New Generalized Second-Order Contingent ... - Springer Link
J Optim Theory Appl (2012) 152:587–604 DOI 10.1007/s10957-011-9915-2
New Generalized Second-Order Contingent Epiderivatives and Set-Valued Optimization Problems S.J. Li · S.K. Zhu · K.L. Teo
Received: 13 December 2010 / Accepted: 3 August 2011 / Published online: 8 September 2011 © Springer Science+Business Media, LLC 2011
Abstract In this paper, we introduce the concept of a generalized second-order composed contingent epiderivative for set-valued maps and discuss its relationship to the generalized second-order contingent epiderivative. We also investigate some of its properties. Then, by virtue of the generalized second-order composed contingent epiderivative, we establish a unified second-order sufficient and necessary optimality condition for set-valued optimization problems, which is a generalization of the corresponding results in the literature. Keywords Set-valued optimization · Generalized second-order composed contingent epiderivative · Optimality conditions 1 Introduction In the last years, the set-valued optimization problems have been of great interest in research, and a lot of notions about derivatives for set-valued maps have been introduced and applied to set up the optimality conditions; see [1, 2, 4–6, 14, 15]. As we know, Aubin [1] first introduced a concept of a contingent derivative for set-valued maps, which is essentially a natural extension of tangents and plays an Communicated by Guang-ya Chen. S.J. Li () · S.K. Zhu College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China e-mail: [email protected] S.K. Zhu e-mail: [email protected] K.L. Teo Department of Mathematics and Statistics, Curtin University of Technology, G.P.O. Box U1987, Perth, WA 6845, Australia e-mail: [email protected]
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important role in set-valued analysis. Simultaneously, the contingent derivative of set-valued maps has also been generally used to express the first-order optimality conditions for set-valued optimization problems ever since the path-breaking paper [2]. However, the necessary conditions and sufficient conditions (see [2–4]) are not unified under standard assumptions. For solving this problem, Jahn and Rauh [5] proposed another derivative called the contingent epiderivative, which is actually an extension of directional derivatives in the single-valued and convex case. Moreover, they applied the contingent epiderivative to establish some unified sufficient and necessary conditions under suitable assumptions. But, unfortunately, since the contingent epiderivative of a set-valued map is a single-valued map, the conditions assuring the existence of the contingent epiderivative are hard to be satisfied; see [5–8]. In view of this problem, Chen and Jahn [6] introduced a generalized contingent epiderivative of set-valued maps and obtained the existence of the generalized contingent epiderivative with standard assumptions. Besides, they also established a unified necessary and sufficient optimality condition in terms of the generalized contingent epiderivative. Recently, the second-order optimality conditions in scalar and vector optimization problems have been widely investigated; for example, see [9–12]. If we analyze these developments, we can easily found that a second-order contingent set, introduced by Aubin [13], and a second-order asymptotic contingent cone, introduced by Penot [12], play a key role in establishing second-order optimality conditions. However, the second-order optimality conditions for set-valued optimization problems still need to be addressed. In [14], Durea used various derivative-like objects for set-valued maps to establish some second-order optimality conditions. Jahn et al. [15] proposed (generalized) second-order contingent epiderivatives for set-valued maps and applied these concepts to establish second-order optimality conditions. It is worth noting that (generalized) second-order contingent epiderivatives are proposed through the second-order contingent set, which is only a closed set and may not be a cone in general cases. Moreover, the second-order contingent set may not be convex even though the set what we consider be a convex set. Therefore, comparing with the (generalized) contingent epiderivative, we can not obtain some similar properties for the (generalized) second-order contingent epiderivative. Motivated by the work reported in [1, 2, 4–6, 15], we introduce a new generalized second-order contingent epiderivative called generalized second-order composed contingent epiderivative for set-valued maps, discuss its existence under standard assumptions and obtain some special properties. By virtue of the generalized second-order composed contingent epiderivative, we obtain a unified sufficient and necessary optimality condition for set-valued optimization problems, which is a generalization of corresponding results in [6]. Moreover, we also give an example to explain that the sufficient optimality condition improves and generalizes the corresponding result in [15]. The organization of this paper is as follows. In Sect. 2, we recall some basic concepts, introduce the generalized second-order composed contingent epiderivative and compare the generalized second-order composed contingent epiderivative with the generalized second-order contingent epiderivative. In Sect. 3, we give some special properties of the generalized second-order composed contingent epiderivative. In Sect. 4, we establish a unified second-order sufficient and necessary optimality condition for set-valued optimization problems.
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2 Notations and Preliminaries Throughout this paper, let X, Y be real normed spaces and S be a nonempty subset of X. 0X and 0Y denote the origins of X and Y , respectively. As usual, we denote by int S, cl S and cone S the interior, closure, and cone hull of S, respectively. Let C ⊂ Y be a pointed, closed and convex cone with apex at the origin and a nonempty interior int C, and let Y be partially ordered by C. Let F : X ⇒ Y be a set-valued map. The domain, graph, and epigraph of F are defined respectively by dom F := {x ∈ X | F (x) = ∅}, graph F := {(x, y) ∈ X × Y | y ∈ F (x)} and epi F := {(x, y) ∈ X × Y | y ∈ F (x) + C}. Now, we recall notions of a contingent cone and a second-order contingent set. Definition 2.1 [13] Let S be a nonempty subset of X, xˆ ∈ cl S, and ω ∈ X. (i) The contingent cone of S at xˆ is T (S, x) ˆ := v ∈ X | ∃tn ↓ 0, ∃vn → v, such that xˆ + tn vn ∈ S, ∀n ∈ N . (ii) The second-order contingent set of S at xˆ in the direction ω ∈ X is ˆ ω) T 2 (S, x, 1 2 := v ∈ X | ∃tn ↓ 0, ∃vn → v, such that xˆ + tn ω + tn vn ∈ S, ∀n ∈ N . 2 Proposition 2.1 [11] Let S ⊂ X be a convex set, xˆ ∈ S, and ω ∈ T (S, x). ˆ Then T T (S, x), ˆ ω = clcone cone(S − x) ˆ −ω and
T 2 (S, x, ˆ ω) ⊂ T T (S, x), ˆ ω .
ˆ ω), then Moreover (see Sect. 2.1 of Chap. 3 in [16]), if 0X ∈ T 2 (S, x, T 2 (S, x, ˆ ω) = T T (S, x), ˆ ω . Remark 2.1 T (S, x) ˆ is a cone, T 2 (S, x, ˆ ω) is a closed set but it is not necessarily a cone, and T (T (S, x), ˆ ω) is a closed cone. Obviously, T 2 (S, x, ˆ 0X ) = T (S, x) ˆ and also T (T (S, x), ˆ 0X ) = T (S, x) ˆ since T (S, x) ˆ is a closed cone. It follows from Proposition 2.1 that T (T (S, x), ˆ ω) is convex and is larger than T 2 (S, x, ˆ ω) when S is a 2 ˆ ω) and T (T (S, x), ˆ ω) may be nonempty only if ω ∈ T (S, x). ˆ convex subset. T (S, x, Furthermore (see more details in [16]), T 2 (S, x, ˆ ω) may be properly contained in T (T (S, x), ˆ ω) and not be convex even though S be a convex set. Next, we collect some equivalent characterizations for these contingent sets. Proposition 2.2 [17] The following statements are equivalent: (i) v ∈ T (S, x). ˆ (ii) There exist sequences λn → +∞ and xn ∈ S, such that xn → x and λn (xn − x) → v.
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Proposition 2.3 [17, 18] The following statements are equivalent: (i) v ∈ T 2 (S, x, ˆ ω). (ii) For every r > 0, there exist sequences tn ↓ 0, γn ↓ 0 and vn → v, such that tn 1 γn → r and xn := xˆ + tn ω + 2 rtn γn vn ∈ S. (iii) There exist sequences αn → +∞, βn → +∞ and xn ∈ S, such that xn → x, ˆ αn (xn − x) ˆ → ω and βn (αn (xn − x) ˆ − ω) → v.
βn αn
→ 2,
Now, we recall some standard notions. Definition 2.2 [18] Let Y be a topological linear space and be partially ordered by a convex cone C ⊂ Y with apex at the origin. (i) A sequence {yn } ⊂ Y is said to be C-decreasing iff ∀i, j ∈ N, i ≤ j implies yj ≤C yi . (ii) A subset D ⊂ Y is said to be C-lower bounded iff there exists a y ∈ Y , such that D ⊂ {y} + C. (iii) The convex cone C is said to be Daniel iff every C-decreasing and C-lower bounded sequence in Y converges to its infimum. Definition 2.3 [18] Let F : S ⇒ Y be a set-valued map and S ⊂ X be convex. Then F is said to be a C-function iff for every x1 , x2 ∈ S and every λ ∈ [0, 1], λF (x1 ) + (1 − λ)F (x2 ) ⊂ F λx1 + (1 − λ)x2 + C. It is obvious that F is a C-function if and only if epi F is convex. Definition 2.4 [6] Let X be a real linear space, and let Y be a real linear space and partially ordered by a convex cone C ⊂ Y with apex at the origin. A set-valued map f : X ⇒ Y is said to be (i) strictly positive homogeneous iff f (αx) = αf (x),
∀α > 0, ∀x ∈ X.
(ii) subadditive iff f (x1 ) + f (x2 ) ⊂ f (x1 + x2 ) + C,
∀x1 , x2 ∈ X.
Iff the properties (i) holds with α ≥ 0 and (ii) holds, f is said to be sublinear. Definition 2.5 [18] Let E be a nonempty subset of Y . A point y ∈ E is called a minimal element of E iff {y} − C \ {0Y } ∩ E = ∅. A point y ∈ E is called a weak minimal element of E iff {y}−int C ∩ E = ∅.
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As usual, we denote by Min E (WMin E) the set of all the minimal (weak minimal) elements of E. Specially, we set Min E = ∅ (WMin E = ∅) if E = ∅. As we know, the notion of the contingent epiderivative of a set-valued map, introduced by Jahn and Rauh [5], has some important properties, and unifies the necessary and sufficient optimality conditions for set-valued optimization problems. However, the existence of the contingent epiderivative in a general case is still an open question. In view of this problem, Chen and Jahn [6] introduced a generalized contingent epiderivative of set-valued maps. Also, Jahn et al. [15] introduced a generalized second-order contingent epiderivative of set-valued maps. Definition 2.6 [6] Let F : S ⇒ Y be a set-valued map and (x, ˆ y) ˆ ∈ graph F . The ˆ y) ˆ : generalized contingent epiderivative of F at (x, ˆ y) ˆ is the set-valued map Dg F (x, X ⇒ Y defined by ˆ y)(x) ˆ := Min y ∈ Y | (x, y) ∈ T epi F, (x, ˆ y) ˆ , ∀x ∈ X. Dg F (x, Definition 2.7 [15] Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . The generalized second-order contingent epiderivative of F at (x, ˆ y) ˆ in the direction (u, ˆ v) ˆ is the set-valued map Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ : X ⇒ Y defined by ˆ y, ˆ u, ˆ v)(x) ˆ := Min y ∈ Y | (x, y) ∈ T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ , Dg2 F (x,
∀x ∈ X.
Note that, it follows from Proposition 2.1 and Remark 2.1 that the second-order contingent set T 2 (S, x, ˆ ω), introduced by Aubin [13], is only a closed set, but it is not ˆ ω) may not be convex even if S be convex. necessarily a cone. Moreover, T 2 (S, x, Thus, comparing with the generalized contingent epiderivative, we can not obtain some similar properties of the generalized second-order contingent epiderivative. It is worth noting that T (T (S, x), ˆ ω) is always a closed cone. Moreover, T (T (S, x), ˆ ω) is a closed and convex cone when S is convex. Motivated by this idea, and Definitions 2.6 and 2.7, we introduce a new generalized second-order contingent epiderivative for set-valued maps as following. Definition 2.8 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . The generalized second-order composed contingent epiderivative of F at ˆ y, ˆ u, ˆ v) ˆ : X ⇒ Y defined (x, ˆ y) ˆ in the direction (u, ˆ v) ˆ is the set-valued map Dg F (x, by ˆ y, ˆ u, ˆ v)(x) ˆ := Min y ∈ Y | (x, y) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ , ∀x ∈ X. Dg F (x, Next, we discuss some relationships among these generalized epiderivatives. For this purpose, we need the following lemma. Lemma 2.1 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . Then we have the following statements: (i) T (epi F, (x, ˆ y)) ˆ + {0X } × C = T (epi F, (x, ˆ y)). ˆ
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(ii) T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ + {0X } × C = T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)). ˆ (iii) T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ + {0X } × C = T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ Proof Since (ii) or (iii) with (u, ˆ v) ˆ = (0X , 0Y ) implies (i), we only need to prove (ii) and (iii). For (ii), it is obvious that T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ ⊂ T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ 2 + {0X } × C. Conversely, for every (x, y) ∈ T (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ + {0X } × C, there exists ˆ y), ˆ (u, ˆ v) ˆ and c ∈ C, (x, ¯ y) ¯ ∈ T 2 epi F, (x, such that ¯ y¯ + c). (x, y) = (x, ¯ y) ¯ + (0X , c) = (x, ˆ y), ˆ (u, ˆ v)), ˆ it follows from Definition 2.1 that there are Since (x, ¯ y) ¯ ∈ T 2 (epi F, (x, sequences (xn , yn ) → (x, ¯ y) ¯ and tn ↓ 0, such that 1 ˆ v) ˆ + tn2 (xn , yn ) ∈ epi F, (x, ˆ y) ˆ + tn (u, 2
∀n ∈ N.
Then we have
1 1 yˆ + tn vˆ + tn2 yn ∈ F xˆ + tn uˆ + tn2 xn + C, 2 2
∀n ∈ N.
(1)
Notice that C is a convex cone and c ∈ C. Together with (1), we have 1 1 1 yˆ + tn vˆ + tn2 (yn + c) = yˆ + tn vˆ + tn2 yn + tn2 c 2 2 2 1 2 ∈ F xˆ + tn uˆ + tn xn + C, 2
∀n ∈ N,
that is, (x, ˆ y) ˆ + tn (u, ˆ v) ˆ + 12 tn2 (xn , yn + c) ∈ epi F, ∀n ∈ N. Moreover, (xn , yn + c) → (x, ¯ y¯ + c) since (xn , yn ) → (x, ¯ y). ¯ Thus, (x, y) = (x, ¯ y¯ + c) belongs to T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ and the proof of (ii) is completed. Similarly, for (iii), we only need to prove that T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ + {0X } × C ⊂ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ . For every (x, y) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ + {0X } × C, there exist (x, ¯ y) ¯ ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ and c ∈ C, such that (x, y) = (x, ¯ y¯ + c). Since (x, ¯ y) ¯ ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)), ˆ by Definition 2.1, there are sequences (xn , yn ) → (x, ¯ y) ¯ and tn ↓ 0, such that (u, ˆ v) ˆ + tn (xn , yn ) ∈ T epi F, (x, ˆ y) ˆ , ∀n ∈ N. ˆ v) ˆ + tn (xn , yn ) and tnk ↓ 0, Moreover, ∀n ∈ N, there exist sequences (xnk , ynk ) → (u, k k k such that (x, ˆ y) ˆ + tn (xn , yn ) ∈ epi F, ∀k ∈ N. Then we have (2) yˆ + tnk ynk ∈ F xˆ + tnk xnk + C, ∀n, k ∈ N.
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Since C is a cone and c ∈ C, together with (2), we have yˆ + tnk ynk + tn c = yˆ + tnk ynk + tnk tn c ∈ F xˆ + tnk xnk + C, ∀n, k ∈ N, that is, (x, ˆ y) ˆ + tnk (xnk , ynk + tn c) ∈ epi F, ∀n, k ∈ N. Since (xnk , ynk ) → (u, ˆ v) ˆ + tn (xn , yn ), we have (xnk , ynk + tn c) → (u, ˆ v) ˆ + tn (xn , yn + c) as k → +∞. Thus, ˆ y) ˆ , ∀n ∈ N. (u, ˆ v) ˆ + tn (xn , yn + c) ∈ T epi F, (x, ¯ y¯ + c) since (xn , yn ) → (x, ¯ y) ¯ as n → +∞. ToSimultaneously, (xn , yn + c) → (x, gether with (x, y) = (x, ¯ y¯ + c), we have (x, y) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ This completes the proof of (iii). Proposition 2.4 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . Then we have the following statements: (i) If the generalized contingent epiderivative Dg F (x, ˆ y) ˆ exists and the set G(x) := y ∈ Y | (x, y) ∈ T epi F, (x, ˆ y) ˆ fulfills the domination property (i.e., G(x) ⊂ Min G(x) + C) for all x ∈ X, then ˆ y) ˆ = T epi F, (x, ˆ y) ˆ . epi Dg F (x, (ii) If the generalized second-order contingent epiderivative Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ exists and the set
:= y ∈ Y | (x, y) ∈ T 2 epi F, (x, G(x) ˆ y), ˆ (u, ˆ v) ˆ (3) fulfills the domination property for all x ∈ X, then ˆ y), ˆ (u, ˆ v) ˆ . ˆ y, ˆ u, ˆ v) ˆ = T 2 epi F, (x, epi Dg2 F (x, ˆ y, ˆ (iii) If the generalized second-order composed contingent epiderivative Dg F (x, u, ˆ v) ˆ exists and the set := y ∈ Y | (x, y) ∈ T T epi F, (x, G(x) ˆ y) ˆ , (u, ˆ v) ˆ (4) fulfills the domination property for all x ∈ X, then epi Dg F (x, ˆ y, ˆ u, ˆ v) ˆ = T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ . Proof We only prove part (i) since parts (ii) and (iii) are similar. In fact, for every (x, y) ∈ epi Dg F (x, ˆ y), ˆ there exist y¯ ∈ Dg F (x, ˆ y)(x) ˆ and c ∈ C, such that y = y¯ + c. Since y¯ ∈ Dg F (x, ˆ y)(x), ˆ it follows from Definition 2.6 that (x, y) ¯ ∈ T (epi F, (x, ˆ y)). ˆ Then, by Lemma 2.1, we get
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(x, y) = (x, y) ¯ + (0X , c) ∈ T epi F, (x, ˆ y) ˆ + {0X } × C = T epi F, (x, ˆ y) ˆ , ˆ y) ˆ ⊂ T (epi F, (x, ˆ y)). ˆ which implies epi Dg F (x, Conversely, suppose that (x, y) ∈ T (epi F, (x, ˆ y)), ˆ then y ∈ G(x). Since G(x) fulfills the domination property, there exists some y¯ ∈ Min G(x), such that y ∈ {y} ¯ + C. By Definition 2.6, we have Min G(x) = Dg F (x, ˆ y)(x). ˆ ˆ y)(x) ˆ + C, that is, (x, y) ∈ epi Dg F (x, ˆ y). ˆ Then we can conclude Thus, y ∈ Dg F (x, that epi Dg F (x, ˆ y) ˆ = T (epi F, (x, ˆ y)). ˆ Proposition 2.5 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . Then (i) ∀x ∈ X, Dg F (x, ˆ y)(x) ˆ = Dg2 F (x, ˆ y, ˆ 0X , 0Y )(x) = Dg F (x, ˆ y, ˆ 0X , 0Y )(x). Moreover, let F be a C-function.
(ii) If for every x ∈ X, the sets G(x) and G(x), defined respectively by (3) and (4), fulfill the domination property, then Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ + C,
∀x ∈ X.
ˆ y), ˆ (u, ˆ v)), ˆ then (iii) If (0X , 0Y ) ∈ T 2 (epi F, (x, ˆ y, ˆ u, ˆ v)(x) ˆ = Dg2 F (x, ˆ y, ˆ u, ˆ v)(x), ˆ Dg F (x,
∀x ∈ X.
Proof It follows from Remark 2.1 that ˆ y), ˆ (0X , 0Y ) = T T epi F, (x, T 2 epi F, (x, ˆ y) ˆ , (0X , 0Y ) = T epi F, (x, ˆ y) ˆ . Thus, part (i) obviously holds from Definitions 2.6, 2.7, and 2.8. Since F is a C-function, epi F is a convex set. For part (ii), by Proposition 2.1, we have T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ ⊂ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ . (5)
Since for every x ∈ X, G(x) and G(x) fulfill the domination property, respectively, from Proposition 2.4, (5) implies epi Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ ⊂ epi Dg F (x, ˆ y, ˆ u, ˆ v), ˆ resulting ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ + C, Dg2 F (x,
∀x ∈ X.
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For part (iii), since (0X , 0Y ) ∈ T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)), ˆ it follows from Proposition 2.1 that T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ = T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ .
Therefore, from Definitions 2.7 and 2.8, part (iii) holds. The following example illustrates Proposition 2.5. Example 2.1 Consider the set-valued map F : R+ ⇒ R2 defined by F (x) := y = (y1 , y2 ) ∈ R2 y1 ≥ x 2 , y1 + y2 ≥ x , ∀x ∈ R+ .
Let C = R2+ . It is easy to verify that F is a C-function. Let (x, ˆ y) ˆ = (0, (0, 0)) ∈ graph F . Then, by Definition 2.1, we have T epi F, (x, ˆ y) ˆ = (x, y) ∈ R × R2 x ≥ 0, y1 ≥ 0, y1 + y2 ≥ x . On the one hand, if we take (u, ˆ v) ˆ = (1, (0, 1)) ∈ T (epi F, (x, ˆ y)), ˆ by Definition 2.1 and directly calculating, we have T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 ≥ 2, y1 + y2 ≥ x , and T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 ≥ 0, y1 + y2 ≥ x . Then, for every x ∈ R, we have
= y ∈ R2 y1 ≥ 2, y1 + y2 ≥ x , G(x) and
= y ∈ R2 y1 ≥ 0, y1 + y2 ≥ x . G(x)
Obviously, G(x) and G(x) fulfill the domination property for every x ∈ R, respectively. It follows from Definitions 2.7 and 2.8 that Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ = y ∈ R2 y1 ≥ 2, y1 + y2 = x , ∀x ∈ R, and
ˆ y, ˆ u, ˆ v)(x) ˆ = y ∈ R2 y1 ≥ 0, y1 + y2 = x , Dg F (x,
∀x ∈ R.
Thus, it is evident that ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ + C, Dg2 F (x, and part (ii) of Proposition 2.5 is satisfied.
∀x ∈ R,
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On the other hand, if we take (u, ˆ v) ˆ = (1, (1, 0)) ∈ T (epi F, (x, ˆ y)), ˆ by the similar method, we have T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 + y2 ≥ x . Obviously,
0, (0, 0) ∈ T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ ,
and Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ = Dg F (x, ˆ y, ˆ u, ˆ v)(x), ˆ
∀x ∈ R.
Thus, part (iii) of Proposition 2.5 is satisfied.
3 Properties of Generalized Second-Order Composed Contingent Epiderivatives In this section, we discuss some properties of generalized second-order composed contingent epiderivatives. Simultaneously, we also show that the corresponding properties of generalized contingent epiderivatives in [6] are specializations of generalized second-order composed contingent epiderivatives. In the remaining sections, we always use the following standard assumption. Assumption 3.1 Let X, Y be real normed spaces and S be a nonempty subset of X. Let Y be partially ordered by a pointed, closed, and convex cone C ⊂ Y with apex at origin and a nonempty interior int C. Let F : S ⇒ Y be a set-valued map, xˆ ∈ S, yˆ ∈ F (x), ˆ and (u, ˆ v) ˆ ∈ X ×Y. Firstly, we establish an existence theorem for generalized second-order composed contingent epiderivatives. In fact, the conditions of Theorem 2 in Chen and Jahn [6] assuring the existence of generalized contingent epiderivatives are also sufficient for the existence of generalized second-order composed contingent epiderivatives. Theorem 3.1 Let Assumption 3.1 be satisfied, C be Daniel, and (u, ˆ v) ˆ ∈ T (epi F, (x, ˆ y)). ˆ If for every x ∈ X, the set G(x), defined by (4), is C-lower bounded, then for all x ∈ X, Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ exists. Proof Since G(x) = T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ is always a closed cone in a normed space, and for every x ∈ X, G(x) is C-lower bounded, it follows from the existence theorem of minimal elements in Luc [3] that MinG(x) is nonempty, that is, for all x ∈ X, Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ exists. Next, we give the following important property of generalized second-order composed contingent epiderivatives for the C-function.
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Theorem 3.2 Let Assumption 3.1 be satisfied and let for every x ∈ X, the generalized second-order composed contingent epiderivative Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ = ∅. Then Dg F (x, ˆ y, ˆ u, ˆ v) ˆ is strictly positive homogeneous. Moreover, if F is a C-function and the set G(x), defined by (4), fulfills the domination property for all x ∈ X, then Dg F (x, ˆ y, ˆ u, ˆ v) ˆ is subadditive. Proof (a) (strictly positive homogenity) For every λ > 0 and x ∈ X, we have ˆ y, ˆ u, ˆ v)(λx) ˆ = Min y ∈ Y | (λx, y) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ Dg F (x, = Min λu ∈ Y | (λx, λu) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = λ Min u ∈ Y | (x, u) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ ˆ y, ˆ u, ˆ v)(x). ˆ = λDg F (x, (b) (subadditivity) It follows from Definition 2.8 that for every x1 , x2 ∈ X, and y1 ∈ Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 ),
y2 ∈ Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 2 ),
we have (x1 , y1 ) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)), ˆ and (x2 , y2 ) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ Since F is a C-function, epi F is convex. Then T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ is a closed and convex cone. Thus, we have 12 (x1 , y1 ) + 12 (x2 , y2 ) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)), ˆ that is, (x1 + x2 , y1 + y2 ) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ , which implies 1 + x2 ). Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 ) + Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 2 ) ⊂ G(x It follows from the domination property and Definition 2.8 that 1 + x2 ) ⊂ Min G(x 1 + x2 ) + C = Dg F (x, G(x ˆ y, ˆ u, ˆ v)(x ˆ 1 + x2 ) + C. Therefore, we have Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 ) + Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 2 ) ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 + x2 ) + C. Remark 3.1 Notice that in Theorem 3.2, we establish a special property of the generalized second-order composed contingent epiderivative, which is similar to the corresponding property of the generalized contingent epiderivative in [6]. But we can not obtain the similar property for the generalized second-order contingent epiderivative,
even though F be a C-function and the set G(x), defined by (3), fulfill the domination property for all x ∈ X. The following example explains Theorem 3.2 and the case mentioned in Remark 3.1.
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Example 3.1 Consider Example 2.1. Obviously, F is a C-function and the sets G(x) and G(x), defined respectively by (3) and (4), fulfill the domination property for all x ∈ X. Moreover, for (x, ˆ y) ˆ = (0, (0, 0)), and (u, ˆ v) ˆ = (1, (0, 1)), we have Dg F (x, ˆ y, ˆ u, ˆ v) ˆ = y ∈ R2 y1 ≥ 0, y1 + y2 = x , ∀x ∈ R, which is strictly positive homogeneous and subadditive. But Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ = y ∈ R2 y1 ≥ 2, y1 + y2 = x ,
∀x ∈ R,
which is merely subadditive. Remark 3.2 Since (0X , 0Y ) always belongs to T (epi F, (x, ˆ y)) ˆ and T epi F, (x, ˆ y) ˆ = T T epi F, (x, ˆ y) ˆ , (0X , 0Y ) , Theorem 3.1 and Theorem 3.2, respectively, reduce to Theorem 2 and Theorem 1 in [6] when we take (u, ˆ v) ˆ = (0X , 0Y ). 4 Optimality Conditions for Set-Valued Optimization Problems In this section, we consider the following set-valued optimization problem: (P)
min F (x),
s.t. x ∈ S,
where S is a nonempty subset of X and F : S ⇒ Y is a set-valued map. Now, we recall some optimality notions for the problem (P). A pair (x, ˆ y) ˆ ∈ graph F is called a weak minimizer of the problem (P) iff yˆ is a weak minimal element of the set F (S), i.e.,
F (S) ∩ {y}−int ˆ C = ∅, where F (S) := F (x). x∈S
A pair (x, ˆ y) ˆ ∈ graph F is called a local weak minimizer of the problem (P) iff there is a neighborhood U of xˆ such that yˆ is a weak minimal element of the set F (S ∩ U ), i.e.,
F (S ∩ U ) ∩ {y}−int ˆ C = ∅, where F (S ∩ U ) := F (x). x∈S∩U
Next, we apply generalized second-order composed contingent epiderivatives for set-valued maps to discuss some second-order optimality conditions for the setvalued optimization problem (P). Theorem 4.1 Let Assumption 3.1 be satisfied and let (x, ˆ y) ˆ ∈ graph F be a local weak minimizer of the problem (P). Then, for every (u, ˆ v) ˆ ∈ T (epi F, (x, ˆ y)) ˆ such that vˆ ∈ −C and every x ∈ dom Dg F (x, ˆ y, ˆ u, ˆ v), ˆ we have Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Y \ −int C − {v} ˆ . (6)
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Proof Suppose that there exists some x ∈ dom Dg F (x, ˆ y, ˆ u, ˆ v) ˆ such that (6) is not satisfied, that is, there exits some y ∈ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ∩ −int C − {v} ˆ . (7) By Definition 2.8, we have (x, y) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ By Proposition 2.2, there exist sequences λn → +∞ and (un , vn ) in T (epi F, (x, ˆ y)), ˆ such that (un , vn ) → (u, ˆ v) ˆ and λn (un , vn ) − (u, ˆ v) ˆ → (x, y), as n → +∞. (8) ˆ y)), ˆ for every n ∈ N, there exist a sequence λkn → +∞ Since (un , vn ) ∈ T (epi F, (x, k k and a sequence (xn , yn ) in epi F , such that (xnk , ynk ) → (x, ˆ y) ˆ and ˆ y) ˆ → (un , vn ), λkn xnk , ynk − (x,
as k → +∞.
(9)
It follows from (7) and (8) that y ∈ −int C − {v} ˆ and λn (vn − v) ˆ → y as n → +∞. Then there exists N1 ∈ N such that λn (vn − v) ˆ ∈ −int C − {v}, ˆ ∀n ≥ N1 , that is, 1 vˆ ∈ −int C, ∀n ≥ N1 . vn − 1 − (10) λn Since λn → +∞ and vˆ ∈ −C, there exists N2 ∈ N, such that λn 1 for every n ≥ N2 , and then 1 1− vˆ ∈ −C, ∀n ≥ N2 . (11) λn Thus, by (10) and (11), we have for every n ≥ max(N1 , N2 ), 1 1 vˆ + 1 − vˆ ∈ −int C − C vn = vn − 1 − λn λn = −int C.
(12)
Moreover, by (9), we have λkn (ynk − y) ˆ → vn as k → +∞. Together with (12), we have for every n ∈ N and n ≥ max(N1 , N2 ), there exists K1 (n) ∈ N such that λkn ynk − yˆ ∈ −int C, ∀k ≥ K1 (n), that is, ynk − yˆ ∈ −int C,
∀n ≥ max(N1 , N2 ), ∀k ≥ K1 (n).
(13)
Since (xnk , ynk ) ∈ epi F , there exists some y¯nk ∈ F (xnk ), such that ynk ∈ {y¯nk } + C. Together with (13), we have ∀n ≥ max(N1 , N2 ), ∀k ≥ K1 (n), y¯nk − yˆ ∈ ynk − {y} ˆ −C ⊂ −int C − C = −int C.
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Consequently, y¯nk ∈ {y}−int ˆ C,
∀n ≥ max(N1 , N2 ), ∀k ≥ K1 (n).
Since ∀n ∈ N, xnk ∈ S and xnk → xˆ as k → +∞, for every neighborhood U of x, ˆ there exists K2 (n) ∈ N, such that xnk ∈ S ∩ U, ∀k ≥ K2 (n). Therefore, we can conclude that for every neighborhood U of x, ˆ and sufficiently large n, k ∈ N, n ≥ max(N1 , N2 ) and k ≥ max(K1 (n), K2 (n)), there are xnk ∈ S ∩ U , such that ˆ C = ∅. F xnk ∩ {y}−int This is a contradiction to the assumption that (x, ˆ y) ˆ ∈ graph F is a local weak minimizer of the problem (P). Remark 4.1 It follows from what we have mentioned in Remark 3.2 that Theorem 4.1 is a generalization of Theorem 5 in [6]. As we know, Theorems 3.1 in [15] still holds for generalized second-order contingent epiderivatives. However, if the assumptions in Proposition 2.5 hold, then Theorem 4.1 is extensive than Theorems 3.1 in [15] for the case of generalized second-order contingent epiderivatives. The following example illustrates Theorem 4.1 and the case mentioned in Remark 4.1. Example 4.1 Consider Example 2.1. Let F : S ⇒ R2 with F (x) := y = (y1 , y2 ) ∈ R2 y1 ≥ x 2 , y1 + y2 ≥ x ,
∀x ∈ S,
ˆ y) ˆ = (0, (0, 0)) is a local weak minwhere S = [−1, 1]. Let C = R2+ . Obviously, (x, imizer of the following set-valued optimization problem: (P∗ )
min F (x),
s.t. x ∈ S.
By Definition 2.1, we have T epi F, (x, ˆ y) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 ≥ 0, y1 + y2 ≥ x . We denote the set H (x, ˆ y) ˆ = (u, ˆ v) ˆ ∈ R × R2 (u, ˆ v) ˆ ∈ T epi F, (x, ˆ y) ˆ with vˆ ∈ −C = u, ˆ (0, vˆ2 ) uˆ ∈ R, uˆ ≤ vˆ2 ≤ 0 . Moreover, by directly calculating, we have for every (u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ if 0 ≥ vˆ2 > u; ˆ {(x, y) | x ∈ R, y1 ≥ 0, y2 ∈ R}, T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = ˆ {(x, y) | x ∈ R, y1 ≥ 0, y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u, and
if 0 ≥ vˆ2 > u; ˆ {(x, y) | x ∈ R, y1 ≥ 2uˆ 2 , y2 ∈ R}, ˆ y), ˆ (u, ˆ v) ˆ = T epi F, (x, {(x, y) | x ∈ R, y1 ≥ 2uˆ 2 , y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u. ˆ 2
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Then, for every x ∈ R and every (u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ we get if 0 ≥ vˆ2 > u; ˆ = {y | y1 ≥ 0, y2 ∈ R}, G(x) ˆ {y | y1 ≥ 0, y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u, and
2 if 0 ≥ vˆ2 > u; ˆ
= {y | y1 ≥ 2uˆ 2 , y2 ∈ R}, G(x) {y | y1 ≥ 2uˆ , y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u. ˆ
Obviously, G(x) and G(x) respectively fulfill the domination property for every x ∈ R and every (u, ˆ v) ˆ ∈ H (x, ˆ y). ˆ It follows from Definitions 2.7 and 2.8 that for every x ∈ R, we have ˆ ∅, if 0 ≥ vˆ2 > u; ˆ y, ˆ u, ˆ v)(x) ˆ = Dg F (x, {y ∈ R2 | y1 ≥ 0, y1 + y2 = x}, if 0 ≥ vˆ2 = u, ˆ and Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ =
∅, {y ∈ R2 | y1 ≥ 2uˆ 2 , y1 + y2 = x},
ˆ if 0 ≥ vˆ2 > u; if 0 ≥ vˆ2 = u. ˆ
On the one hand, −int C − {v} ˆ = (y1 , y2 ) ∈ R2 y1 < 0, y2 < −vˆ2 ,
∀(u, ˆ v) ˆ ∈ H (x, ˆ y). ˆ
ˆ y, ˆ u, ˆ v), ˆ we have Thus, it is evident that for every x ∈ dom Dg F (x, Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Y \ −int C − {v} ˆ , and the necessary conditions in Theorem 4.1 are satisfied. On the other hand, for every (u, ˆ v) ˆ ∈ H (x, ˆ y) ˆ with uˆ = 0, we have ˆ y, ˆ u, ˆ v)(x) ˆ Dg F (x, ˆ y, ˆ u, ˆ v)(x), ˆ Dg2 F (x,
∀x ∈ R,
ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Y \ −int C − {v} ˆ , Dg2 F (x,
∀x ∈ R.
which implies
Theorem 4.2 Let Assumption 3.1 be satisfied, S be convex, and F be a C-function. − xˆ − u), If for every (u, ˆ v) ˆ ∈ T (epi F, (x, ˆ y)) ˆ with vˆ ∈ −C, the set G(x ˆ defined by (4), fulfills the domination property for all x ∈ S and ∅ = Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ − xˆ − u) ˆ ⊂ Y \ −int C − {v} ˆ , ∀x ∈ S, (14) then (x, ˆ y) ˆ ∈ graph F is a weak minimizer of the problem (P). Proof Since (0X , 0Y ) always belongs to T (epi F, (x, ˆ y)), ˆ 0Y ∈ −C, and T epi F, (x, ˆ y) ˆ = T T epi F, (x, ˆ y) ˆ , (0X , 0Y ) ,
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it follows from Definitions 2.6 and 2.8 that the condition (14) implies ∅ = Dg F (x, ˆ y)(x ˆ − x) ˆ ⊂ Y \ (−int C),
∀x ∈ S.
By Theorem 6 in [6], we can conclude that (x, ˆ y) ˆ ∈ graph F is a weak minimizer of the problem (P). Remark 4.2 Obviously, Theorem 4.2 generalizes Theorem 6 in [6]. Simultaneously, Theorem 4.2 is different from Theorem 3.2 in [15] since the existences of the contingent epiderivative DF (x, ˆ y) ˆ (see [5]) and the second-order contingent epiderivative ˆ y, ˆ u, ˆ v) ˆ (see [15]) are hard to be satisfied in general cases. D 2 F (x, The following example explains Theorem 4.2 and the case mentioned in Remark 4.2. Example 4.2 Consider Example 2.1 and the following set-valued optimization problem: (P∗∗ )
min F (x),
s.t.
x ∈ R+ .
Let (x, ˆ y) ˆ = (0, (0, 0)) and C = R2+ . Similarly, we have T epi F, (x, ˆ y) ˆ = (x, y) ∈ R × R2 x ≥ 0, y1 ≥ 0, y1 + y2 ≥ x .
(15)
Denote the set H (x, ˆ y) ˆ = (u, ˆ v) ˆ ∈ R × R2 (u, ˆ v) ˆ ∈ T epi F, (x, ˆ y) ˆ with vˆ ∈ −C = 0, (0, 0) . Thus, for every (u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ we have T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = T epi F, (x, ˆ y) ˆ
(16)
and for every x ∈ R+ , we have
− xˆ − u) G(x ˆ = y ∈ R2 | y1 ≥ 0, y1 + y2 ≥ x ,
which satisfies the domination property. Moreover, it follows from (15) and (16) that for every (u, ˆ v) ˆ ∈ H (x, ˆ y) ˆ and every x ∈ R+ , Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ − xˆ − u) ˆ = Dg F (x, ˆ y)(x ˆ − x) ˆ = y ∈ R2 y1 ≥ 0, y1 + y2 = x . Then we have ∅ = Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ − xˆ − u) ˆ ⊂ Y \ −int C −{v} ˆ ,
∀(u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ ∀x ∈ R+ .
Therefore, the sufficient conditions in Theorem 4.2 are fulfilled, and then (x, ˆ y) ˆ is a weak minimizer of the problem (P∗∗ ). But it can be seen from (15) that DF (x, ˆ y) ˆ
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does not exist. In fact, if DF (x, ˆ y) ˆ exists, by the definition of the contingent epiderivative in [5], we have for every x¯ ≥ 0, y ∈ R2 y1 ≥ 0, y1 + y2 ≥ x¯ ⊂ DF (x, ˆ y)( ˆ x) ¯ + R2+ . This is impossible. Therefore, Theorem 3.2 in [15] is not applicable.
5 Concluding Remarks In this paper, we propose a new concept of a second-order derivative for set-valued maps, which is called the generalized second-order composed contingent epiderivative, and discuss some relationships between the derivative and the generalized second-order contingent epiderivative. We also investigate some of its properties. Simultaneously, by virtue of the derivative, we obtain a unified second-order sufficient and necessary optimality condition for set-valued optimization problems, which improves existing results. Acknowledgements This research was supported by the National Natural Science Foundation of China (Grant No. 10871216). The authors thank the two anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper.
References 1. Aubin, J.P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Mathematical Analysis and Applications, Part A, pp. 160–229. Academic Press, New York (1981) 2. Corley, H.W.: Optimality conditions for maximizations of set-valued functions. J. Optim. Theory Appl. 58, 1–10 (1988) 3. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) 4. Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991) 5. Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46, 193–211 (1997) 6. Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48, 187–200 (1998) 7. Jahn, J., Khan, A.A.: The existence of contingent epiderivatives for set-valued maps. Appl. Math. Lett. 16, 1179–1185 (2003) 8. Luis, R.M., Miguel, S.: About contingent epiderivatives. J. Math. Anal. Appl. 327, 745–762 (2007) 9. Bigi, G.: On sufficient second order optimality conditions in multiobjective optimization. Math. Methods Oper. Res. 63, 77–85 (2006) 10. Bonnans, J.F., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9, 466–492 (1999) 11. Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004) 12. Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1998) 13. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) 14. Durea, M.: First and second order optimality conditions for set-valued optimization problems. Rend. Circ. Mat. Palermo 2, 451–468 (2004)
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15. Jahn, J., Khan, A.A., Zeilinger, P.: Second-order optimality conditions in set optimization. J. Optim. Theory Appl. 125, 331–347 (2005) 16. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) 17. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984) 18. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)
New Generalized Second-Order Contingent Epiderivatives and Set-Valued Optimization Problems S.J. Li · S.K. Zhu · K.L. Teo
Received: 13 December 2010 / Accepted: 3 August 2011 / Published online: 8 September 2011 © Springer Science+Business Media, LLC 2011
Abstract In this paper, we introduce the concept of a generalized second-order composed contingent epiderivative for set-valued maps and discuss its relationship to the generalized second-order contingent epiderivative. We also investigate some of its properties. Then, by virtue of the generalized second-order composed contingent epiderivative, we establish a unified second-order sufficient and necessary optimality condition for set-valued optimization problems, which is a generalization of the corresponding results in the literature. Keywords Set-valued optimization · Generalized second-order composed contingent epiderivative · Optimality conditions 1 Introduction In the last years, the set-valued optimization problems have been of great interest in research, and a lot of notions about derivatives for set-valued maps have been introduced and applied to set up the optimality conditions; see [1, 2, 4–6, 14, 15]. As we know, Aubin [1] first introduced a concept of a contingent derivative for set-valued maps, which is essentially a natural extension of tangents and plays an Communicated by Guang-ya Chen. S.J. Li () · S.K. Zhu College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China e-mail: [email protected] S.K. Zhu e-mail: [email protected] K.L. Teo Department of Mathematics and Statistics, Curtin University of Technology, G.P.O. Box U1987, Perth, WA 6845, Australia e-mail: [email protected]
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important role in set-valued analysis. Simultaneously, the contingent derivative of set-valued maps has also been generally used to express the first-order optimality conditions for set-valued optimization problems ever since the path-breaking paper [2]. However, the necessary conditions and sufficient conditions (see [2–4]) are not unified under standard assumptions. For solving this problem, Jahn and Rauh [5] proposed another derivative called the contingent epiderivative, which is actually an extension of directional derivatives in the single-valued and convex case. Moreover, they applied the contingent epiderivative to establish some unified sufficient and necessary conditions under suitable assumptions. But, unfortunately, since the contingent epiderivative of a set-valued map is a single-valued map, the conditions assuring the existence of the contingent epiderivative are hard to be satisfied; see [5–8]. In view of this problem, Chen and Jahn [6] introduced a generalized contingent epiderivative of set-valued maps and obtained the existence of the generalized contingent epiderivative with standard assumptions. Besides, they also established a unified necessary and sufficient optimality condition in terms of the generalized contingent epiderivative. Recently, the second-order optimality conditions in scalar and vector optimization problems have been widely investigated; for example, see [9–12]. If we analyze these developments, we can easily found that a second-order contingent set, introduced by Aubin [13], and a second-order asymptotic contingent cone, introduced by Penot [12], play a key role in establishing second-order optimality conditions. However, the second-order optimality conditions for set-valued optimization problems still need to be addressed. In [14], Durea used various derivative-like objects for set-valued maps to establish some second-order optimality conditions. Jahn et al. [15] proposed (generalized) second-order contingent epiderivatives for set-valued maps and applied these concepts to establish second-order optimality conditions. It is worth noting that (generalized) second-order contingent epiderivatives are proposed through the second-order contingent set, which is only a closed set and may not be a cone in general cases. Moreover, the second-order contingent set may not be convex even though the set what we consider be a convex set. Therefore, comparing with the (generalized) contingent epiderivative, we can not obtain some similar properties for the (generalized) second-order contingent epiderivative. Motivated by the work reported in [1, 2, 4–6, 15], we introduce a new generalized second-order contingent epiderivative called generalized second-order composed contingent epiderivative for set-valued maps, discuss its existence under standard assumptions and obtain some special properties. By virtue of the generalized second-order composed contingent epiderivative, we obtain a unified sufficient and necessary optimality condition for set-valued optimization problems, which is a generalization of corresponding results in [6]. Moreover, we also give an example to explain that the sufficient optimality condition improves and generalizes the corresponding result in [15]. The organization of this paper is as follows. In Sect. 2, we recall some basic concepts, introduce the generalized second-order composed contingent epiderivative and compare the generalized second-order composed contingent epiderivative with the generalized second-order contingent epiderivative. In Sect. 3, we give some special properties of the generalized second-order composed contingent epiderivative. In Sect. 4, we establish a unified second-order sufficient and necessary optimality condition for set-valued optimization problems.
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2 Notations and Preliminaries Throughout this paper, let X, Y be real normed spaces and S be a nonempty subset of X. 0X and 0Y denote the origins of X and Y , respectively. As usual, we denote by int S, cl S and cone S the interior, closure, and cone hull of S, respectively. Let C ⊂ Y be a pointed, closed and convex cone with apex at the origin and a nonempty interior int C, and let Y be partially ordered by C. Let F : X ⇒ Y be a set-valued map. The domain, graph, and epigraph of F are defined respectively by dom F := {x ∈ X | F (x) = ∅}, graph F := {(x, y) ∈ X × Y | y ∈ F (x)} and epi F := {(x, y) ∈ X × Y | y ∈ F (x) + C}. Now, we recall notions of a contingent cone and a second-order contingent set. Definition 2.1 [13] Let S be a nonempty subset of X, xˆ ∈ cl S, and ω ∈ X. (i) The contingent cone of S at xˆ is T (S, x) ˆ := v ∈ X | ∃tn ↓ 0, ∃vn → v, such that xˆ + tn vn ∈ S, ∀n ∈ N . (ii) The second-order contingent set of S at xˆ in the direction ω ∈ X is ˆ ω) T 2 (S, x, 1 2 := v ∈ X | ∃tn ↓ 0, ∃vn → v, such that xˆ + tn ω + tn vn ∈ S, ∀n ∈ N . 2 Proposition 2.1 [11] Let S ⊂ X be a convex set, xˆ ∈ S, and ω ∈ T (S, x). ˆ Then T T (S, x), ˆ ω = clcone cone(S − x) ˆ −ω and
T 2 (S, x, ˆ ω) ⊂ T T (S, x), ˆ ω .
ˆ ω), then Moreover (see Sect. 2.1 of Chap. 3 in [16]), if 0X ∈ T 2 (S, x, T 2 (S, x, ˆ ω) = T T (S, x), ˆ ω . Remark 2.1 T (S, x) ˆ is a cone, T 2 (S, x, ˆ ω) is a closed set but it is not necessarily a cone, and T (T (S, x), ˆ ω) is a closed cone. Obviously, T 2 (S, x, ˆ 0X ) = T (S, x) ˆ and also T (T (S, x), ˆ 0X ) = T (S, x) ˆ since T (S, x) ˆ is a closed cone. It follows from Proposition 2.1 that T (T (S, x), ˆ ω) is convex and is larger than T 2 (S, x, ˆ ω) when S is a 2 ˆ ω) and T (T (S, x), ˆ ω) may be nonempty only if ω ∈ T (S, x). ˆ convex subset. T (S, x, Furthermore (see more details in [16]), T 2 (S, x, ˆ ω) may be properly contained in T (T (S, x), ˆ ω) and not be convex even though S be a convex set. Next, we collect some equivalent characterizations for these contingent sets. Proposition 2.2 [17] The following statements are equivalent: (i) v ∈ T (S, x). ˆ (ii) There exist sequences λn → +∞ and xn ∈ S, such that xn → x and λn (xn − x) → v.
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Proposition 2.3 [17, 18] The following statements are equivalent: (i) v ∈ T 2 (S, x, ˆ ω). (ii) For every r > 0, there exist sequences tn ↓ 0, γn ↓ 0 and vn → v, such that tn 1 γn → r and xn := xˆ + tn ω + 2 rtn γn vn ∈ S. (iii) There exist sequences αn → +∞, βn → +∞ and xn ∈ S, such that xn → x, ˆ αn (xn − x) ˆ → ω and βn (αn (xn − x) ˆ − ω) → v.
βn αn
→ 2,
Now, we recall some standard notions. Definition 2.2 [18] Let Y be a topological linear space and be partially ordered by a convex cone C ⊂ Y with apex at the origin. (i) A sequence {yn } ⊂ Y is said to be C-decreasing iff ∀i, j ∈ N, i ≤ j implies yj ≤C yi . (ii) A subset D ⊂ Y is said to be C-lower bounded iff there exists a y ∈ Y , such that D ⊂ {y} + C. (iii) The convex cone C is said to be Daniel iff every C-decreasing and C-lower bounded sequence in Y converges to its infimum. Definition 2.3 [18] Let F : S ⇒ Y be a set-valued map and S ⊂ X be convex. Then F is said to be a C-function iff for every x1 , x2 ∈ S and every λ ∈ [0, 1], λF (x1 ) + (1 − λ)F (x2 ) ⊂ F λx1 + (1 − λ)x2 + C. It is obvious that F is a C-function if and only if epi F is convex. Definition 2.4 [6] Let X be a real linear space, and let Y be a real linear space and partially ordered by a convex cone C ⊂ Y with apex at the origin. A set-valued map f : X ⇒ Y is said to be (i) strictly positive homogeneous iff f (αx) = αf (x),
∀α > 0, ∀x ∈ X.
(ii) subadditive iff f (x1 ) + f (x2 ) ⊂ f (x1 + x2 ) + C,
∀x1 , x2 ∈ X.
Iff the properties (i) holds with α ≥ 0 and (ii) holds, f is said to be sublinear. Definition 2.5 [18] Let E be a nonempty subset of Y . A point y ∈ E is called a minimal element of E iff {y} − C \ {0Y } ∩ E = ∅. A point y ∈ E is called a weak minimal element of E iff {y}−int C ∩ E = ∅.
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As usual, we denote by Min E (WMin E) the set of all the minimal (weak minimal) elements of E. Specially, we set Min E = ∅ (WMin E = ∅) if E = ∅. As we know, the notion of the contingent epiderivative of a set-valued map, introduced by Jahn and Rauh [5], has some important properties, and unifies the necessary and sufficient optimality conditions for set-valued optimization problems. However, the existence of the contingent epiderivative in a general case is still an open question. In view of this problem, Chen and Jahn [6] introduced a generalized contingent epiderivative of set-valued maps. Also, Jahn et al. [15] introduced a generalized second-order contingent epiderivative of set-valued maps. Definition 2.6 [6] Let F : S ⇒ Y be a set-valued map and (x, ˆ y) ˆ ∈ graph F . The ˆ y) ˆ : generalized contingent epiderivative of F at (x, ˆ y) ˆ is the set-valued map Dg F (x, X ⇒ Y defined by ˆ y)(x) ˆ := Min y ∈ Y | (x, y) ∈ T epi F, (x, ˆ y) ˆ , ∀x ∈ X. Dg F (x, Definition 2.7 [15] Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . The generalized second-order contingent epiderivative of F at (x, ˆ y) ˆ in the direction (u, ˆ v) ˆ is the set-valued map Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ : X ⇒ Y defined by ˆ y, ˆ u, ˆ v)(x) ˆ := Min y ∈ Y | (x, y) ∈ T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ , Dg2 F (x,
∀x ∈ X.
Note that, it follows from Proposition 2.1 and Remark 2.1 that the second-order contingent set T 2 (S, x, ˆ ω), introduced by Aubin [13], is only a closed set, but it is not ˆ ω) may not be convex even if S be convex. necessarily a cone. Moreover, T 2 (S, x, Thus, comparing with the generalized contingent epiderivative, we can not obtain some similar properties of the generalized second-order contingent epiderivative. It is worth noting that T (T (S, x), ˆ ω) is always a closed cone. Moreover, T (T (S, x), ˆ ω) is a closed and convex cone when S is convex. Motivated by this idea, and Definitions 2.6 and 2.7, we introduce a new generalized second-order contingent epiderivative for set-valued maps as following. Definition 2.8 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . The generalized second-order composed contingent epiderivative of F at ˆ y, ˆ u, ˆ v) ˆ : X ⇒ Y defined (x, ˆ y) ˆ in the direction (u, ˆ v) ˆ is the set-valued map Dg F (x, by ˆ y, ˆ u, ˆ v)(x) ˆ := Min y ∈ Y | (x, y) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ , ∀x ∈ X. Dg F (x, Next, we discuss some relationships among these generalized epiderivatives. For this purpose, we need the following lemma. Lemma 2.1 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . Then we have the following statements: (i) T (epi F, (x, ˆ y)) ˆ + {0X } × C = T (epi F, (x, ˆ y)). ˆ
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(ii) T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ + {0X } × C = T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)). ˆ (iii) T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ + {0X } × C = T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ Proof Since (ii) or (iii) with (u, ˆ v) ˆ = (0X , 0Y ) implies (i), we only need to prove (ii) and (iii). For (ii), it is obvious that T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ ⊂ T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ 2 + {0X } × C. Conversely, for every (x, y) ∈ T (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ + {0X } × C, there exists ˆ y), ˆ (u, ˆ v) ˆ and c ∈ C, (x, ¯ y) ¯ ∈ T 2 epi F, (x, such that ¯ y¯ + c). (x, y) = (x, ¯ y) ¯ + (0X , c) = (x, ˆ y), ˆ (u, ˆ v)), ˆ it follows from Definition 2.1 that there are Since (x, ¯ y) ¯ ∈ T 2 (epi F, (x, sequences (xn , yn ) → (x, ¯ y) ¯ and tn ↓ 0, such that 1 ˆ v) ˆ + tn2 (xn , yn ) ∈ epi F, (x, ˆ y) ˆ + tn (u, 2
∀n ∈ N.
Then we have
1 1 yˆ + tn vˆ + tn2 yn ∈ F xˆ + tn uˆ + tn2 xn + C, 2 2
∀n ∈ N.
(1)
Notice that C is a convex cone and c ∈ C. Together with (1), we have 1 1 1 yˆ + tn vˆ + tn2 (yn + c) = yˆ + tn vˆ + tn2 yn + tn2 c 2 2 2 1 2 ∈ F xˆ + tn uˆ + tn xn + C, 2
∀n ∈ N,
that is, (x, ˆ y) ˆ + tn (u, ˆ v) ˆ + 12 tn2 (xn , yn + c) ∈ epi F, ∀n ∈ N. Moreover, (xn , yn + c) → (x, ¯ y¯ + c) since (xn , yn ) → (x, ¯ y). ¯ Thus, (x, y) = (x, ¯ y¯ + c) belongs to T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)) ˆ and the proof of (ii) is completed. Similarly, for (iii), we only need to prove that T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ + {0X } × C ⊂ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ . For every (x, y) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ + {0X } × C, there exist (x, ¯ y) ¯ ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ and c ∈ C, such that (x, y) = (x, ¯ y¯ + c). Since (x, ¯ y) ¯ ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)), ˆ by Definition 2.1, there are sequences (xn , yn ) → (x, ¯ y) ¯ and tn ↓ 0, such that (u, ˆ v) ˆ + tn (xn , yn ) ∈ T epi F, (x, ˆ y) ˆ , ∀n ∈ N. ˆ v) ˆ + tn (xn , yn ) and tnk ↓ 0, Moreover, ∀n ∈ N, there exist sequences (xnk , ynk ) → (u, k k k such that (x, ˆ y) ˆ + tn (xn , yn ) ∈ epi F, ∀k ∈ N. Then we have (2) yˆ + tnk ynk ∈ F xˆ + tnk xnk + C, ∀n, k ∈ N.
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Since C is a cone and c ∈ C, together with (2), we have yˆ + tnk ynk + tn c = yˆ + tnk ynk + tnk tn c ∈ F xˆ + tnk xnk + C, ∀n, k ∈ N, that is, (x, ˆ y) ˆ + tnk (xnk , ynk + tn c) ∈ epi F, ∀n, k ∈ N. Since (xnk , ynk ) → (u, ˆ v) ˆ + tn (xn , yn ), we have (xnk , ynk + tn c) → (u, ˆ v) ˆ + tn (xn , yn + c) as k → +∞. Thus, ˆ y) ˆ , ∀n ∈ N. (u, ˆ v) ˆ + tn (xn , yn + c) ∈ T epi F, (x, ¯ y¯ + c) since (xn , yn ) → (x, ¯ y) ¯ as n → +∞. ToSimultaneously, (xn , yn + c) → (x, gether with (x, y) = (x, ¯ y¯ + c), we have (x, y) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ This completes the proof of (iii). Proposition 2.4 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . Then we have the following statements: (i) If the generalized contingent epiderivative Dg F (x, ˆ y) ˆ exists and the set G(x) := y ∈ Y | (x, y) ∈ T epi F, (x, ˆ y) ˆ fulfills the domination property (i.e., G(x) ⊂ Min G(x) + C) for all x ∈ X, then ˆ y) ˆ = T epi F, (x, ˆ y) ˆ . epi Dg F (x, (ii) If the generalized second-order contingent epiderivative Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ exists and the set
:= y ∈ Y | (x, y) ∈ T 2 epi F, (x, G(x) ˆ y), ˆ (u, ˆ v) ˆ (3) fulfills the domination property for all x ∈ X, then ˆ y), ˆ (u, ˆ v) ˆ . ˆ y, ˆ u, ˆ v) ˆ = T 2 epi F, (x, epi Dg2 F (x, ˆ y, ˆ (iii) If the generalized second-order composed contingent epiderivative Dg F (x, u, ˆ v) ˆ exists and the set := y ∈ Y | (x, y) ∈ T T epi F, (x, G(x) ˆ y) ˆ , (u, ˆ v) ˆ (4) fulfills the domination property for all x ∈ X, then epi Dg F (x, ˆ y, ˆ u, ˆ v) ˆ = T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ . Proof We only prove part (i) since parts (ii) and (iii) are similar. In fact, for every (x, y) ∈ epi Dg F (x, ˆ y), ˆ there exist y¯ ∈ Dg F (x, ˆ y)(x) ˆ and c ∈ C, such that y = y¯ + c. Since y¯ ∈ Dg F (x, ˆ y)(x), ˆ it follows from Definition 2.6 that (x, y) ¯ ∈ T (epi F, (x, ˆ y)). ˆ Then, by Lemma 2.1, we get
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(x, y) = (x, y) ¯ + (0X , c) ∈ T epi F, (x, ˆ y) ˆ + {0X } × C = T epi F, (x, ˆ y) ˆ , ˆ y) ˆ ⊂ T (epi F, (x, ˆ y)). ˆ which implies epi Dg F (x, Conversely, suppose that (x, y) ∈ T (epi F, (x, ˆ y)), ˆ then y ∈ G(x). Since G(x) fulfills the domination property, there exists some y¯ ∈ Min G(x), such that y ∈ {y} ¯ + C. By Definition 2.6, we have Min G(x) = Dg F (x, ˆ y)(x). ˆ ˆ y)(x) ˆ + C, that is, (x, y) ∈ epi Dg F (x, ˆ y). ˆ Then we can conclude Thus, y ∈ Dg F (x, that epi Dg F (x, ˆ y) ˆ = T (epi F, (x, ˆ y)). ˆ Proposition 2.5 Let F : S ⇒ Y be a set-valued map, (x, ˆ y) ˆ ∈ graph F , and (u, ˆ v) ˆ ∈ X × Y . Then (i) ∀x ∈ X, Dg F (x, ˆ y)(x) ˆ = Dg2 F (x, ˆ y, ˆ 0X , 0Y )(x) = Dg F (x, ˆ y, ˆ 0X , 0Y )(x). Moreover, let F be a C-function.
(ii) If for every x ∈ X, the sets G(x) and G(x), defined respectively by (3) and (4), fulfill the domination property, then Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ + C,
∀x ∈ X.
ˆ y), ˆ (u, ˆ v)), ˆ then (iii) If (0X , 0Y ) ∈ T 2 (epi F, (x, ˆ y, ˆ u, ˆ v)(x) ˆ = Dg2 F (x, ˆ y, ˆ u, ˆ v)(x), ˆ Dg F (x,
∀x ∈ X.
Proof It follows from Remark 2.1 that ˆ y), ˆ (0X , 0Y ) = T T epi F, (x, T 2 epi F, (x, ˆ y) ˆ , (0X , 0Y ) = T epi F, (x, ˆ y) ˆ . Thus, part (i) obviously holds from Definitions 2.6, 2.7, and 2.8. Since F is a C-function, epi F is a convex set. For part (ii), by Proposition 2.1, we have T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ ⊂ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ . (5)
Since for every x ∈ X, G(x) and G(x) fulfill the domination property, respectively, from Proposition 2.4, (5) implies epi Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ ⊂ epi Dg F (x, ˆ y, ˆ u, ˆ v), ˆ resulting ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ + C, Dg2 F (x,
∀x ∈ X.
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For part (iii), since (0X , 0Y ) ∈ T 2 (epi F, (x, ˆ y), ˆ (u, ˆ v)), ˆ it follows from Proposition 2.1 that T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ = T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ .
Therefore, from Definitions 2.7 and 2.8, part (iii) holds. The following example illustrates Proposition 2.5. Example 2.1 Consider the set-valued map F : R+ ⇒ R2 defined by F (x) := y = (y1 , y2 ) ∈ R2 y1 ≥ x 2 , y1 + y2 ≥ x , ∀x ∈ R+ .
Let C = R2+ . It is easy to verify that F is a C-function. Let (x, ˆ y) ˆ = (0, (0, 0)) ∈ graph F . Then, by Definition 2.1, we have T epi F, (x, ˆ y) ˆ = (x, y) ∈ R × R2 x ≥ 0, y1 ≥ 0, y1 + y2 ≥ x . On the one hand, if we take (u, ˆ v) ˆ = (1, (0, 1)) ∈ T (epi F, (x, ˆ y)), ˆ by Definition 2.1 and directly calculating, we have T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 ≥ 2, y1 + y2 ≥ x , and T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 ≥ 0, y1 + y2 ≥ x . Then, for every x ∈ R, we have
= y ∈ R2 y1 ≥ 2, y1 + y2 ≥ x , G(x) and
= y ∈ R2 y1 ≥ 0, y1 + y2 ≥ x . G(x)
Obviously, G(x) and G(x) fulfill the domination property for every x ∈ R, respectively. It follows from Definitions 2.7 and 2.8 that Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ = y ∈ R2 y1 ≥ 2, y1 + y2 = x , ∀x ∈ R, and
ˆ y, ˆ u, ˆ v)(x) ˆ = y ∈ R2 y1 ≥ 0, y1 + y2 = x , Dg F (x,
∀x ∈ R.
Thus, it is evident that ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ + C, Dg2 F (x, and part (ii) of Proposition 2.5 is satisfied.
∀x ∈ R,
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On the other hand, if we take (u, ˆ v) ˆ = (1, (1, 0)) ∈ T (epi F, (x, ˆ y)), ˆ by the similar method, we have T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 + y2 ≥ x . Obviously,
0, (0, 0) ∈ T 2 epi F, (x, ˆ y), ˆ (u, ˆ v) ˆ ,
and Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ = Dg F (x, ˆ y, ˆ u, ˆ v)(x), ˆ
∀x ∈ R.
Thus, part (iii) of Proposition 2.5 is satisfied.
3 Properties of Generalized Second-Order Composed Contingent Epiderivatives In this section, we discuss some properties of generalized second-order composed contingent epiderivatives. Simultaneously, we also show that the corresponding properties of generalized contingent epiderivatives in [6] are specializations of generalized second-order composed contingent epiderivatives. In the remaining sections, we always use the following standard assumption. Assumption 3.1 Let X, Y be real normed spaces and S be a nonempty subset of X. Let Y be partially ordered by a pointed, closed, and convex cone C ⊂ Y with apex at origin and a nonempty interior int C. Let F : S ⇒ Y be a set-valued map, xˆ ∈ S, yˆ ∈ F (x), ˆ and (u, ˆ v) ˆ ∈ X ×Y. Firstly, we establish an existence theorem for generalized second-order composed contingent epiderivatives. In fact, the conditions of Theorem 2 in Chen and Jahn [6] assuring the existence of generalized contingent epiderivatives are also sufficient for the existence of generalized second-order composed contingent epiderivatives. Theorem 3.1 Let Assumption 3.1 be satisfied, C be Daniel, and (u, ˆ v) ˆ ∈ T (epi F, (x, ˆ y)). ˆ If for every x ∈ X, the set G(x), defined by (4), is C-lower bounded, then for all x ∈ X, Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ exists. Proof Since G(x) = T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ is always a closed cone in a normed space, and for every x ∈ X, G(x) is C-lower bounded, it follows from the existence theorem of minimal elements in Luc [3] that MinG(x) is nonempty, that is, for all x ∈ X, Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ exists. Next, we give the following important property of generalized second-order composed contingent epiderivatives for the C-function.
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Theorem 3.2 Let Assumption 3.1 be satisfied and let for every x ∈ X, the generalized second-order composed contingent epiderivative Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ = ∅. Then Dg F (x, ˆ y, ˆ u, ˆ v) ˆ is strictly positive homogeneous. Moreover, if F is a C-function and the set G(x), defined by (4), fulfills the domination property for all x ∈ X, then Dg F (x, ˆ y, ˆ u, ˆ v) ˆ is subadditive. Proof (a) (strictly positive homogenity) For every λ > 0 and x ∈ X, we have ˆ y, ˆ u, ˆ v)(λx) ˆ = Min y ∈ Y | (λx, y) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ Dg F (x, = Min λu ∈ Y | (λx, λu) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = λ Min u ∈ Y | (x, u) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ ˆ y, ˆ u, ˆ v)(x). ˆ = λDg F (x, (b) (subadditivity) It follows from Definition 2.8 that for every x1 , x2 ∈ X, and y1 ∈ Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 ),
y2 ∈ Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 2 ),
we have (x1 , y1 ) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)), ˆ and (x2 , y2 ) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ Since F is a C-function, epi F is convex. Then T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)) ˆ is a closed and convex cone. Thus, we have 12 (x1 , y1 ) + 12 (x2 , y2 ) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)), ˆ that is, (x1 + x2 , y1 + y2 ) ∈ T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ , which implies 1 + x2 ). Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 ) + Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 2 ) ⊂ G(x It follows from the domination property and Definition 2.8 that 1 + x2 ) ⊂ Min G(x 1 + x2 ) + C = Dg F (x, G(x ˆ y, ˆ u, ˆ v)(x ˆ 1 + x2 ) + C. Therefore, we have Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 ) + Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 2 ) ⊂ Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ 1 + x2 ) + C. Remark 3.1 Notice that in Theorem 3.2, we establish a special property of the generalized second-order composed contingent epiderivative, which is similar to the corresponding property of the generalized contingent epiderivative in [6]. But we can not obtain the similar property for the generalized second-order contingent epiderivative,
even though F be a C-function and the set G(x), defined by (3), fulfill the domination property for all x ∈ X. The following example explains Theorem 3.2 and the case mentioned in Remark 3.1.
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Example 3.1 Consider Example 2.1. Obviously, F is a C-function and the sets G(x) and G(x), defined respectively by (3) and (4), fulfill the domination property for all x ∈ X. Moreover, for (x, ˆ y) ˆ = (0, (0, 0)), and (u, ˆ v) ˆ = (1, (0, 1)), we have Dg F (x, ˆ y, ˆ u, ˆ v) ˆ = y ∈ R2 y1 ≥ 0, y1 + y2 = x , ∀x ∈ R, which is strictly positive homogeneous and subadditive. But Dg2 F (x, ˆ y, ˆ u, ˆ v) ˆ = y ∈ R2 y1 ≥ 2, y1 + y2 = x ,
∀x ∈ R,
which is merely subadditive. Remark 3.2 Since (0X , 0Y ) always belongs to T (epi F, (x, ˆ y)) ˆ and T epi F, (x, ˆ y) ˆ = T T epi F, (x, ˆ y) ˆ , (0X , 0Y ) , Theorem 3.1 and Theorem 3.2, respectively, reduce to Theorem 2 and Theorem 1 in [6] when we take (u, ˆ v) ˆ = (0X , 0Y ). 4 Optimality Conditions for Set-Valued Optimization Problems In this section, we consider the following set-valued optimization problem: (P)
min F (x),
s.t. x ∈ S,
where S is a nonempty subset of X and F : S ⇒ Y is a set-valued map. Now, we recall some optimality notions for the problem (P). A pair (x, ˆ y) ˆ ∈ graph F is called a weak minimizer of the problem (P) iff yˆ is a weak minimal element of the set F (S), i.e.,
F (S) ∩ {y}−int ˆ C = ∅, where F (S) := F (x). x∈S
A pair (x, ˆ y) ˆ ∈ graph F is called a local weak minimizer of the problem (P) iff there is a neighborhood U of xˆ such that yˆ is a weak minimal element of the set F (S ∩ U ), i.e.,
F (S ∩ U ) ∩ {y}−int ˆ C = ∅, where F (S ∩ U ) := F (x). x∈S∩U
Next, we apply generalized second-order composed contingent epiderivatives for set-valued maps to discuss some second-order optimality conditions for the setvalued optimization problem (P). Theorem 4.1 Let Assumption 3.1 be satisfied and let (x, ˆ y) ˆ ∈ graph F be a local weak minimizer of the problem (P). Then, for every (u, ˆ v) ˆ ∈ T (epi F, (x, ˆ y)) ˆ such that vˆ ∈ −C and every x ∈ dom Dg F (x, ˆ y, ˆ u, ˆ v), ˆ we have Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Y \ −int C − {v} ˆ . (6)
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Proof Suppose that there exists some x ∈ dom Dg F (x, ˆ y, ˆ u, ˆ v) ˆ such that (6) is not satisfied, that is, there exits some y ∈ Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ∩ −int C − {v} ˆ . (7) By Definition 2.8, we have (x, y) ∈ T (T (epi F, (x, ˆ y)), ˆ (u, ˆ v)). ˆ By Proposition 2.2, there exist sequences λn → +∞ and (un , vn ) in T (epi F, (x, ˆ y)), ˆ such that (un , vn ) → (u, ˆ v) ˆ and λn (un , vn ) − (u, ˆ v) ˆ → (x, y), as n → +∞. (8) ˆ y)), ˆ for every n ∈ N, there exist a sequence λkn → +∞ Since (un , vn ) ∈ T (epi F, (x, k k and a sequence (xn , yn ) in epi F , such that (xnk , ynk ) → (x, ˆ y) ˆ and ˆ y) ˆ → (un , vn ), λkn xnk , ynk − (x,
as k → +∞.
(9)
It follows from (7) and (8) that y ∈ −int C − {v} ˆ and λn (vn − v) ˆ → y as n → +∞. Then there exists N1 ∈ N such that λn (vn − v) ˆ ∈ −int C − {v}, ˆ ∀n ≥ N1 , that is, 1 vˆ ∈ −int C, ∀n ≥ N1 . vn − 1 − (10) λn Since λn → +∞ and vˆ ∈ −C, there exists N2 ∈ N, such that λn 1 for every n ≥ N2 , and then 1 1− vˆ ∈ −C, ∀n ≥ N2 . (11) λn Thus, by (10) and (11), we have for every n ≥ max(N1 , N2 ), 1 1 vˆ + 1 − vˆ ∈ −int C − C vn = vn − 1 − λn λn = −int C.
(12)
Moreover, by (9), we have λkn (ynk − y) ˆ → vn as k → +∞. Together with (12), we have for every n ∈ N and n ≥ max(N1 , N2 ), there exists K1 (n) ∈ N such that λkn ynk − yˆ ∈ −int C, ∀k ≥ K1 (n), that is, ynk − yˆ ∈ −int C,
∀n ≥ max(N1 , N2 ), ∀k ≥ K1 (n).
(13)
Since (xnk , ynk ) ∈ epi F , there exists some y¯nk ∈ F (xnk ), such that ynk ∈ {y¯nk } + C. Together with (13), we have ∀n ≥ max(N1 , N2 ), ∀k ≥ K1 (n), y¯nk − yˆ ∈ ynk − {y} ˆ −C ⊂ −int C − C = −int C.
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Consequently, y¯nk ∈ {y}−int ˆ C,
∀n ≥ max(N1 , N2 ), ∀k ≥ K1 (n).
Since ∀n ∈ N, xnk ∈ S and xnk → xˆ as k → +∞, for every neighborhood U of x, ˆ there exists K2 (n) ∈ N, such that xnk ∈ S ∩ U, ∀k ≥ K2 (n). Therefore, we can conclude that for every neighborhood U of x, ˆ and sufficiently large n, k ∈ N, n ≥ max(N1 , N2 ) and k ≥ max(K1 (n), K2 (n)), there are xnk ∈ S ∩ U , such that ˆ C = ∅. F xnk ∩ {y}−int This is a contradiction to the assumption that (x, ˆ y) ˆ ∈ graph F is a local weak minimizer of the problem (P). Remark 4.1 It follows from what we have mentioned in Remark 3.2 that Theorem 4.1 is a generalization of Theorem 5 in [6]. As we know, Theorems 3.1 in [15] still holds for generalized second-order contingent epiderivatives. However, if the assumptions in Proposition 2.5 hold, then Theorem 4.1 is extensive than Theorems 3.1 in [15] for the case of generalized second-order contingent epiderivatives. The following example illustrates Theorem 4.1 and the case mentioned in Remark 4.1. Example 4.1 Consider Example 2.1. Let F : S ⇒ R2 with F (x) := y = (y1 , y2 ) ∈ R2 y1 ≥ x 2 , y1 + y2 ≥ x ,
∀x ∈ S,
ˆ y) ˆ = (0, (0, 0)) is a local weak minwhere S = [−1, 1]. Let C = R2+ . Obviously, (x, imizer of the following set-valued optimization problem: (P∗ )
min F (x),
s.t. x ∈ S.
By Definition 2.1, we have T epi F, (x, ˆ y) ˆ = (x, y) ∈ R × R2 x ∈ R, y1 ≥ 0, y1 + y2 ≥ x . We denote the set H (x, ˆ y) ˆ = (u, ˆ v) ˆ ∈ R × R2 (u, ˆ v) ˆ ∈ T epi F, (x, ˆ y) ˆ with vˆ ∈ −C = u, ˆ (0, vˆ2 ) uˆ ∈ R, uˆ ≤ vˆ2 ≤ 0 . Moreover, by directly calculating, we have for every (u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ if 0 ≥ vˆ2 > u; ˆ {(x, y) | x ∈ R, y1 ≥ 0, y2 ∈ R}, T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = ˆ {(x, y) | x ∈ R, y1 ≥ 0, y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u, and
if 0 ≥ vˆ2 > u; ˆ {(x, y) | x ∈ R, y1 ≥ 2uˆ 2 , y2 ∈ R}, ˆ y), ˆ (u, ˆ v) ˆ = T epi F, (x, {(x, y) | x ∈ R, y1 ≥ 2uˆ 2 , y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u. ˆ 2
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Then, for every x ∈ R and every (u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ we get if 0 ≥ vˆ2 > u; ˆ = {y | y1 ≥ 0, y2 ∈ R}, G(x) ˆ {y | y1 ≥ 0, y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u, and
2 if 0 ≥ vˆ2 > u; ˆ
= {y | y1 ≥ 2uˆ 2 , y2 ∈ R}, G(x) {y | y1 ≥ 2uˆ , y1 + y2 ≥ x}, if 0 ≥ vˆ2 = u. ˆ
Obviously, G(x) and G(x) respectively fulfill the domination property for every x ∈ R and every (u, ˆ v) ˆ ∈ H (x, ˆ y). ˆ It follows from Definitions 2.7 and 2.8 that for every x ∈ R, we have ˆ ∅, if 0 ≥ vˆ2 > u; ˆ y, ˆ u, ˆ v)(x) ˆ = Dg F (x, {y ∈ R2 | y1 ≥ 0, y1 + y2 = x}, if 0 ≥ vˆ2 = u, ˆ and Dg2 F (x, ˆ y, ˆ u, ˆ v)(x) ˆ =
∅, {y ∈ R2 | y1 ≥ 2uˆ 2 , y1 + y2 = x},
ˆ if 0 ≥ vˆ2 > u; if 0 ≥ vˆ2 = u. ˆ
On the one hand, −int C − {v} ˆ = (y1 , y2 ) ∈ R2 y1 < 0, y2 < −vˆ2 ,
∀(u, ˆ v) ˆ ∈ H (x, ˆ y). ˆ
ˆ y, ˆ u, ˆ v), ˆ we have Thus, it is evident that for every x ∈ dom Dg F (x, Dg F (x, ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Y \ −int C − {v} ˆ , and the necessary conditions in Theorem 4.1 are satisfied. On the other hand, for every (u, ˆ v) ˆ ∈ H (x, ˆ y) ˆ with uˆ = 0, we have ˆ y, ˆ u, ˆ v)(x) ˆ Dg F (x, ˆ y, ˆ u, ˆ v)(x), ˆ Dg2 F (x,
∀x ∈ R,
ˆ y, ˆ u, ˆ v)(x) ˆ ⊂ Y \ −int C − {v} ˆ , Dg2 F (x,
∀x ∈ R.
which implies
Theorem 4.2 Let Assumption 3.1 be satisfied, S be convex, and F be a C-function. − xˆ − u), If for every (u, ˆ v) ˆ ∈ T (epi F, (x, ˆ y)) ˆ with vˆ ∈ −C, the set G(x ˆ defined by (4), fulfills the domination property for all x ∈ S and ∅ = Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ − xˆ − u) ˆ ⊂ Y \ −int C − {v} ˆ , ∀x ∈ S, (14) then (x, ˆ y) ˆ ∈ graph F is a weak minimizer of the problem (P). Proof Since (0X , 0Y ) always belongs to T (epi F, (x, ˆ y)), ˆ 0Y ∈ −C, and T epi F, (x, ˆ y) ˆ = T T epi F, (x, ˆ y) ˆ , (0X , 0Y ) ,
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it follows from Definitions 2.6 and 2.8 that the condition (14) implies ∅ = Dg F (x, ˆ y)(x ˆ − x) ˆ ⊂ Y \ (−int C),
∀x ∈ S.
By Theorem 6 in [6], we can conclude that (x, ˆ y) ˆ ∈ graph F is a weak minimizer of the problem (P). Remark 4.2 Obviously, Theorem 4.2 generalizes Theorem 6 in [6]. Simultaneously, Theorem 4.2 is different from Theorem 3.2 in [15] since the existences of the contingent epiderivative DF (x, ˆ y) ˆ (see [5]) and the second-order contingent epiderivative ˆ y, ˆ u, ˆ v) ˆ (see [15]) are hard to be satisfied in general cases. D 2 F (x, The following example explains Theorem 4.2 and the case mentioned in Remark 4.2. Example 4.2 Consider Example 2.1 and the following set-valued optimization problem: (P∗∗ )
min F (x),
s.t.
x ∈ R+ .
Let (x, ˆ y) ˆ = (0, (0, 0)) and C = R2+ . Similarly, we have T epi F, (x, ˆ y) ˆ = (x, y) ∈ R × R2 x ≥ 0, y1 ≥ 0, y1 + y2 ≥ x .
(15)
Denote the set H (x, ˆ y) ˆ = (u, ˆ v) ˆ ∈ R × R2 (u, ˆ v) ˆ ∈ T epi F, (x, ˆ y) ˆ with vˆ ∈ −C = 0, (0, 0) . Thus, for every (u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ we have T T epi F, (x, ˆ y) ˆ , (u, ˆ v) ˆ = T epi F, (x, ˆ y) ˆ
(16)
and for every x ∈ R+ , we have
− xˆ − u) G(x ˆ = y ∈ R2 | y1 ≥ 0, y1 + y2 ≥ x ,
which satisfies the domination property. Moreover, it follows from (15) and (16) that for every (u, ˆ v) ˆ ∈ H (x, ˆ y) ˆ and every x ∈ R+ , Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ − xˆ − u) ˆ = Dg F (x, ˆ y)(x ˆ − x) ˆ = y ∈ R2 y1 ≥ 0, y1 + y2 = x . Then we have ∅ = Dg F (x, ˆ y, ˆ u, ˆ v)(x ˆ − xˆ − u) ˆ ⊂ Y \ −int C −{v} ˆ ,
∀(u, ˆ v) ˆ ∈ H (x, ˆ y), ˆ ∀x ∈ R+ .
Therefore, the sufficient conditions in Theorem 4.2 are fulfilled, and then (x, ˆ y) ˆ is a weak minimizer of the problem (P∗∗ ). But it can be seen from (15) that DF (x, ˆ y) ˆ
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does not exist. In fact, if DF (x, ˆ y) ˆ exists, by the definition of the contingent epiderivative in [5], we have for every x¯ ≥ 0, y ∈ R2 y1 ≥ 0, y1 + y2 ≥ x¯ ⊂ DF (x, ˆ y)( ˆ x) ¯ + R2+ . This is impossible. Therefore, Theorem 3.2 in [15] is not applicable.
5 Concluding Remarks In this paper, we propose a new concept of a second-order derivative for set-valued maps, which is called the generalized second-order composed contingent epiderivative, and discuss some relationships between the derivative and the generalized second-order contingent epiderivative. We also investigate some of its properties. Simultaneously, by virtue of the derivative, we obtain a unified second-order sufficient and necessary optimality condition for set-valued optimization problems, which improves existing results. Acknowledgements This research was supported by the National Natural Science Foundation of China (Grant No. 10871216). The authors thank the two anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper.
References 1. Aubin, J.P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Mathematical Analysis and Applications, Part A, pp. 160–229. Academic Press, New York (1981) 2. Corley, H.W.: Optimality conditions for maximizations of set-valued functions. J. Optim. Theory Appl. 58, 1–10 (1988) 3. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) 4. Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991) 5. Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46, 193–211 (1997) 6. Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res. 48, 187–200 (1998) 7. Jahn, J., Khan, A.A.: The existence of contingent epiderivatives for set-valued maps. Appl. Math. Lett. 16, 1179–1185 (2003) 8. Luis, R.M., Miguel, S.: About contingent epiderivatives. J. Math. Anal. Appl. 327, 745–762 (2007) 9. Bigi, G.: On sufficient second order optimality conditions in multiobjective optimization. Math. Methods Oper. Res. 63, 77–85 (2006) 10. Bonnans, J.F., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9, 466–492 (1999) 11. Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004) 12. Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1998) 13. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) 14. Durea, M.: First and second order optimality conditions for set-valued optimization problems. Rend. Circ. Mat. Palermo 2, 451–468 (2004)
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15. Jahn, J., Khan, A.A., Zeilinger, P.: Second-order optimality conditions in set optimization. J. Optim. Theory Appl. 125, 331–347 (2005) 16. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) 17. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984) 18. Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)
