ACTA MATHEMATICAE APPLICATAE SINICA Nov., ( µ ) ( (

35 Þ 6 Ð Å Vol. 35 No. 6 2012 11 ACTA MATHEMATICAE APPLICATAE SINICA Nov., 2012 È ÄÎ Ç ÓÑ ( µ 266590) (E-mail: jgzhu980@yahoo.com.cn) Ð ( Æ (Í ), µ 266555) (E-mail: bbhao981@yahoo.com.cn) Þ» ½ α- Ð Æ Ä Õ Å α- Ð Ø Æ Ä ½ Ö Ö» Ï ½ Ó Á Ï α- Ð Ø Æ Ä Ö ¾ Ó MR(2000) Å 90C26; 90C29 É O224 1 Å ËÄÜÅ Ú Ð ±Ô¾ Ý Â«Å ËÄ ÜÅ ¾ÒÛ ÐÅÎ ±Ô¾ Ù Đ ¾  ߱ ² Ç ÉÐ ¾Â ¼ ±«¾ÒÈ º [1 5].  ¾ Ï Đ ¾ ±º ±Ô¾ Ù Sheng [6] ß¼ α- ѹ ÅÉ Ü É ¾ Â Ê α- ѹ ÅÉ ¾ÇÄ Ìß¼ Đ ¾ α- ѹ Å À ¾Å Ö «α- Ñ Ù ¹ Å Đ ¾ ¼ ¾ ÔÂ Ë 2009 10 19 ¹ 2011 7 15 ¹ ƽ»µ (61101208).

1092 35 Þ 2 Ã Ì X, Y, Z Ü ÅÈ «X, Y, Z Ê X, Y, Z ¾È S Y, K Z ÆŹ S = { s Y : s (s) 0, s S } Ê S ¾ ¹ s (s) Ê s s ¾ Ü S ¾Ñ ¹Ê S i = { s Y : s (s) > 0, s S \ {θ Y } } ( θ Y Ê Y ¾ «). D X ¾ ¼ F : D 2 Y Ë G : D 2 Z Đ ³ F(D) = F(x). F ¾Æ Ë Æ ÜÊ x D graphf = { (x, y) D Y : x D, y F(x) }, epi D F = { (x, y) D Y : x D, y F(x) + S }. 1.1 [6] η : X X X «É D X η- ž α > 0 É F : D 2 Y α- Ñ S- ž ½ x 1, x 2 D, λ [0, 1], λ α F(x 1 ) + (1 λ α )F(x 2 ) F(x 2 + λη(x 1, x 2 )) + S. 1.2 η : X X X «É D X η- ž α > 0 É F : D 2 Y α- Ñ S- ž ½ θ ints, ¼ x 1, x 2 D, λ [0, 1], ε > 0 εθ + λ α F(x 1 ) + (1 λ α )F(x 2 ) F(x 2 + λη(x 1, x 2 )) + S. 1.3 [7] C X Y, C, (x 0, y 0 ) cl C, C (x 0, y 0 ) ¾ (1, α)- Ù ²¹ T (1,α) C ((x 0, y 0 )) ÜÊ X Y ¾ ¼ ³ (x, y) T (1,α) C ((x 0, y 0 )) ³ Ø h n 0 +, (x n, y n ) C, (x n, y n ) (x 0, y 0 ) (n + ), { xn x 0 (x, y) = lim, y n y } 0 n + h n h α. n 1.4 [8] F : S 2 Y, (x 0, y 0 ) graphf. epi S D α F((x 0, y 0 )) = T (1,α) epi SF ((x 0, y 0 )) ¾ É D α F((x 0, y 0 )) «Ê F (x 0, y 0 ) ¾ α- Ñ Ù 1.5 [6] C X Y. C (1, α)- Å ½ Û¾ (x 1, y 1 ), (x 2, y 2 ) C, λ [0, 1], (λx 1 + (1 λ)x 2, λ α y 1 + (1 λ α )y 2 ) C. Ì N ¾ ¾ Ì ³ Ì A, B D, A + B = {a + b : a A, b B}, A B = {(a, b) : a A, b B}. A R ( R = (, + )), A 0, Û a A, a 0. A B ² a A, b B a b.

6 ³ Ã Ê Æ Ä Ï Á 1093 3 Í À Á 2.1 η : X X X «É D X η- ž F : D 2 Y α- Ñ S- ž epi D F α- Ñ Å Ò µ (x 1, y 1 ), (x 2, y 2 ) epi D F, y 1 F(x 1 )+S, y 2 F(x 2 )+S. ÝÊ D X η- ž Ú x 2 +λη(x 1, x 2 ) D. ÝÊ F : D 2 Y α- Ñ S- ž Ú θ ints, ¼ x 1, x 2 D, λ [0, 1], ε > 0 εθ+λ α F(x 1 )+(1 λ α )F(x 2 ) F(x 2 + λη(x 1, x 2 )) + S. θ ints, ³ S ÆŹ Á εθ ints, Y ¾ ¾ V ¼ εθ + V ints. V ¾ ÐÚ ¹ ¾ ε ¼ 2εθ V. Ý εθ 2εθ = εθ ints. Ú Á λ α y 1 + (1 λ α )y 2 λ α (F(x 1 ) + S) + (1 λ α )(F(x 2 ) + S) =λ α F(x 1 ) + (1 λ α )F(x 2 ) + S =εθ + λ α F(x 1 ) + (1 λ α )F(x 2 ) + S εθ εθ + λ α F(x 1 ) + (1 λ α )F(x 2 ) + S + ints F(x 2 + λη(x 1, x 2 )) + S + S + ints F(x 2 + λη(x 1, x 2 )) + S + ints F(x 2 + λη(x 1, x 2 )) + ints F(x 2 + λη(x 1, x 2 )) + S [x 2 + λη(x 1, x 2 ), λ α y 1 + (1 λ α )y 2 ] epi D F, Õ epi D F α- Ñ Å Á 2.2 [6] C X Y «η ( [6] ¾Â C 3 ) ¾ α- Ñ Å (x 0, y 0 ) clc, T (1,α) C ((x 0, y 0 )) Õ Â«η ¾ α- Ñ Å Á 2.3 X, Y Ü ÅÈ «S Y ÆŹ³ int S. D X η- ž F : D 2 Y D α- Ñ S- ž D α F((x 0, y 0 ))(D)+intS Å D α F((x 0, y 0 ))(D) = D α F((x 0, y 0 ))(x). x D Ò λ (0, 1) λ α (0, 1). v 1, v 2 D α F((x 0, y 0 ))(D) + ints, x i D, y i D α F((x 0, y 0 ))(x i ), s i ints ¼ v i = y i + s i, i = 1, 2. s 0 = λ α s 1 + (1 λ α )s 2, ÝÊ int S Å Á s 0 ints. x i D, y i D α F((x 0, y 0 ))(x i ), i = 1, 2. Î (x 1, y 1 ) T (1,α) epi ((x DF 0, y 0 )), (x 2, y 2 ) T (1,α) epi ((x DF 0, y 0 )). F D S- Å ¾ ß 2.1 ß 2.2 T (1,α) epi ((x DF 0, y 0 )) Â«η ¾ α- Ñ Å Á Ú [x 2 + λη(x 1, x 2 ), λ α y 1 + (1 λ α )y 2 ] T (1,α) epi DF ((x 0, y 0 )). λ α D α F((x 0, y 0 ))(x 1 ) + (1 λ α )D α F((x 0, y 0 ))(x 2 ) D α F((x 0, y 0 ))(x 2 + λη(x 1, x 2 )) + S.

1094 35 Þ λ α v 1 + (1 λ α )v 2 =λ α y 1 + (1 λ α )y 2 + S 0 λ α D α F((x 0, y 0 ))(x 1 ) + (1 λ α )D α F((x 0, y 0 ))(x 2 ) + ints D α F((x 0, y 0 ))(x 2 + λη(x 1, x 2 )) + S + ints D α F((x 0, y 0 ))(x 2 + λη(x 1, x 2 )) + ints D α F((x 0, y 0 ))(D) + ints. Á D α F((x 0, y 0 ))(D) + ints Å Á 2.4 ( ) D X η- ž S Y ÆŹ³ int S. ½ F : D 2 Y D α- Ñ S- ž ¾ (i) Ë (ii) ³ à (i) x D, ¼ D α F((x 0, y 0 ))(x) ( ints) ; (ii) y S \ {0}, ¼ D α F((x 0, y 0 ))(x), y 0, x D. Ò (i) Ë (ii) à [2] ¾ß 1.1 x D, y S \ {0}, µ D α F((x 0, y 0 ))(x) ( ints) ¼ 0 y, µ < 0, Á (i) Ë (ii) à (i) D α F((x 0, y 0 ))(x) ( ints) =, x D. (3.1) 0 D α F((x 0, y 0 ))(D) + ints. ½ 0 D α F((x 0, y 0 ))(D) + ints, x D, y D α F((x 0, y 0 ))(x) ¼ 0 y + ints, y ints, ÐÚ¼¹ y D α F((x 0, y 0 ))(x) (int S), (3.1) Ú 0 D α F((x 0, y 0 ))(D) + ints. ß 2.3 D α F((x 0, y 0 ))(D) + ints Å Ú Å y S \ {0} ¼ y + εθ, y 0, θ ints, ε > 0, y D α F((x 0, y 0 ))(D). (3.2) (3.2) ε + ¼ θ, y 0, θ ints. Û¾ θ S = cls = clints, θ, y 0. Ý y S \ {0}. (3.2) ε 0, y, y 0, y D α F((x 0, y 0 ))(D). Ý (ii) 4 ÏÆ ß º ÍÀ (VOP) min F(x), s.t. x A

6 ³ Ã Ê Æ Ä Ï Á 1095 A = { x X : G(x) ( K) }, F(A) = F(x). x A 4.1 [2] x 0 A Ê (VOP) ¾ Ô ºÈ y 0 F(x 0 ) ¼ (F(A) y 0 ) ( ints) =. Á 4.2 D X η- ž F : D 2 Y, G : D 2 ÊÅ Ý Z Ù ¾ Đ x 0 A Ê (VOP) ¾ Ô z 0 G(x 0 ) ( K), ϕ (x) = D α F((x 0, y 0 ))(x) (D α G((x 0, z 0 ))(x) + G(x 0 ) ( K)) : D 2 Y Z Ê α- Ñ S K- ž (s, k ) S K, ³ (s, k ) (θ Y, θ Z ) ( θ Y Ê Y ¾ «), ¼ [ inf s (D α F((x 0, y 0 ))(x)) + k (D α G((x 0, z 0 ))(x)) ] 0, (4.3) x D ³ k (G(x 0 ) ( K)) = {0}, (4.4) s (D α F((x 0, y 0 ))(x)) = s (y), k (D α G((x 0, z 0 ))(x)) = y D α F((x 0,y 0))(x) z D α G((x 0,z 0))(x) k (z), D α F((x 0, y 0 ))(x) (D α G(x 0, z 0 )(x) + G(x 0 ) ( K)) = (y, z). y D α F((x 0,y 0))(x),z D α G(x 0,z 0)(x)+G(x 0) ( K) Ò Ü 4.1 º y F(x 0 ), ¼ (F(A) y 0 ) ( ints) =. ϕ (X) = ϕ (x). ϕ (X) [ (ints intk) ] =. x X ϕ (X) [ (int S intk) ], y D α F((x 0, y 0 ))(x), z D α G((x 0, z 0 ))(x), z 0 G(x 0) ( K) ¼ (y, z + z 0) int(s K). z D α G((x 0, z 0 ))(x), Æ {t n }, t n 0, x n X, z n G(x n ) + K, x n x 0, z n z 0, ¼ (x, z) = lim n (t n(x n x 0 ), t α n(z n z 0 )). Ú z n z 0 +z 0 intk, z n z 0 z 0 intk K(n N). µ z n = z n+k n G(x n )+ K, k n K, z n G(x n)(n N), z n K k n z n K, Ú G(x n ) ( K), x n A (n N). y D α F((x 0, y 0 ))(x) ints, Ú t n, x n A, y n F(x n ) + S, ¼ (x, y) = lim n (t n(x n x 0 ), t α n (y n y 0 )).

1096 35 Þ Ú M N, y n {y 0 } ints, n M. (F(A) y 0 ) ( ints) = Á φ (X) [ (ints intk)] =. ß 2.4 (s, k ) S K, ³ (s, k ) (θ Y, θ Z ) ¼ s ( D α F((x 0, y 0 ))(x)) + k (D α G((x 0, z 0 ))(x) + G(x 0 ) ( K)) 0, x D. (4.5) (4.5) µ x = θ Y, ¼¹ k (G(x 0 ) ( K)) 0. x 0 A, k ¾ Ü k (G(x 0 ) ( K)) 0. Ý (4.4) (4.4) ²¼ (4.5) ¼ (4.3) Ê 4.3 D X η- ž F : D 2 Y Ê Ù ¾ α- Ñ S- Å É G : D 2 Z Ê Ù ¾ α- Ñ K- Å É x 0 A Ê (VOP) ¾ Ô z 0 G(x 0 ) ( K), (s, k ) S K, ³ (s, k ) (θ Y, θ Z ), ¼ inf x D [s (D α F((x 0, y 0 ))(x)) + k (D α G((x 0, z 0 ))(x))] 0 (4.6) ³ k (G(x 0 ) ( K)) = {0}, (4.7) s (D α F((x 0, y 0 ))(x)) = s (y), k (D α G((x 0, z 0 ))(x)) = y D α F((x 0,y 0))(x) z D α G((x 0,z 0))(x) k (z). Ò φ (x) = D α F((x 0, y 0 ))(x) (D α G((x 0, z 0 ))(x) + G(x 0 ) ( K)). x i D, y i D α F((x 0, y 0 ))(x i ). ß 2.1 ß 2.2 T (1,α) T (1,α) epi DG (x 0, z 0 ) α- Ñ Å Á Õ Ë [x 2 + λη(x 1, x 2 ), λ α y 1 + (1 λ α )y 2 ] T (1,α) epi DF (x 0, y 0 ), [x 2 + λη(x 1, x 2 ), λ α z 1 + (1 λ α )z 2 ] T (1,α) epi DG (x 0, z 0 ), λ α D α F((x 0, y 0 ))(x 1 ) + (1 λ α )D α F((x 0, y 0 ))(x 2 ) epi DF (x 0, y 0 ) Ë D α F((x 0, y 0 ))(x 2 + λη(x 1, x 2 )) + S (4.8) λ α D α G((x 0, z 0 ))(x 1 ) + (1 λ α )D α G((x 0, z 0 ))(x 2 ) D α G((x 0, z 0 ))(x 2 + λη(x 1, x 2 )) + K. (4.9)

6 ³ Ã Ê Æ Ä Ï Á 1097 λ α (D α F((x 0, y 0 ))(x 1 ) (D α G((x 0, z 0 ))(x 1 ) + G(x 0 ) ( K))) + (1 λ α )(D α F((x 0, y 0 ))(x 2 ) (D α G((x 0, z 0 ))(x 2 ) + G(x 0 ) ( K))) =(λ α D α F((x 0, y 0 ))(x 1 ) + (1 λ α )D α F((x 0, y 0 ))(x 2 )) (λ α D α G((x 0, y 0 ))(x 1 ) + G(x 0 ) ( K)) + (1 λ α )(D α G((x 0, y 0 ))(x 2 ) + G(x 0 ) ( K)) =(λ α D α F((x 0, y 0 ))(x 1 ) + (1 λ α )D α F((x 0, y 0 ))(x 2 ))) (λ α D α G((x 0, y 0 ))(x 1 ) + (1 λ α )D α G((x 0, y 0 ))(x 2 ) + G(x 0 ) ( K)). (4.10) Ý (4.8), (4.9) Ë (4.10) ¼ λ α φ (x 1 ) + (1 λ α )φ (x 2 ) ( D α F((x 0, y 0 ))(x 2 + λη(x 1, x 2 )) + S) (D α G((x 0, z 0 ))(x 2 + λη(x 1, x 2 )) + G(x 0 ) ( K) + K ) =D α F((x 0, y 0 ))(x 2 + λη(x 1, x 2 )) (D α G((x 0, z 0 ))(x 2 + λη(x 1, x 2 )) + G(x 0 ) ( K)) + S K. Ú φ (x) α- Ñ S K- Å É 4.2 м Ë Ì [1] Jahn J, Raul R. Contingent Epiderivalives and Set-valued Optimization. Math. Meth. Oper. Res., 1997, 46: 193 211 [2] Li Z. A Theorem of the Alternative and Its Application to the Optimization of Set-valued Maps. J. Optim. Theory Appl., 1999, 100(2): 365 375 [3] Li Z F. Benson Proper Efficieny in the Vector Optimization of Set-valued Maps. J. Optim. Theory Appl., 1998, 98(3): 623 649 [4] Yang X M, Yang X Q, Chen G Y. Theorems of the Alternative and Optimization with Set-valued Maps. J. Optim. Theory Appl., 2000, 107(3): 627 640 [5] Lin L J. Optimization of Set-valued Functions. J. Math. Anal. Appl., 1994, 186: 30 51 [6] Sheng B H, Liu S Y. Kuhn-tucker Condition and Wolfe Duality of Preinvex Set-valued Optimization. Appl. Math. Mech. (Engl. Ed.), 2006, 27(12): 1655 1664 [7] Sheng B H, Liu S Y. The Generalized Optimality Conditions of Set-valued Optimization with Benson Proper Efficiency. Acta Math. Sci., 2003, 46(3): 611 620 [8] Sheng B H, Liu S Y. The Optimality Conditions of Nonconvex Set-valued Vector Optimization. Acta Math. Sci. B, 2002, 22(1): 47 55

1098 35 Þ The Derivative Type Optimality Conditions of Subpreinvex Set-valued Optimization ZHU Jianguang (College of Science, Shandong University of Science and Technology, Qingdao 266590) (E-mail: jgzhu980@yahoo.com.cn) Hao Binbin (College of Science, China University of Petroleum, Qingdao 266555) (E-mail: bbhao981@yahoo.com.cn) Abstract In this paper, the concept of α-order cone subpreinvex of set-valued maps is introduced, and a derivative type theorem of the alternative for cone subpreinvex set-valued maps by the α order tangent derivative; using this theorem, the derivative type necessary optimality condition of set-valued maps are given. Key words set-valued optimization; α order tangent derivative; cone subpreinvex; theorem of the alternative; weak efficient solution MR(2000) Subject Classification 90C26; 90C29 Chinese Library Classification O224