Finally, we will show that aff(k) k + aff(k K), k K. (15) k + aff(k K) aff(k), k K. (16) By Lemma 2.3, we only need to show that k + aff(k) aff(k) aff(k), k K. (17) Let y k + aff(k) aff(k). Then, there exist k i 1 K, k j 2 K, λ i 1 R, λ j 2 R(i = 1,, m 1 ; j = 1,, m 2 ) with m 1 λ i 1 = 1 and m 2 j=1 λ j 2 = 1 such that m 1 m 2 y = k + λ i 1k1 i λ j 2k2. j (18) j=1 7

8 Clearly, m 1 m λ i 1 λ j 2 = 1. (19) j=1 By (18) and (19), y aff(k). Thus, (17) holds. Hence, (16) holds. By (15) and (16), we obtain aff(k) = k + aff(k K), k K. LEMMA 2.5 Let K 1 and K 2 be two nonempty subsets in Y. Then aff(k 1 K 2 ) = aff(k 1 ) aff(k 2 ). (20) Proof Clearly, It follows from Lemma 2.3 that aff(k 1 K 2 ) = aff(k 1 {0} + {0} K 2 ). (21) aff(k 1 {0} + {0} K 2 ) = aff(k 1 {0}) + aff({0} K 2 ) = aff(k 1 ) {0} + {0} aff(k 2 ) = aff(k 1 ) aff(k 2 ) (22) By (21) and (22), (20) holds. LEMMA 2.6 Let K 1 and K 2 be two nontrivial convex cone in Y and Z. If icr(k 1 ) and icr(k 2 ), then icr(k 1 K 2 ) = icr(k 1 ) icr(k 2 ). (23) Proof Firstly, we will show that icr(k 1 ) icr(k 2 ) icr(k 1 K 2 ). (24) Let (k 1, k 2 ) icr(k 1 ) icr(k 2 ). Clearly, (k 1, k 2 ) K 1 K 2. Let (v 1, v 2 ) aff(k 1 K 2 ) (k 1, k 2 ). By Lemma 2.5, (v 1, v 2 ) (aff(k 1 ) k 1 ) (aff(k 2 ) k 2 ). Since k 1 icr(k 1 ), for 8

9 v 1 aff(k 1 ) k 1, there exists ε 1 0 > 0 such that k 1 + εv 1 K 1, ε [0, ε 1 0]. (25) Since k 2 icr(k 2 ), for v 2 aff(k 2 ) k 2, there exists ε 2 0 > 0 such that k 2 + εv 2 K 2,, ε [0, ε 2 0]. (26) Choose ε 3 0 = min{ε 1 0, ε 2 0}. It follows from (25) and (26) that there exists ε 3 0 > 0 such that (k 1, k 2 ) + ε(v 1, v 2 ) K 1 K 2, ε [0, ε 3 0], which implies (k 1, k 2 ) icr(k 1 K 2 ). Thus, (24) holds. Finally, we will show that icr(k 1 K 2 ) icr(k 1 ) icr(k 2 ). (27) Since icr(k 1 ) and icr(k 2 ), it follows from (24) that icr(k 1 K 2 ). Let (k 1, k 2 ) icr(k 1 K 2 ). Clearly, (k 1, k 2 ) K 1 K 2. Let (v 1, v 2 ) (aff(k 1 ) k 1 ) (aff(k 2 ) k 2 ). Clearly, (v 1, v 2 ) aff(k 1 K 2 ) (k 1, k 2 ). Therefore, there exists ε 0 > 0 such that (k 1, k 2 ) + ε(v 1, v 2 ) K 1 K 2, ε [0, ε 0 ], which implies k 1 + εv 1 K 1 (28) and k 2 + εv 2 K 2. (29) 9

10 By (28) and (29), we have (k 1, k 2 ) icr(k 1 ) icr(k 2 ). Therefore, (27) holds. By (24) and (27), (23) holds. LEMMA 2.7 [15,16] Let K be a convex set with icr(k) in Y. If 0 / icr(k), then there exists y Y \ {0} such that k, y 0, k K. 3 Generalized subconvexlike set-valued map In this section, firstly, we will introduce several classes of generalized convex set-valued maps. Secondly, we will discuss their relationships. Finally, we will obtain some properties of the generalized cone subconvexlike set-valued maps. From now on, let A be a nonempty subset in X, and let F : A 2 Y be a set-valued map. Write F (A) = F (x). We suppose that icr(c) icr(d). x A Definition 3.1 [11] A set-valued map F : A 2 Y is called C-convexlike if, x 1, x 2 A and λ (0, 1), λf (x 1 ) + (1 λ)f (x 2 ) F (A) + C, Remark 3.1 It follows from [11] that F : A 2 Y is C-convexlike if and only if F (A) + C is a convex set. Definition 3.2 [9] A set-valued map F : A 2 Y is called C-subconvexlike if, c icr(c) such that, x 1, x 2 A, λ (0, 1), ε > 0, εc + λf (x 1 ) + (1 λ)f (x 2 ) F (A) + C. Remark 3.2 Proposition 3.2 of [9] shows that F : A 2 Y is C-subconvexlike if and only if 10

11 F (A) + icr(c) is a convex set. Definition 3.3 A set-valued map F : A 2 Y is called generalized C-subconvexlike if cone(f (A)) + icr(c) is a convex set. THEOREM 3.1 If the set-valued map F : A 2 Y is C-convexlike, then F is C-subconvexlike. Proof It follows from the C-convexlikeness of F that F (A) + C is a convex set. We need to prove that F (A) + icr(c) is a convex set. Indeed, let m i F (A) + icr(c)(i = 1, 2), λ (0, 1). Then, there exist x i A, c i icr(c)(i = 1, 2) such that m i = y i + c i, y i F (x i ), i = 1, 2. By Remark 3.1 and Lemma 2.2, λm 1 + (1 λ)m 2 = λ(y 1 + c 1 ) + (1 λ)(y 2 + c 2 ) = [λy 1 + (1 λ)y 2 ] + [λc 1 + (1 λ)c 2 ] (F (A) + C) + icr(c) = F (A) + (C + icr(c)) = F (A) + icr(c) Remark 3.3 The following example shows that a C-subconvexlike set-valued map may not be C-convexlike. Thus, the C-subconvexlikeness is a generalization of C-convexlikeness. Example 3.1 Let X = Y = R 2, C = {(y 1, 0) y 1 0} and A = {(1, 0), (0, 2)}. The set-valued map F : A 2 Y is defined as follows: F (1, 0) = {(y 1, y 2 ) 1 < y 1 2, 0 y 2 1} {(1, 0), (1, 1)}; F (0, 2) = {(y 1, y 2 ) 1 < y 1 2, 1 y 2 2} {(1, 2), (1, 1)}. Clearly, F (A) + icr(c) is a convex set, and F (A) + C is not a convex set. Therefore, F is C-subconvexlike. However, F is not C-convexlike. THEOREM 3.2 If the set-valued map F : A 2 Y is C-subconvexlike, then F is generalized C-subconvexlike. Proof It follows from the C-subconvexlikeness of F that F (A)+icr(C) is a convex set. We need 11

12 to prove that cone(f (A)) + icr(c) is a convex set. Indeed, let m i cone(f (A)) + icr(c)(i = 1, 2), λ (0, 1). Then, there exist ρ i 0, x i A, c i icr(c)(i = 1, 2) such that m i = ρ i y i + c i, y i F (x i ), i = 1, 2. Case 1. ρ 1 = 0 or ρ 2 = 0. Clearly, λm 1 + (1 λ)m 2 cone(f (A)) + icr(c). Case 2. ρ 1 > 0 and ρ 2 > 0. Since F (A) + icr(c) is a convex set, we have λm 1 + (1 λ)m 2 = λ(ρ 1 y 1 + c 1 ) + (1 λ)(ρ 2 y 2 + c 2 ) = λρ 1 (y ρ 1 c 1 ) + (1 λ)ρ 2 (y ρ 2 c 2 ) λρ 1 = [λρ 1 + (1 λ)ρ 2 ]{ λρ 1 +(1 λ)ρ 2 (y ρ 1 c 1 ) + (1 λ)ρ 2 λρ 1 +(1 λ)ρ 2 (y ρ 2 c 2 )} [λρ 1 + (1 λ)ρ 2 ](F (A) + icr(c)) cone(f (A)) + icr(c). Case 1 and Case 2 show that cone(f (A)) + icr(c) is a convex set. Remark 3.4 The following example shows that a generalized C-subconvexlike set-valued map may be not C-subconvexlike. Thus, generalized C-subconvexlikeness is a generalization of C-subconvexlikeness. Example 3.2 Let X = Y = R 2, C = {(y 1, 0) y 1 0} and A = {(1, 0), (0, 2)}. The set-valued map F : A 2 Y is defined as follows: F (1, 0) = {(y 1, y 2 ) y 2 y 1 + 2, y 1 0, y 2 1}; F (0, 2) = {(y 1, y 2 ) y 2 y 1 + 2, y 1 1, y 2 0}. Clearly, cone(f (A))+icr(C) is a convex set, and F (A)+icr(C) is not a convex set. Therefore, F is generalized C-subconvexlike. However, F is not C-subconvexlike. Next, we give some equivalent properties of generalized C-subconvexlike set-valued maps. THEOREM 3.3 A set-valued map F : A 2 Y is generalized C-subconvexlike if and only if, c icr(c), x 1, x 2 A, λ (0, 1), c + λf (x 1 ) + (1 λ)f (x 2 ) cone(f (A)) + icr(c). (30) 12

13 Proof Necessity. Let c icr(c), x 1, x 2 A, y 1 F (x 1 ), y 2 F (x 2 ) and λ (0, 1). Clearly, y 1 + c cone(f (A)) + icr(c) (31) and y 2 + c cone(f (A)) + icr(c). (32) Since F is generalized C-subconvexlike, it follows from (31) and (32) that c + λy 1 + (1 λ)y 2 = λ(y 1 + c) + (1 λ)(y 2 + c) cone(f (A)) + icr(c). Therefore, (30) holds. Sufficiency. Let m i cone(f (A)) + icr(c)(i = 1, 2), λ (0, 1). Then, there exist ρ i 0, x i A, c i icr(c)(i = 1, 2) such that m i = ρ i y i + c i, y i F (x i ), i = 1, 2. Case 1. ρ 1 = 0 or ρ 2 = 0. Clearly, λm 1 + (1 λ)m 2 cone(f (A)) + icr(c). Case 2. ρ 1 > 0 and ρ 2 > 0. Since F (A) + icr(c) is a convex set, we have λm 1 + (1 λ)m 2 = λ(ρ 1 y 1 + c 1 ) + (1 λ)(ρ 2 y 2 + c 2 ) = [λc 1 + (1 λ)c 2 ] + [λρ 1 y 1 + (1 λ)ρ 2 y 2 ] 1 λρ 1 (1 λ)ρ 2 = [λρ 1 +(1 λ)ρ 2 ]{ [λc 1 +(1 λ)c 2 ]+ y 1 + y 2 }. λρ 1 + (1 λ)ρ 2 λρ 1 + (1 λ)ρ 2 λρ 1 + (1 λ)ρ 2 (33) Clearly, 1 λρ 1 + (1 λ)ρ 2 [λc 1 + (1 λ)c 2 ] icr(c), λρ 1 λρ 1 + (1 λ)ρ 2 (0, 1), (1 λ)ρ 2 λρ 1 + (1 λ)ρ 2 (0, 1) 13

14 and By (30) and (33), we obtain λρ 1 (1 λ)ρ 2 + = 1. λρ 1 + (1 λ)ρ 2 λρ 1 + (1 λ)ρ 2 λm 1 + (1 λ)m 2 [λρ 1 + (1 λ)ρ 2 ](cone(f (A)) + icr(c)) cone(f (A)) + icr(c). Case 1 and Case 2 shows that cone(f (A))+icr(C) is a convex set. Therefore, F is generalized C-subconvexlike. THEOREM 3.4 The following statements are equivalent: (a) F : A 2 Y is generalized C-subconvexlike; (b) c icr(c), x 1, x 2 A, λ (0, 1), c + λf (x 1 ) + (1 λ)f (x 2 ) cone(f (A)) + icr(c); (c) c icr(c), x 1, x 2 A, λ (0, 1), ε > 0, εc + λf (x 1 ) + (1 λ)f (x 2 ) cone(f (A)) + C; (d) x 1, x 2 A, λ (0, 1), c C, ε > 0, εc + λf (x 1 ) + (1 λ)f (x 2 ) cone(f (A)) + C. Proof By Theorem 3.3, (a) (b). The implications (b) (c) (d) are clear. Therefore, we need to prove that (d) (b). Let c icr(c), x 1, x 2 A and λ (0, 1). Then, c C, ε > 0, εc + λf (x 1 ) + (1 λ)f (x 2 ) cone(f (A)) + C. (34) Since c icr(c) = icr(icr(c)), by Lemma 2.4 (a) and (c), for c = 0 c C C aff(c C) = aff(c) c = aff(icr(c)) c, there exists λ 0 > 0 such that c + λ 0 ( c ) icr(c). 14

15 Write c = c + λ 0 ( c ). Clearly, c = c + λ 0 c. By (34) and Lemma 2.2, we obtain c+λf (x 1 )+(1 λ)f (x 2 ) = (c+λ 0 c )+λf (x 1 )+(1 λ)f (x 2 ) = [λ 0 c +λf (x 1 )+(1 λ)f (x 2 )]+c cone(f (A)) + C + c cone(f (A)) + C + icr(c) = cone(f (A)) + icr(c). 4 Optimality conditions In order to establish optimality conditions for vector optimization problems with set-valued maps, we need to present a separation property for a generalized C-subconvexlike set-valued map. Now, we consider the following two systems: System 1: There exists x 0 A such that F (x 0 ) ( icr(c)) ; System 2: There exists y C + \ {0} such that y, y 0, y F (A). THEOREM 4.1 (i) Suppose that F : A 2 Y is generalized C-subconvexlike and icr(conef (A)+ icr(c)). If System 1 has no solutions, then System 2 has a solution. (ii) If y C +i is a solution of System 2, then System 1 has no solutions. Proof (i) Firstly, we assert that 0 / cone(f (A))+icr(C). Otherwise, there exist x 0 A, α 0 such that 0 αf (x 0 ) + icr(c). Case 1. If α = 0, then 0 icr(c), which contradicts 0 / icr(c) (see Remark 2.3). Case 2. If α > 0, there exists y 0 F (x 0 ) such that y 0 1 icr(c) icr(c), which contradicts α that System 1 has no solutions. Thus, we obtain 0 / cone(f (A)) + icr(c). 15

16 Since F is generalized C-subconvexlike, cone(f (A)) + icr(c) is a convex set. By Lemma 2.7, there exists y Y \ {0} such that y, y 0, y cone(f (A)) + icr(c). So, Letting α = 0 in (35), we obtain αf (x) + c, y 0, x A, c icr(c), α 0. (35) c, y 0, c icr(c). (36) We will show that y C +. Otherwise, there exists c C such that c, y < 0. Hence, θc, y < 0, θ > 0. By Lemma 2.2, we have θc + c icr(c), θ > 0, c icr(c). (37) It follows from (36) and (37) that θ c, y + c, y 0, θ > 0, c icr(c). (38) We note that (38) does not hold when θ > c,y c,y 0. Therefore, y C +. Letting α = 1 in (35), we have F (x) + c, y 0, x A, c icr(c). Letting c 0 icr(c) and λ n > 0 with lim n λ n = 0, we have F (x) + λ n c 0, y 0, x A, n N. (39) It follows from (39) that F (x), y 0, x A. 16

17 (ii) Since y C +i is a solution of System 2, we have y, y 0, y F (A). (40) Now, we suppose that System 1 has a solution. Then, there exists x 0 A such that F (x 0 ) ( icr(c)). Thus, there exists y 0 F (x 0 ) such that y 0 icr(c). Clearly, y 0 0. So, we have y 0, y < 0, which contradicts (40). Remark 4.1 In (i), the condition icr(cone(f (A)) + icr(c)) can be replaced by the condition aff(c)) = aff(cone(f (A)) + icr(c)). In fact, when aff(c) = aff(cone(f (A)) + icr(c)), icr(c) icr(cone(f (A)) + icr(c)). Since icr(c), icr(cone(f (A)) + icr(c)). However, the following example shows that the condition icr(cone(f (A)) + icr(c)) is weaker than the condition aff(c) = aff(cone(f (A)) + icr(c)). Example 4.1 In Example 3.2, it is clear that icr(cone(f (A))+icr(C)) = {(y 1, y 2 ) y 1 > 0, y 2 0} =. However, aff(c) = {(y 1, y 2 ) y 1 R, y 2 = 0} = R 2 = aff(cone(f (A)) + icr(c)). Remark 4.2 Since cone(f (A)) + icr(c) is a nonempty convex subset in Y, The condition icr(cone(f (A)) + icr(c)) can be deleted if Y is a finite-dimensional space. Remark 4.3 Theorem 4.1 improves Theorem 2.1 of [1], Theorem 3.5 of [5] and Theorem 3.9 of [9]. Let F : A 2 Y and G : A 2 Z be two set-valued maps. Now, we consider the following vector optimization problem with set-valued maps: (VP) min F (x) s.t. G(x) D. The feasible set of (VP) is denoted by S = {x A G(x) D }. Now, we define W Min(F (A), C) = {y 0 F (A) (y 0 F (A)) icr(c) = }. 17

18 Definition 4.1 A point x 0 is called a weakly efficient solution of (VP) if there exists x 0 S such that F (x 0 ) W Min(F (S), C). A point pair (x 0, y 0 ) is called a weak minimizer of (VP) if y 0 F (x 0 ) W Min(F (S), C). Let I(x) = F (x) G(x), x A. It is clear that I is a set-valued map from A to Y Z, where Y Z is a linear space with nontrivial pointed convex cone C D. The algebraic dual space of Y Z is Y Z, and the algebraic dual cone of C D is C + D +. By Definition 3.3, we say that the set-valued map I : A 2 Y Z is generalized C D- subconvexlike if cone(i(a)) + icr(c D) is a convex set in Y Z. Now we present a necessary optimality condition for (VP) as follows: THEOREM 4.2 Suppose that the following conditions hold: (i) (x 0, y 0 ) is a weak minimizer of (VP); (ii) I 1 : A 2 Y Z is generalized C D-subconvexlike, where I 1 (x) = (F (x) y 0 ) G(x), x A; (iii) icr(cone(i 1 (A)) + icr(c D)). Then, there exists (y, z ) C + D + with (y, z ) (0, 0) such that Proof According to Definition 4.1, we have inf ( F (x), x A y + G(x), z ) = y 0, y, inf G(x 0 ), z = 0. (y 0 F (S)) icr(c) =. (41) Clearly, I 1 (x) = I(x) (y 0, 0), x A. We assert that I 1 (x) icr(c D) =, x A. (42) 18

19 Otherwise, there exists x A such that I 1 (x) icr(c D). By Lemma 2.6, icr(c D) = icr(c) icr(d). Therefore, I 1 (x) (icr(c) icr(d)). (43) By (43), we obtain (y 0 F (x)) icr(c) (44) and G(x) icr(d). (45) It follows from (45) that x S. Thus, by (44), we have (y 0 F (S)) icr(c), which contradicts (41). Therefore, (42) holds. By Theorem 4.1, there exists (y, z ) C + D + with (y, z ) (0, 0) such that I 1 (x), (y, z ) 0, x A, i.e., F (x), y + G(x), z y 0, y, x A. (46) Since x 0 S, there exists p G(x 0 ) such that p D. Because z D +, we obtain p, z 0. Taking x = x 0 in (46), we get y 0, y + p, z y 0, y It follows that p, z 0. So, p, z = 0. Thus, we have y 0, y F (x 0 ), y + G(x 0 ), z. (47) 19

20 Therefore, it follows from (46) and (47) that Taking again x = x 0 in (46), we obtain inf ( F (x), x A y + G(x), z ) = y 0, y. y 0, y + G(x 0 ), z y 0, y. So, G(x 0 ), z 0. We have shown that there exists p G(x 0 ) such that p, z = 0. Thus, we have inf G(x 0 ), z = 0. The following example shows that the conditions of Theorem 4.2 can be satisfied. Example 4.2 Let X = Y = Z = R 2, C = D = {(y 1, 0) y 1 0} and A = {(1, 0), (1, 2)}. The set-valued map F : X 2 Y on A is defined as follows: F (1, 0) = {(y 1, y 2 ) y 1 = 1, 0 y 2 1}, F (1, 2) = {(y 1, y 2 ) y 1 > 1, 0 y 2 y 1 + 2}. The set-valued map G : X 2 Y on A is defined as follows: G(1, 0) = {(y 1, y 2 ) y 1 0, 0 y 2 y 1 + 1}, G(1, 2) = {(y 1, y 2 ) y 1 1, y y 2 1}. Let x 0 = (1, 0) and y 0 = (1, 0) F (x 0 ). Clearly, all conditions of Theorem 4.2 are satisfied. Therefore, there exist y : (y 1, y 2 ), y = y 1 + y 2 and z : (y 1, y 2 ), z = y 1 + y 2 such that and inf x A ( F (x), y + G(x), z ) = y 0, y inf G(x 0 ), z = 0. Remark 4.4 Theorem 4.2 improves Theorem 3.2 of [1], Theorem 3.1 of [3] and Theorem 4.2 of [9]. 20

21 We also show a sufficient optimality condition for (VP) as follows: THEOREM 4.3 Suppose that the following conditions hold: (i) x 0 S; (ii) there exist y 0 F (x 0 ) and (y, z ) C +i D + such that Then (x 0, y 0 ) is a weak minimizer of (VP). Proof By condition (ii), we have inf ( F (x), x A y + G(x), z ) y 0, y. F (x) y 0, y + G(x), z 0, x A. (48) We assert (x 0, y 0 ) is a weak minimizer of (VP). Otherwise, there exists x S such that (y 0 F (x)) icr(c). Therefore, there exists y F (x) such that y 0 y icr(c) C \{0}. Thus, we obtain y y 0, y < 0. (49) Since x S, there exists q G(x) such that q D. Hence, q, z 0. (50) Adding (49) to (50), we have y y 0, y + q, z < 0, which contradicts (48). Therefore, (x 0, y 0 ) is a weak minimizer of (VP). The following example shows that the conditions of Theorem 4.3 can be satisfied. Example 4.3 Let X = Y = Z = R 2, C = D = {(y 1, 0) y 1 0} and A = {(1, 0), (1, 2)}. The set-valued map F : X 2 Y on A is defined as follows: F (1, 0) = {(y 1, y 2 ) y 1 1, y 1 y 2 2}, F (1, 2) = {(y 1, y 2 ) y 1 2, 1 y 2 y 1 }. 21

22 The set-valued map G : X 2 Y on A is defined as follows: G(1, 0) = {(y 1, y 2 ) 1 y 1 0, y 2 = 0}, G(1, 2) = {(y 1, y 2 ) 1 y 1 0, 0 y 2 1}. Let x 0 = (1, 0), y 0 = (1, 1) F (x 0 ), (y 1, y 2 ), y = y 1 + y 2 and (y 1, y 2 ), z = y 1. Clearly, all conditions of Theorem 4.3 are satisfied. Therefore, (x 0, y 0 ) is a weak minimizer of (VP). Acknowledgements This work was supported by the National Nature Science Foundation of China (Grant No ) and the Natural Science Foundation of Chongqing (Grant No. CSTC, 2009BB8240). References [1] Z.M. Li, The optimality conditions for vector optimization of set-valued maps, J. Math. Anal. Appl. 237 (1999), pp [2] Z.M. Li, A theorem of the alternative and its Application to the optimization of set-valued maps, J. Optim. Theory Appl. 100 (1999), pp [3] Y.W. Huang and Z.M. Li, Optimality condition and Lagrangian multipliers of vector optimization with set-valued maps in linear spaces, Oper. Res. Tran. 5 (2001), pp [4] Y.W. Huang, Generalized cone-subconvexlike set-valued maps and applications to vector optimization, J. Chongqing. University (Eng.Ed). 1 (2002), pp [5] M. Adán and V. Novo, Partial and generalized subconvexity in vector optimization problems, J. Convex. Anal. 8 (2001), pp [6] M. Adán and V. Novo, Efficient and weak efficient points in vector optimization with generalized cone convexity, Appl. Math. Lett. 16 (2003), pp

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