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Generalized Constraint Qualifications and Optimality Conditions for Set-Valued Optimization Problems
Journal of Mathematical Analysis and Applications 265, 309᎐321 Ž2002. doi:10.1006rjmaa.2001.7705, available online at http:rrwww.idealibrary.com on
Generalized Constraint Qualifications and Optimality Conditions for Set-Valued Optimization Problems Huang Yong-Wei1 Institute of Applied Mathematics, Chongqing Uni¨ ersity (Campus B), Chongqing, 400045, People’s Republic of China E-mail: [email protected] Submitted by Koichi Mizukami Received July 26, 2000
In this paper we discuss the connections of four generalized constraint qualifications for set-valued vector optimization problems with constraints. Then some K-T type necessary and sufficient optimality conditions are derived, in terms of the contingent epiderivatives. 䊚 2002 Elsevier Science Key Words: set-valued maps; generalized constraint qualifications; optimality conditions; contingent epiderivatives.
1. INTRODUCTION In recent years, there has been an increasing amount of attention to optimality conditions for set-valued optimization problems. For instance, Jahn and Rauh w1x introduced the notion of the contingent epiderivative of a set-valued map and derived the formulation of optimality conditions in the unconstrained set-valued optimization. Li w3x obtained the optimality conditions for set-valued optimization by applying the alternative theorem in real linear space without any topology. Lin w5x defined a weak subdifferential of set-valued maps in real linear topological spaces, and the Moreau᎐Rockafellar type and the Farkas᎐Minkowski type theorems for set-valued maps were generalized. Through these, the K-T type necessary conditions for the existence of a weak minimal point have been given. Yang w7x introduced Dini directional derivatives for set-valued maps, in terms of which a derivative concept of a Jacobificator for set-valued maps 1 Current address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, People’s Republic of China.
309 0022-247Xr02 $35.00 䊚 2002 Elsevier Science All rights reserved.
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is introduced. Their applications were given to present optimality conditions and mean value theorems. In this paper we are concerned with the generalized constraint qualifications and the optimality conditions for set-valued optimization problems. In Section 2, we give some notations and preliminaries. In Section 3, we introduce four constraint qualifications and investigate the connections between them. In Section 4, we obtain some Kuhn᎐Tucker type sufficient and necessary conditions for set-valued optimization problems with constraints, in terms of contingent epiderivatives.
2. NOTATIONS AND PRELIMINARIES Throughout this paper, let X, Y, Z, and W be real normed linear spaces. Let Y, Z, and W be partially ordered by pointed convex cones Yq, Zq, Wq, respectively. Assume that the interiors of Yq, Zq, denoted by int Yq, int Zq respectively, are both nonempty. However, the interior of Wq is not necessary to be nonempty. We denote by Y U , ZU , W U the dual U spaces of Y, Z, W, respectively. The dual cone Yq of Yq is defined by U U U U Yq s y g Y ¬ ² y, y : G 0, ᭙ y g Yq 4 , where ² y, yU : denotes the value of the linear continuous functional yU at the point y. The meanings of U U Zq , Wq are similar. Let D be a nonempty subset of X, let F: D ª 2 Y, G: D ª 2 Z , H: D ª 2 W be set-valued maps such that F Ž x . / ⭋, G Ž x . / ⭋, H Ž x . / ⭋, ᭙ x g D. Let F Ž D. s
D FŽ x. ,
² F Ž x . , yU : s ² y, yU : ¬ y g F Ž x . 4 ,
xgD
² F Ž D . , yU : s
D ² F Ž x . , yU : .
xgD
We denote by R the set of real numbers, by N the set of natural numbers, by O the null element of every space. For A ; R, b g R, write AG ŽF , ) , -. b, iff a G ŽF , ) , -. b, ᭙a g A. DEFINITION 2.1. Ža. A subset M in Y is called nearly convex if ᭚␣ g Ž0, 1., such that for ᭙ y 1 , y 2 g M, ␣ y 1 q Ž1 y ␣ . y 2 g M. Žb. The set-valued map F: D ª 2 Y is Yq-convex on D if D is convex, and for x 1 , x 2 g D and ᭙ g Ž0, 1., F Ž x 1 . q Ž1 y . F Ž x 2 . ; F Ž x 1 q Ž1 y . x 2 . q Yq. Žc. The set-valued map F: D ª 2 Y is nearly Yq-convexlike on D, if there is g Ž0, 1. such that for ᭙ x 1 , x 2 g D, F Ž x 1 . q Ž1 y . F Ž x 2 . ; F Ž D . q Yq.
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It is well known that Yq-convex « Yq-convexlike Ž see w 8 x . « Yq-subconvexlike Ž see w 8 x . « generalized Yq-subconvexlike Žsee w17x., and that Yq-convexlike « nearly Yq-convexlike. DEFINITION 2.2. Ža. The set epi F s Ž x, y . g X = Y ¬ x g D, y g F Ž x . q Yq 4 is called the epigraph of F on D. Žb. Let S be a nonempty subset of W, and let s0 g cl S Žclosure of S . be a given element. A vector h g W is called a tangent vector to S at s0 , if there are a sequence Ž z n . ng N of elements in S and a sequence Ž n . ng N of positive real numbers with s0 s lim nª⬁ z n and h s lim nª⬁ nŽ z n y s0 .. The set T Ž S, s0 . of all tangent vectors to S at s0 is called a contingent cone to S at s0 . Žc. Let a pair Ž x 0 , y 0 . g X = Y with x 0 g D and y 0 g F Ž x 0 . be given. Denote by TX ŽepiŽ F ., Ž x 0 , y 0 .. the projection of T ŽepiŽ F ., Ž x 0 , y 0 .. on X. A single-valued map DF Ž x 0 , y 0 .: TX ŽepiŽ F ., Ž x 0 , y 0 .. ; X ª Y, whose epigraph equals the contingent cone to the epigraph of F at Ž x 0 , y 0 ., i.e., epiŽ DF Ž x 0 , y 0 .. s T ŽepiŽ F ., Ž x 0 , y 0 .., is called contingent epiderivative of F at Ž x 0 , y 0 .. Consider the set-valued optimization problem ŽP.
Ž P . min F Ž x .
s.t. x g K ,
where the feasible set K s x g D ¬ yGŽ x . l Zq/ ⭋, yH Ž x . l Wq / ⭋4 . DEFINITION 2.3. A point Ž x 0 , y 0 , z 0 , w 0 . is called a feasible point of ŽP. if x 0 g K, y 0 g F Ž x 0 ., z 0 g G Ž x 0 . l y Zq, w 0 g H Ž x 0 . l y Wq. A point x 0 g K is called a weakly efficient solution of ŽP., if ᭚ y 0 g F Ž x 0 . such that Ž y 0 y F Ž K .. l int Yqs ⭋. In this case, the point Ž x 0 , y 0 , z 0 , w 0 . is called a weak minimizer of ŽP.. Remark 2.1. It is obvious that when Wqs O 4 , ŽP. can be written as ŽPX .,
Ž PX . min F Ž x .
s.t. x g K X ,
where the set K X s x g D ¬ yGŽ x . l Zq/ ⭋, O g H Ž x .4 . Since ŽPX . is a particular case of ŽP., we are concerned in this paper with Problem ŽP.. The following lemmas will be very useful.
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LEMMA 2.1 Žsee w9x.. con¨ ex set. LEMMA 2.2 Žsee w6x..
If M ; Y is a nearly con¨ ex set, then int M is a U If yU0 g Yq _ O 4 , y 0 g int Yq, then ² y 0 , yU0 : ) 0.
LEMMA 2.3 Žsee w9x.. Let M ; Y be a nearly con¨ ex set, and let int M / ⭋. Suppose that for each yU g Y U _ O4 , ² u, yU : ) 0, ᭙ u g int M. Then ² u, yU : G 0, ᭙ u g M.
3. GENERALIZED CONSTRAINT QUALIFICATIONS In what follows, we put U s Z = W, Uqs Zq= Wq, J s Ž G, H .: D ª 2 U. The notation Ž G, H .Ž x . is used for G Ž x . = H Ž x . here. One can easily U verify that U U s Ž Z = W .U s ZU = W U , and that Uq s Ž Zq= Wq .U s U U Zq= Wq. Now we consider the following four generalized constraint qualifications ŽQ 0 . to ŽQ 3 .. Let Rys r g R ¬ r - 04 . ŽQ 0 . ᭚ xX g D such that yGŽ xX . l int Zq/ ⭋, yint H Ž xX . l Wq / ⭋. ŽQ 1 . Ži. ᭚ xX g D such that yGŽ xX . l int Zq/ ⭋, yH Ž xX . l Wq / ⭋. Žii. yint H Ž D . l Wq/ ⭋. U ŽQ 2 . Ži. For each zU g Zq _ O 4 , there is xX g D such that ² G Ž xX ., X U z : l Ry/ ⭋, yH Ž x . l Wq/ ⭋. U Žii. For each wU g Wq _ O 4 , there is xX g D such that ² H Ž xX ., U w : l Ry/ ⭋. U U ŽQ 3 . For each Ž zU , wU . g Zq = Wq _Ž O, O .4 , there is x⬘ g D such that Ž² G Ž xX . , zU: q² H Ž xX . , wU: . l Ry/ ⭋. THEOREM 3.1. The following implications are true:
Ž Q 0 . « Ž Q1 . « Ž Q 2 . « Ž Q 3 . . Proof. It is easy to show ŽQ 0 . « ŽQ 1 .. Now we prove ŽQ 1 . « ŽQ 2 .. Suppose that ŽQ 1 . holds. U Ža. Let zU g Zq _ O 4 . Hence ᭚ xX g D such that yGŽ x⬘. l int Zq / ⭋. By Lemma 2.2 we have ²int Zq, zU : ) 0. Therefore ² G Ž xX ., zU : l Ry/ ⭋. U Žb. Let wU g Wq _ O4 . Thus ᭚ w 0 g W such that ² w 0 , wU : ) 0. Let w 1 g yint H Ž D . l Wq. So, ᭚ ) 0 such that w 1 q w 0 g yint H Ž D . ; yH Ž D .. Hence ᭚ xX g D such that w 1 q w 0 g yH Ž xX .. Since w 1 g Wq,
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then ² w 1 q w 0 , wU : s ² w 1 , wU : q ² w 0 , wU : ) 0. Therefore ² H Ž xX ., wU : l Ry/ ⭋. U Next we show ŽQ 2 . « ŽQ 3 .. Assume that ŽQ 2 . holds. Let Ž zU , wU . g Zq U = Wq _Ž O, O .4 . If zU s O, then ŽQ 3 . is introduced directly by Žii. of ŽQ 2 .. If zU / O, then from Ži. of ŽQ 2 ., ᭚ xX g D, such that ² G Ž xX ., zU : l Ry/ ⭋, yH Ž xX . l Wq/ ⭋. Thus ² H Ž xX ., wU : l Ž Ryj 04. / ⭋. Hence ᭚ r 1 g ² G Ž xX ., zU : l Ry, r 2 g ² H Ž xX ., wU : l Ž Ryj 04. such that r 1 q r 2 - 0. This means Ž² G Ž xX ., zU : q ² H Ž xX ., wU :. l Ry/ ⭋. LEMMA 3.1. intŽ J Ž D . q Uq . / ⭋, if and only if intŽ J Ž D . q Žint Zq . = Wq . / ⭋. Indeed, the proof of the last lemma is similar to the proof of Lemma 2.2 w9x, when J is a vector-valued map. U U U LEMMA 3.2. If uU s Ž zU , wU . g Uq s Zq = Wq , zU / O, u s Ž z, w . g Žint Zq . = Wq, then ² u, uU : ) 0. U Proof. Since zU g Zq _ O 4 , and since z g int Zq, thus ² z, zU : ) 0. U U Hence ² u, u : s ² z, z : q ² w, wU : ) 0, because of ² w, wU : G 0.
LEMMA 3.3. If the set-¨ alued map J: D ª 2 U is nearly Uq-con¨ exlike on D, then M s J Ž D . q Žint Zq . = Wq is nearly con¨ ex. Proof. Let m1 , m 2 g M; then ᭚ x i g D, u i g Žint Zq . = Wq, such that m i g J Ž x i . q u i , i s 1, 2. Since J is nearly Uq-convexlike, there exists ␣ g Ž0, 1.; for the previous x 1 , x 2 g D, we have
␣ J Ž x 1 . q Ž 1 y ␣ . J Ž x 2 . ; J Ž D . q Uq . Thus ␣ Ž m1 y u1 . q Ž1 y ␣ .Ž m 2 y u 2 . g J Ž D . q Uq. Because the set Žint Zq . = Wq is convex, so we have u 0 s ␣ u1 q Ž1 y ␣ . u 2 g Žint Zq . = Wq. Thereby, m [ ␣ m1 q Ž 1 y ␣ . m 2 g ␣ J Ž x 1 . q Ž 1 y ␣ . J Ž x 2 . q u 0 . Because of int Zqq Zq; int Zq, so m g J Ž D . q Uqq u 0 ; J Ž D . q Ž int Zq . = Wqs M. Therefore M is nearly convex. THEOREM 3.2. Let intŽ J Ž D . q Uq . / ⭋. Suppose that J is nearly Uqcon¨ exlike on D. Let H Ž D . be a con¨ ex set with a nonempty topological interior. Then
Ž Q1 . m Ž Q 2 . m Ž Q 3 . .
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Proof. We need only show ŽQ 3 . « ŽQ 1 .. Suppose that Ži. of ŽQ 1 . is not true. Then for any x g D, we have O f G Ž x . q int Zq or O f H Ž x . q Wq, i.e., Ž O, O . f M s J Ž D . q Žint Zq . = Wq. According to Lemma 3.1 and the assumption of intŽ J Ž D . q Uq . / ⭋, we have int M / ⭋. Furthermore, Ž O, O . f int M. Since J is nearly Uq-convexlike on D, thus by Lemma 3.3, M is a nearly convex set. Hence, int M is convex. By applying the separation theorem of convex sets of linear topological spaces Žsee w19x., there is a hyperplane E properly separating O 4 and int M, i.e., ᭚ uU s Ž zU , wU . g ZU = W U _Ž O, O .4 , a g R, such that ² u, uU : G a G 0,
᭙ u g int M,
Ž 3.1.
U
where the hyperplane E s y g U ¬ ² y, u : s a4 . Next, we show that ² u, uU : ) 0,
᭙ u g int M.
Ž 3.2.
We have two cases. One case is a ) 0. Then it follows by Ž3.1. and Ž3.2. holds. The other case is a s 0. We have ² u, uU : G 0,
᭙ u g int M.
Ž 3.3.
Suppose that Ž3.2. does not hold. According to Ž3.3., there is u 0 g int M such that ² u 0 , uU : s 0. Let ¨ g int M. Then ᭚ ) 0 such that u 0 y ¨ g int M. Thus by Ž3.3., we have ² u 0 y ¨ , uU : G 0, i.e., ² u 0 , uU : G ² ¨ , uU :. So, ² ¨ , uU : F 0. On the other hand, again by Ž3.3. we get ² ¨ , uU : G 0. Therefore, ² ¨ , uU : s 0, ᭙ ¨ g int M. This is absurd since the hyperplane E separates O 4 and int M properly. Hence the proof that Ž3.3. holds is complete. By Lemma 2.3 we obtain ² u, uU : G 0, U
᭙ u g M. U
U
U Zq = U
Ž 3.4. U Wq ;
In the following, we prove u s Ž z , w . g indeed, assume U z U f Zq . Then there is z1 g Zq such that ² z1 , z : - 0. By Ž3.4., for any x g D, any zX g int Zq, any wX g Wq, we have  s ² p q zX , zU : q ² q q wX , wU : G 0, ᭙ p g G Ž x ., ᭙q g H Ž x .. Since z1 g Zq, then z1 q zX g int Zq. Also by Ž3.4., we have ² p q z1 q zX , zU : q ² q q wX , wU : G 0, i.e., ² z1 , zU : q  G 0, ᭙ ) 0. Ž 3.5. However, Ž3.5. does not hold when ) yr² z1 , zU : G 0. This contradicU U tion illustrates that zU g Zq . Similarly, we can also prove wU g Wq . Thus, U U U U U U ᭚ u s Ž z , w . g Zq= Wq _Ž O, O .4 , such that ² u, u : G 0, ᭙ u g M, i.e.,
² G Ž x . q z, zU: q² H Ž x . q w, wU: G 0, ᭙ x g D, ᭙ z g int Zq , ᭙ w g Wq .
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Take w s O in the previous expression. Then
² G Ž x . , zU: q ² z, zU : q² H Ž x . , wU: G 0, ᭙ x g D, ᭙ z g int Zq , ᭙ ) 0.
Ž 3.6.
If zU s O, then from Ž3.6. we have ² H Ž x ., wU : G 0, ᭙ x g D, which contradicts ŽQ 3 .. Suppose zU / 0. According to ŽQ 3 ., we have that ᭚ xX g D, p g G Ž xX ., q g H Ž xX . such that ² p, zU : q ² q, wU : - 0. Therefore there is a sufficiently small ␦ ) 0 such that ² p, zU : q ² q, wU : q ␦ - 0.
Ž 3.7.
However, if we take s ␦r² z, zU : ) 0 in Ž3.6., then Ž3.7. is in contradiction to Ž3.6.. Therefore Ži. of ŽQ 1 . holds. In the following we prove that Žii. of ŽQ 1 . is true. Otherwise int H Ž D . l ŽyWq . s ⭋. Thus again by the separation theorem of convex sets, there is wU g W U _ O 4 such that ² H Ž D ., wU : G ² y Wq, wU :, i.e., ² H Ž D ., wU : U q ²Wq, wU : G 0. It is easy to check that wU g Wq . Furthermore U ² H Ž D ., w : G 0, which contradicts ŽQ 3 .. THEOREM 3.3. If ŽQ 0 . holds, then there exists y 1 g Y, such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. Proof. According to ŽQ 0 ., we have that there exists xX g D such that O g G Ž xX . q int Zq; intŽ G Ž xX . q Zq ., and O g int H Ž xX . q Wq; intŽ H Ž xX . q Wq .. Since ⭋ / F Ž xX ., thus ⭋ / F Ž xX . q int Yq; intŽ F Ž xX . q Yq .. Hence ᭚ y 1 g Y such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž xX . q Yq= Zq= Wq .. Remark 3.1. In fact, Theorem 3.1, Theorem 3.2, and Theorem 3.3 are all still true when Y, Z, W are respectively assumed to be linear topological spaces.
4. OPTIMALITY CONDITIONS In this section, we establish some K-T-type sufficient and necessary optimality conditions for set-valued optimization in terms of contingent epiderivatives which were introduced by Jahn and Rauh w1x. THEOREM 4.1 ŽSufficiency.. Let D ; X be a con¨ ex set, let Ž F, G, H .Ž x . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a feasible point of ŽP.. Suppose that ŽQ 1 . holds. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , U U U y 0 , z 0 , w 0 . exists and there is Ž yU , zU , wU . g Yq = Zq = Wq _Ž O, O, O .4 ,
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HUANG YONG-WEI
such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0, U
᭙ x g D,
U
² z 0 , z : q ² w 0 , w : s 0, then x 0 is a weakly efficient solution of ŽP.. Proof. According to Lemma 3 of w1x, for each x g D we have
Ž F , G, H . Ž x . y Ž y 0 , z 0 , w 0 . 4 ; D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . 4 q Yq= Zq= Wq . Thus for each x g D, each Ž y, z, w . g Ž F, G, H .Ž x ., there c g Yq, d g Zq, e g Wq such that
Ž y y y 0 y c, z y z 0 y d, w y w 0 y e . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . . Hence, by assumption, we obtain ² y y y 0 y c, yU : q ² z y z 0 y d, zU : q ² w y w 0 y e, wU : G 0, or ² y y y 0 , yU : q ² z, zU : q ² w, wU : G ² c, yU : q ² z 0 , zU : q ² d, zU : q ² w 0 , wU : q ² e, wU : s ² c, yU : q ² d, zU : q ² e, wU : G 0. Therefore, ² y y y 0 , yU : q ² z, zU : q ² w, wU : G 0, ᭙ Ž y, z, w . g Ž F , G, H . Ž x . , ᭙ x g D.
Ž 4.1.
Next we show yU / O. Suppose that yU s O. Then Ž zU , wU . / Ž O, O ., and Ž4.1. can be written as
² G Ž x . , zU: q² H Ž x . , wU: G 0,
᭙ x g D.
Ž 4.2.
Thus we have two cases. One case is z / 0. By Q 1 , ᭚ x g D, ᭚ z g G Ž xX ., wX g H Ž xX ., such that yzX g int Zq, ywX g Wq. Hence, it follows by Lemma 3.2 that ² zX , zU : q ² wX , wU : - 0, which contradicts Ž4.2.. The other case is zU s 0. Then wU / O. Again by Q 1 , there is w 噛 g Wq such that yw 噛 g int H Ž D .. Then for any w g W, ᭚ ) 0 such that yw 噛 " w g H Ž D .. So, ² y w 噛 " w, wU : G 0, i.e., "² w, wU : G ² w 噛 , wU : G 0. Therefore, ² w, wU : s 0. It follows that wU s 0. This is a contradiction. Therefore we have yU / 0. U
X
X
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Finally, we prove x 0 is a weakly efficient solution of ŽP1.. Otherwise, ᭚ x 噛 g K such that Ž y 0 y F Ž x 噛 .. l int Yq/ ⭋. Thus, ᭚ t g F Ž x 噛 . such that y 0 y t g int Yq. By Lemma 2.2, we have ² t y y 0 , yU : - 0.
Ž 4.3.
Since x 噛 g K, there are p g G Ž x 噛 ., q g H Ž x 噛 . such that yp g Zq, yq g Wq. Taking Ž4.3. into account, we get ² t y y 0 , yU : q ² p, zU : q ² q, wU : - 0, which contradicts Ž4.1.. In the remainder of this paper, we assume that int Wq/ ⭋. THEOREM 4.2 ŽNecessity.. Let D ; Y be a con¨ ex set, let Ž F, G, H . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a weak minimizer of ŽP.. Suppose that Q 0 is fulfilled. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 . exists, and TX ŽepiŽ F, G, H ., Ž x 0 , y 0 , z 0 , w 0 .. s X, then U U U there is Ž yU , zU , wU . g Yq = Zq = Wq , with yU / O, such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0,
᭙ x g D,
² z 0 , zU : q ² w 0 , wU : s 0. Proof. Set P Ž x . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . ,
x g D.
We show first that P Ž x . is a Yq= Zq= Wq-convex single-valued map on D. Let x i g D, i s 1, 2, g Ž0, 1.. Thus by positive homogeneity, we have
P Ž x 1 . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x 1 y x 0 . ,
Ž 1 y . P Ž x 2 . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , W0 . Ž Ž 1 y . x 2 y x 0 . . By subadditivity, we obtain
P Ž x 1 . q Ž 1 y . P Ž x 2 . g P Ž x 1 q Ž 1 y . x 2 . q Yq= Zq= Wq . This illustrates that P Ž x . is Yq= Zq= Wq-convex on D. Set RŽ x . s P Ž x . q Ž O, z 0 , w 0 ., ᭙ x g D. Obviously, RŽ x . is also Yq= Zq= Wq-convex on D. Next we prove that yR Ž x . f int Yq= int Zq= int Wq ,
᭙ x g D.
Otherwise, there is xX g D such that yR Ž xX . [ y Ž yX , zX , wX . y Ž O, z 0 , w 0 . g int Yq= int Zq= int W .
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By definition of the contingent epiderivative,
Ž xX y x 0 , yX , zX , wX . g epi Ž D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . . s T Ž epi Ž F , G, H . , Ž x 0 , y 0 , z 0 , w 0 . . . According to the definition of the contingent cone, the following sequences exist: x n g D, c n g Yq, d n g Zq, e n g Wq, yn g F Ž x n . q c n , z n g GŽ x n . q d n , wn g H Ž x n . q e n , satisfying x n ª x 0 , yn ª y 0 , z n ª z 0 ; n ) 0, such that
n Ž x n y x 0 , yn y y 0 , z n y z 0 , wn y w 0 . ª Ž xX y x 0 , yX , zX , wX . . Thus, nŽ y 0 y yn . ª yyX , nŽ z 0 y z n . ª yzX , nŽ w 0 y wn . ª ywX . Because yyX g int Yq, so ᭚M1 g N such that nŽ y 0 y yn . g int Yq, ᭙ n G M1. Since int Yq is a cone, and since n ) 0, hence y 0 y yn g int Yq ,
᭙ n G M1 .
Ž 4.4.
Because yzX y z 0 g int Zq, so ᭚M2 g N such that nŽ z 0 y z n . y z 0 g int Zq, ᭙ n G M2 . That is nŽŽ1 y 1r n . z 0 y z n . g int Zq, ᭙ n G M2 , or Ž1 y 1r n . z 0 y z n g int Zq, ᭙ n G M2 . In a similar way, ᭚M3 g N such that Ž1 y 1r n . w 0 y wn g int Wq, ᭙ n G M3 . Take M s maxŽ M1 , M2 , M3 .. Whenever n G M, we have n ) k ) 1. Otherwise nŽ y 0 y yn . ª O g int Yq because of yn ª y 0 . This is absurd. Since Ž1 y 1r n . z 0 g yZq, and since int Zqq Zq; int Zq, therefore yz n g int Zq; Zq. So yz n g yŽ G Ž x n . q Zq . l Zq, ᭙ n G M. Since Zq is a convex cone, we get yG Ž x n . l Zq/ ⭋,
᭙ n G M.
Similarly, we have yH Ž x n . l Wq/ ⭋, ᭙ n G M. This means that x n belongs to a feasible set of ŽP.. It follows by Ž4.4. that
Ž y 0 y F Ž x n . y Yq . l int Yq/ ⭋,
᭙ n G M,
since yn g F Ž x n . q Yq. So, Ž y 0 y F Ž x n .. l int Yq/ ⭋, which does not coincide with the supposition that Ž x 0 , y 0 , z 0 , w 0 . is a weak minimizer of ŽP.. Therefore, yR Ž x . f int Yq= int Zq= int Wq ,
᭙ x g D.
By applying Lemma 2.2 of w13x or Theorem 3.1 of w12x, we have that U U U ᭚Ž yU , zU , wU . g Yq = Zq = Wq _Ž O, O, O .4 , such that
² R Ž x . , Ž yU , zU , wU .: G 0,
᭙ x g D.
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Hence,
² P Ž x . , Ž yU , zU , wU .: q ² z 0 , zU : q ² w 0 , wU : G 0,
᭙ x g D,
where P Ž x . s DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 .Ž x y x 0 .. Since Ž O, O, O . s P Ž x 0 ., hence ² z 0 , zU : q ² w 0 , wU : G 0. On the other hand, due to z 0 g yZq, w 0 g yWq, consequently, ² z 0 , zU : q ² w 0 , wU : F 0. Therefore, ² z 0 , zU : q ² w 0 , wU : s 0. Thus ² P Ž x ., Ž yU , zU , wU .: G 0, ᭙ x g D, i.e.,
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0,
᭙ x g D.
Next we show yU / O. Since Q 0 is fulfilled, then from Theorem 3.3, there exists y 1 g Y such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. Again by Lemma 3 of w1x, we have
Ž F , G, H . Ž x . y Ž y 0 , z 0 , w 0 . 4 ; D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . 4 qYq= Zq= Wq ,
᭙ x g D.
Thus ² y y y 0 , yU : q ² z, zU : q ² w, wU : G 0, ᭙ Ž y, z, w . g Ž F , G, H . Ž x . , ᭙ x g D.
Ž 4.5.
Suppose that yU s O. Then Ž zU , wU . / 0, and Ž4.5. can be written as
² G Ž x . , zU: q² H Ž x . , wU: G 0,
᭙ x g D.
Ž 4.6.
Thus,
² F Ž x . q Yq , yU: q² G Ž x . q Zq , zU: q² H Ž x . q Wq , wU: G 0, ᭙ x g D.
Ž 4.7.
Since Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq ., then for any x g D, ¨ 1 g GŽ x ., ¨ 2 g H Ž x ., k 1 g Zq, k 2 g Wq, there is ) 0 such that Ž y 1 , O, O . " Ž O, ¨ 1 q k 1 , ¨ 2 q k 2 . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. By Ž4.7., we have ² ¨ 1 q k 1 , zU : q ² ¨ 2 q k 2 , wU : s 0. Observing Ž4.6., we get ² k 1 , zU : q ² k 2 , wU : s 0,
᭙k 1 g Zq , ᭙k 2 g Wq .
Ž 4.8.
Then we also have two cases. One case is zU / 0. Then Ž4.8. contradicts Lemma 3.2. The other case is zU s 0. Then wU / 0. For any ¨ g W, there is 1 ) 0 such that Ž y 0 , O, O . " 1Ž O, O, ¨ . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. By Ž4.7., we obtain ² ¨ , wU : s 0, ᭙ ¨ g W. Thus wU s 0. This is a contradiction. Therefore we have yU / 0.
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COROLLARY 4.1. Let D ; Y be a con¨ ex set, let Ž F, G, H .Ž x . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a weak minimizer of ŽP.. Suppose that there exists y 1 g Y such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 . exists, and TX ŽepiŽ F, G, H ., Ž x 0 , y 0 , z 0 , w 0 .. s X, then there is Ž yU , zU , wU . U U U g Yq = Zq = Wq , with yU / O, such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0,
᭙ x g D;
² z 0 , zU : q ² w 0 , wU : s 0. COROLLARY 4.2. Let D ; Y be a con¨ ex set, let Ž F, G, H .Ž x . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a weak minimizer of ŽP.. Suppose that Q 1 is fulfilled. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 . exists, and TX ŽepiŽ F, G, H ., Ž x 0 , y 0 , z 0 , w 0 .. s X, then there is U U U Ž yU , zU , wU . g Yq = Zq = Wq , with yU / O, such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0, U
᭙ x g D;
U
² z 0 , z : q ² w 0 , w : s 0. REFERENCES 1. J. Jahn and R. Rauh, Contingent epiderivative and set-valued optimization, Math. Methods Oper. Res. 46 Ž1997., 193᎐211. 2. G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 Ž1998., 187᎐200. 3. Z. M. Li, The optimality conditions for vector optimization of set-valued maps, J. Math. Anal. Appl. 237 Ž1999., 365᎐375. 4. H. W. Corley, Optimality conditions for maximizations of set-valued functions, J. Optim. Theory Appl. 58 Ž1988., 1᎐10. 5. L. J. Lin, Optimization of set-valued functions, J. Math. Anal. Appl. 186 Ž1994., 30᎐51. 6. Y. D. Hu and Z. Q. Meng, On optimality conditions for vector optimization based on convex set-valued mapping, Oper. Res. Trans. 2 Ž1998., 1᎐5. 7. X. Q. Yang, Directional derivatives for set-valued mappings and applications, Math. Methods Oper. Res. 48 Ž1998., 273᎐285. 8. Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps, J. Optim. Theory Appl. 98 Ž1998., 623᎐649. 9. T. Illes and G. Kassay, Theorems of the alternative and optimality conditions for convexlike and general convexlike programming, J. Optim. Theory Appl. 101 Ž1999., 243᎐257. 10. D. T. Luc, Contingent derivatives of set-valued maps and applications to vector optimization, Math. Programming 50 Ž1991., 99᎐111. 11. A. Jourani, Constraint qualifications and Lagrange multipliers in nondifferentiable problems, J. Optim. Theory Appl. 81 Ž1994., 533᎐548. 12. G. Y. Chen and W. D. Rong, Characterizations of Benson proper efficiency for nonconvex vector optimization, J. Optim. Theory Appl. 98 Ž1998., 365᎐384.
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13. Z. M. Li, The optimality conditions of differentiable vector optimization problems, J. Math. Anal. Appl. 201 Ž1996., 35᎐43. 14. J.-P. Penot and P. H. Quang, Generalized convexity of functions and generalized monotonicity of set-valued maps, J. Optim. Theory Appl. 92 Ž1997., 343᎐356. 15. R. T. Rockafellar, Dualization of subgradient conditions for optimality, Nonlinear Anal. 20 Ž1993., 627᎐646. 16. Y. W. Huang, Theorems of the alternative for nearly convexlike set-valued maps, J. Chongqing Jianzhu Uni¨ . 22, No. 4 Ž2000., 65᎐68. 17. Y. W. Huang, Optimality conditions and Lagrangian multipliers of vector optimization with set-valued maps in linear spaces, Oper. Res. Trans. 5, No. 1 Ž2001., 63᎐69. 18. R. T. Rockafellar, ‘‘Convex Analysis,’’ Princeton Univ. Press, Princeton, NJ, 1970. 19. J. V. Tiel, ‘‘Convex Analysis: An Introductory Text,’’ Wiley, New York, 1984. 20. J.-P. Aubin and I. Ekeland, ‘‘Applied Nonlinear Analysis,’’ Wiley, New York, 1984. 21. J.-P. Aubin and H. Frankowska, ‘‘Set-Valued Analysis,’’ Birkhauser, Boston, 1990. ¨
Generalized Constraint Qualifications and Optimality Conditions for Set-Valued Optimization Problems Huang Yong-Wei1 Institute of Applied Mathematics, Chongqing Uni¨ ersity (Campus B), Chongqing, 400045, People’s Republic of China E-mail: [email protected] Submitted by Koichi Mizukami Received July 26, 2000
In this paper we discuss the connections of four generalized constraint qualifications for set-valued vector optimization problems with constraints. Then some K-T type necessary and sufficient optimality conditions are derived, in terms of the contingent epiderivatives. 䊚 2002 Elsevier Science Key Words: set-valued maps; generalized constraint qualifications; optimality conditions; contingent epiderivatives.
1. INTRODUCTION In recent years, there has been an increasing amount of attention to optimality conditions for set-valued optimization problems. For instance, Jahn and Rauh w1x introduced the notion of the contingent epiderivative of a set-valued map and derived the formulation of optimality conditions in the unconstrained set-valued optimization. Li w3x obtained the optimality conditions for set-valued optimization by applying the alternative theorem in real linear space without any topology. Lin w5x defined a weak subdifferential of set-valued maps in real linear topological spaces, and the Moreau᎐Rockafellar type and the Farkas᎐Minkowski type theorems for set-valued maps were generalized. Through these, the K-T type necessary conditions for the existence of a weak minimal point have been given. Yang w7x introduced Dini directional derivatives for set-valued maps, in terms of which a derivative concept of a Jacobificator for set-valued maps 1 Current address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, People’s Republic of China.
309 0022-247Xr02 $35.00 䊚 2002 Elsevier Science All rights reserved.
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is introduced. Their applications were given to present optimality conditions and mean value theorems. In this paper we are concerned with the generalized constraint qualifications and the optimality conditions for set-valued optimization problems. In Section 2, we give some notations and preliminaries. In Section 3, we introduce four constraint qualifications and investigate the connections between them. In Section 4, we obtain some Kuhn᎐Tucker type sufficient and necessary conditions for set-valued optimization problems with constraints, in terms of contingent epiderivatives.
2. NOTATIONS AND PRELIMINARIES Throughout this paper, let X, Y, Z, and W be real normed linear spaces. Let Y, Z, and W be partially ordered by pointed convex cones Yq, Zq, Wq, respectively. Assume that the interiors of Yq, Zq, denoted by int Yq, int Zq respectively, are both nonempty. However, the interior of Wq is not necessary to be nonempty. We denote by Y U , ZU , W U the dual U spaces of Y, Z, W, respectively. The dual cone Yq of Yq is defined by U U U U Yq s y g Y ¬ ² y, y : G 0, ᭙ y g Yq 4 , where ² y, yU : denotes the value of the linear continuous functional yU at the point y. The meanings of U U Zq , Wq are similar. Let D be a nonempty subset of X, let F: D ª 2 Y, G: D ª 2 Z , H: D ª 2 W be set-valued maps such that F Ž x . / ⭋, G Ž x . / ⭋, H Ž x . / ⭋, ᭙ x g D. Let F Ž D. s
D FŽ x. ,
² F Ž x . , yU : s ² y, yU : ¬ y g F Ž x . 4 ,
xgD
² F Ž D . , yU : s
D ² F Ž x . , yU : .
xgD
We denote by R the set of real numbers, by N the set of natural numbers, by O the null element of every space. For A ; R, b g R, write AG ŽF , ) , -. b, iff a G ŽF , ) , -. b, ᭙a g A. DEFINITION 2.1. Ža. A subset M in Y is called nearly convex if ᭚␣ g Ž0, 1., such that for ᭙ y 1 , y 2 g M, ␣ y 1 q Ž1 y ␣ . y 2 g M. Žb. The set-valued map F: D ª 2 Y is Yq-convex on D if D is convex, and for x 1 , x 2 g D and ᭙ g Ž0, 1., F Ž x 1 . q Ž1 y . F Ž x 2 . ; F Ž x 1 q Ž1 y . x 2 . q Yq. Žc. The set-valued map F: D ª 2 Y is nearly Yq-convexlike on D, if there is g Ž0, 1. such that for ᭙ x 1 , x 2 g D, F Ž x 1 . q Ž1 y . F Ž x 2 . ; F Ž D . q Yq.
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It is well known that Yq-convex « Yq-convexlike Ž see w 8 x . « Yq-subconvexlike Ž see w 8 x . « generalized Yq-subconvexlike Žsee w17x., and that Yq-convexlike « nearly Yq-convexlike. DEFINITION 2.2. Ža. The set epi F s Ž x, y . g X = Y ¬ x g D, y g F Ž x . q Yq 4 is called the epigraph of F on D. Žb. Let S be a nonempty subset of W, and let s0 g cl S Žclosure of S . be a given element. A vector h g W is called a tangent vector to S at s0 , if there are a sequence Ž z n . ng N of elements in S and a sequence Ž n . ng N of positive real numbers with s0 s lim nª⬁ z n and h s lim nª⬁ nŽ z n y s0 .. The set T Ž S, s0 . of all tangent vectors to S at s0 is called a contingent cone to S at s0 . Žc. Let a pair Ž x 0 , y 0 . g X = Y with x 0 g D and y 0 g F Ž x 0 . be given. Denote by TX ŽepiŽ F ., Ž x 0 , y 0 .. the projection of T ŽepiŽ F ., Ž x 0 , y 0 .. on X. A single-valued map DF Ž x 0 , y 0 .: TX ŽepiŽ F ., Ž x 0 , y 0 .. ; X ª Y, whose epigraph equals the contingent cone to the epigraph of F at Ž x 0 , y 0 ., i.e., epiŽ DF Ž x 0 , y 0 .. s T ŽepiŽ F ., Ž x 0 , y 0 .., is called contingent epiderivative of F at Ž x 0 , y 0 .. Consider the set-valued optimization problem ŽP.
Ž P . min F Ž x .
s.t. x g K ,
where the feasible set K s x g D ¬ yGŽ x . l Zq/ ⭋, yH Ž x . l Wq / ⭋4 . DEFINITION 2.3. A point Ž x 0 , y 0 , z 0 , w 0 . is called a feasible point of ŽP. if x 0 g K, y 0 g F Ž x 0 ., z 0 g G Ž x 0 . l y Zq, w 0 g H Ž x 0 . l y Wq. A point x 0 g K is called a weakly efficient solution of ŽP., if ᭚ y 0 g F Ž x 0 . such that Ž y 0 y F Ž K .. l int Yqs ⭋. In this case, the point Ž x 0 , y 0 , z 0 , w 0 . is called a weak minimizer of ŽP.. Remark 2.1. It is obvious that when Wqs O 4 , ŽP. can be written as ŽPX .,
Ž PX . min F Ž x .
s.t. x g K X ,
where the set K X s x g D ¬ yGŽ x . l Zq/ ⭋, O g H Ž x .4 . Since ŽPX . is a particular case of ŽP., we are concerned in this paper with Problem ŽP.. The following lemmas will be very useful.
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LEMMA 2.1 Žsee w9x.. con¨ ex set. LEMMA 2.2 Žsee w6x..
If M ; Y is a nearly con¨ ex set, then int M is a U If yU0 g Yq _ O 4 , y 0 g int Yq, then ² y 0 , yU0 : ) 0.
LEMMA 2.3 Žsee w9x.. Let M ; Y be a nearly con¨ ex set, and let int M / ⭋. Suppose that for each yU g Y U _ O4 , ² u, yU : ) 0, ᭙ u g int M. Then ² u, yU : G 0, ᭙ u g M.
3. GENERALIZED CONSTRAINT QUALIFICATIONS In what follows, we put U s Z = W, Uqs Zq= Wq, J s Ž G, H .: D ª 2 U. The notation Ž G, H .Ž x . is used for G Ž x . = H Ž x . here. One can easily U verify that U U s Ž Z = W .U s ZU = W U , and that Uq s Ž Zq= Wq .U s U U Zq= Wq. Now we consider the following four generalized constraint qualifications ŽQ 0 . to ŽQ 3 .. Let Rys r g R ¬ r - 04 . ŽQ 0 . ᭚ xX g D such that yGŽ xX . l int Zq/ ⭋, yint H Ž xX . l Wq / ⭋. ŽQ 1 . Ži. ᭚ xX g D such that yGŽ xX . l int Zq/ ⭋, yH Ž xX . l Wq / ⭋. Žii. yint H Ž D . l Wq/ ⭋. U ŽQ 2 . Ži. For each zU g Zq _ O 4 , there is xX g D such that ² G Ž xX ., X U z : l Ry/ ⭋, yH Ž x . l Wq/ ⭋. U Žii. For each wU g Wq _ O 4 , there is xX g D such that ² H Ž xX ., U w : l Ry/ ⭋. U U ŽQ 3 . For each Ž zU , wU . g Zq = Wq _Ž O, O .4 , there is x⬘ g D such that Ž² G Ž xX . , zU: q² H Ž xX . , wU: . l Ry/ ⭋. THEOREM 3.1. The following implications are true:
Ž Q 0 . « Ž Q1 . « Ž Q 2 . « Ž Q 3 . . Proof. It is easy to show ŽQ 0 . « ŽQ 1 .. Now we prove ŽQ 1 . « ŽQ 2 .. Suppose that ŽQ 1 . holds. U Ža. Let zU g Zq _ O 4 . Hence ᭚ xX g D such that yGŽ x⬘. l int Zq / ⭋. By Lemma 2.2 we have ²int Zq, zU : ) 0. Therefore ² G Ž xX ., zU : l Ry/ ⭋. U Žb. Let wU g Wq _ O4 . Thus ᭚ w 0 g W such that ² w 0 , wU : ) 0. Let w 1 g yint H Ž D . l Wq. So, ᭚ ) 0 such that w 1 q w 0 g yint H Ž D . ; yH Ž D .. Hence ᭚ xX g D such that w 1 q w 0 g yH Ž xX .. Since w 1 g Wq,
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then ² w 1 q w 0 , wU : s ² w 1 , wU : q ² w 0 , wU : ) 0. Therefore ² H Ž xX ., wU : l Ry/ ⭋. U Next we show ŽQ 2 . « ŽQ 3 .. Assume that ŽQ 2 . holds. Let Ž zU , wU . g Zq U = Wq _Ž O, O .4 . If zU s O, then ŽQ 3 . is introduced directly by Žii. of ŽQ 2 .. If zU / O, then from Ži. of ŽQ 2 ., ᭚ xX g D, such that ² G Ž xX ., zU : l Ry/ ⭋, yH Ž xX . l Wq/ ⭋. Thus ² H Ž xX ., wU : l Ž Ryj 04. / ⭋. Hence ᭚ r 1 g ² G Ž xX ., zU : l Ry, r 2 g ² H Ž xX ., wU : l Ž Ryj 04. such that r 1 q r 2 - 0. This means Ž² G Ž xX ., zU : q ² H Ž xX ., wU :. l Ry/ ⭋. LEMMA 3.1. intŽ J Ž D . q Uq . / ⭋, if and only if intŽ J Ž D . q Žint Zq . = Wq . / ⭋. Indeed, the proof of the last lemma is similar to the proof of Lemma 2.2 w9x, when J is a vector-valued map. U U U LEMMA 3.2. If uU s Ž zU , wU . g Uq s Zq = Wq , zU / O, u s Ž z, w . g Žint Zq . = Wq, then ² u, uU : ) 0. U Proof. Since zU g Zq _ O 4 , and since z g int Zq, thus ² z, zU : ) 0. U U Hence ² u, u : s ² z, z : q ² w, wU : ) 0, because of ² w, wU : G 0.
LEMMA 3.3. If the set-¨ alued map J: D ª 2 U is nearly Uq-con¨ exlike on D, then M s J Ž D . q Žint Zq . = Wq is nearly con¨ ex. Proof. Let m1 , m 2 g M; then ᭚ x i g D, u i g Žint Zq . = Wq, such that m i g J Ž x i . q u i , i s 1, 2. Since J is nearly Uq-convexlike, there exists ␣ g Ž0, 1.; for the previous x 1 , x 2 g D, we have
␣ J Ž x 1 . q Ž 1 y ␣ . J Ž x 2 . ; J Ž D . q Uq . Thus ␣ Ž m1 y u1 . q Ž1 y ␣ .Ž m 2 y u 2 . g J Ž D . q Uq. Because the set Žint Zq . = Wq is convex, so we have u 0 s ␣ u1 q Ž1 y ␣ . u 2 g Žint Zq . = Wq. Thereby, m [ ␣ m1 q Ž 1 y ␣ . m 2 g ␣ J Ž x 1 . q Ž 1 y ␣ . J Ž x 2 . q u 0 . Because of int Zqq Zq; int Zq, so m g J Ž D . q Uqq u 0 ; J Ž D . q Ž int Zq . = Wqs M. Therefore M is nearly convex. THEOREM 3.2. Let intŽ J Ž D . q Uq . / ⭋. Suppose that J is nearly Uqcon¨ exlike on D. Let H Ž D . be a con¨ ex set with a nonempty topological interior. Then
Ž Q1 . m Ž Q 2 . m Ž Q 3 . .
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Proof. We need only show ŽQ 3 . « ŽQ 1 .. Suppose that Ži. of ŽQ 1 . is not true. Then for any x g D, we have O f G Ž x . q int Zq or O f H Ž x . q Wq, i.e., Ž O, O . f M s J Ž D . q Žint Zq . = Wq. According to Lemma 3.1 and the assumption of intŽ J Ž D . q Uq . / ⭋, we have int M / ⭋. Furthermore, Ž O, O . f int M. Since J is nearly Uq-convexlike on D, thus by Lemma 3.3, M is a nearly convex set. Hence, int M is convex. By applying the separation theorem of convex sets of linear topological spaces Žsee w19x., there is a hyperplane E properly separating O 4 and int M, i.e., ᭚ uU s Ž zU , wU . g ZU = W U _Ž O, O .4 , a g R, such that ² u, uU : G a G 0,
᭙ u g int M,
Ž 3.1.
U
where the hyperplane E s y g U ¬ ² y, u : s a4 . Next, we show that ² u, uU : ) 0,
᭙ u g int M.
Ž 3.2.
We have two cases. One case is a ) 0. Then it follows by Ž3.1. and Ž3.2. holds. The other case is a s 0. We have ² u, uU : G 0,
᭙ u g int M.
Ž 3.3.
Suppose that Ž3.2. does not hold. According to Ž3.3., there is u 0 g int M such that ² u 0 , uU : s 0. Let ¨ g int M. Then ᭚ ) 0 such that u 0 y ¨ g int M. Thus by Ž3.3., we have ² u 0 y ¨ , uU : G 0, i.e., ² u 0 , uU : G ² ¨ , uU :. So, ² ¨ , uU : F 0. On the other hand, again by Ž3.3. we get ² ¨ , uU : G 0. Therefore, ² ¨ , uU : s 0, ᭙ ¨ g int M. This is absurd since the hyperplane E separates O 4 and int M properly. Hence the proof that Ž3.3. holds is complete. By Lemma 2.3 we obtain ² u, uU : G 0, U
᭙ u g M. U
U
U Zq = U
Ž 3.4. U Wq ;
In the following, we prove u s Ž z , w . g indeed, assume U z U f Zq . Then there is z1 g Zq such that ² z1 , z : - 0. By Ž3.4., for any x g D, any zX g int Zq, any wX g Wq, we have  s ² p q zX , zU : q ² q q wX , wU : G 0, ᭙ p g G Ž x ., ᭙q g H Ž x .. Since z1 g Zq, then z1 q zX g int Zq. Also by Ž3.4., we have ² p q z1 q zX , zU : q ² q q wX , wU : G 0, i.e., ² z1 , zU : q  G 0, ᭙ ) 0. Ž 3.5. However, Ž3.5. does not hold when ) yr² z1 , zU : G 0. This contradicU U tion illustrates that zU g Zq . Similarly, we can also prove wU g Wq . Thus, U U U U U U ᭚ u s Ž z , w . g Zq= Wq _Ž O, O .4 , such that ² u, u : G 0, ᭙ u g M, i.e.,
² G Ž x . q z, zU: q² H Ž x . q w, wU: G 0, ᭙ x g D, ᭙ z g int Zq , ᭙ w g Wq .
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Take w s O in the previous expression. Then
² G Ž x . , zU: q ² z, zU : q² H Ž x . , wU: G 0, ᭙ x g D, ᭙ z g int Zq , ᭙ ) 0.
Ž 3.6.
If zU s O, then from Ž3.6. we have ² H Ž x ., wU : G 0, ᭙ x g D, which contradicts ŽQ 3 .. Suppose zU / 0. According to ŽQ 3 ., we have that ᭚ xX g D, p g G Ž xX ., q g H Ž xX . such that ² p, zU : q ² q, wU : - 0. Therefore there is a sufficiently small ␦ ) 0 such that ² p, zU : q ² q, wU : q ␦ - 0.
Ž 3.7.
However, if we take s ␦r² z, zU : ) 0 in Ž3.6., then Ž3.7. is in contradiction to Ž3.6.. Therefore Ži. of ŽQ 1 . holds. In the following we prove that Žii. of ŽQ 1 . is true. Otherwise int H Ž D . l ŽyWq . s ⭋. Thus again by the separation theorem of convex sets, there is wU g W U _ O 4 such that ² H Ž D ., wU : G ² y Wq, wU :, i.e., ² H Ž D ., wU : U q ²Wq, wU : G 0. It is easy to check that wU g Wq . Furthermore U ² H Ž D ., w : G 0, which contradicts ŽQ 3 .. THEOREM 3.3. If ŽQ 0 . holds, then there exists y 1 g Y, such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. Proof. According to ŽQ 0 ., we have that there exists xX g D such that O g G Ž xX . q int Zq; intŽ G Ž xX . q Zq ., and O g int H Ž xX . q Wq; intŽ H Ž xX . q Wq .. Since ⭋ / F Ž xX ., thus ⭋ / F Ž xX . q int Yq; intŽ F Ž xX . q Yq .. Hence ᭚ y 1 g Y such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž xX . q Yq= Zq= Wq .. Remark 3.1. In fact, Theorem 3.1, Theorem 3.2, and Theorem 3.3 are all still true when Y, Z, W are respectively assumed to be linear topological spaces.
4. OPTIMALITY CONDITIONS In this section, we establish some K-T-type sufficient and necessary optimality conditions for set-valued optimization in terms of contingent epiderivatives which were introduced by Jahn and Rauh w1x. THEOREM 4.1 ŽSufficiency.. Let D ; X be a con¨ ex set, let Ž F, G, H .Ž x . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a feasible point of ŽP.. Suppose that ŽQ 1 . holds. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , U U U y 0 , z 0 , w 0 . exists and there is Ž yU , zU , wU . g Yq = Zq = Wq _Ž O, O, O .4 ,
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HUANG YONG-WEI
such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0, U
᭙ x g D,
U
² z 0 , z : q ² w 0 , w : s 0, then x 0 is a weakly efficient solution of ŽP.. Proof. According to Lemma 3 of w1x, for each x g D we have
Ž F , G, H . Ž x . y Ž y 0 , z 0 , w 0 . 4 ; D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . 4 q Yq= Zq= Wq . Thus for each x g D, each Ž y, z, w . g Ž F, G, H .Ž x ., there c g Yq, d g Zq, e g Wq such that
Ž y y y 0 y c, z y z 0 y d, w y w 0 y e . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . . Hence, by assumption, we obtain ² y y y 0 y c, yU : q ² z y z 0 y d, zU : q ² w y w 0 y e, wU : G 0, or ² y y y 0 , yU : q ² z, zU : q ² w, wU : G ² c, yU : q ² z 0 , zU : q ² d, zU : q ² w 0 , wU : q ² e, wU : s ² c, yU : q ² d, zU : q ² e, wU : G 0. Therefore, ² y y y 0 , yU : q ² z, zU : q ² w, wU : G 0, ᭙ Ž y, z, w . g Ž F , G, H . Ž x . , ᭙ x g D.
Ž 4.1.
Next we show yU / O. Suppose that yU s O. Then Ž zU , wU . / Ž O, O ., and Ž4.1. can be written as
² G Ž x . , zU: q² H Ž x . , wU: G 0,
᭙ x g D.
Ž 4.2.
Thus we have two cases. One case is z / 0. By Q 1 , ᭚ x g D, ᭚ z g G Ž xX ., wX g H Ž xX ., such that yzX g int Zq, ywX g Wq. Hence, it follows by Lemma 3.2 that ² zX , zU : q ² wX , wU : - 0, which contradicts Ž4.2.. The other case is zU s 0. Then wU / O. Again by Q 1 , there is w 噛 g Wq such that yw 噛 g int H Ž D .. Then for any w g W, ᭚ ) 0 such that yw 噛 " w g H Ž D .. So, ² y w 噛 " w, wU : G 0, i.e., "² w, wU : G ² w 噛 , wU : G 0. Therefore, ² w, wU : s 0. It follows that wU s 0. This is a contradiction. Therefore we have yU / 0. U
X
X
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317
Finally, we prove x 0 is a weakly efficient solution of ŽP1.. Otherwise, ᭚ x 噛 g K such that Ž y 0 y F Ž x 噛 .. l int Yq/ ⭋. Thus, ᭚ t g F Ž x 噛 . such that y 0 y t g int Yq. By Lemma 2.2, we have ² t y y 0 , yU : - 0.
Ž 4.3.
Since x 噛 g K, there are p g G Ž x 噛 ., q g H Ž x 噛 . such that yp g Zq, yq g Wq. Taking Ž4.3. into account, we get ² t y y 0 , yU : q ² p, zU : q ² q, wU : - 0, which contradicts Ž4.1.. In the remainder of this paper, we assume that int Wq/ ⭋. THEOREM 4.2 ŽNecessity.. Let D ; Y be a con¨ ex set, let Ž F, G, H . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a weak minimizer of ŽP.. Suppose that Q 0 is fulfilled. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 . exists, and TX ŽepiŽ F, G, H ., Ž x 0 , y 0 , z 0 , w 0 .. s X, then U U U there is Ž yU , zU , wU . g Yq = Zq = Wq , with yU / O, such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0,
᭙ x g D,
² z 0 , zU : q ² w 0 , wU : s 0. Proof. Set P Ž x . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . ,
x g D.
We show first that P Ž x . is a Yq= Zq= Wq-convex single-valued map on D. Let x i g D, i s 1, 2, g Ž0, 1.. Thus by positive homogeneity, we have
P Ž x 1 . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x 1 y x 0 . ,
Ž 1 y . P Ž x 2 . s D Ž F , G, H . Ž x 0 , y 0 , z 0 , W0 . Ž Ž 1 y . x 2 y x 0 . . By subadditivity, we obtain
P Ž x 1 . q Ž 1 y . P Ž x 2 . g P Ž x 1 q Ž 1 y . x 2 . q Yq= Zq= Wq . This illustrates that P Ž x . is Yq= Zq= Wq-convex on D. Set RŽ x . s P Ž x . q Ž O, z 0 , w 0 ., ᭙ x g D. Obviously, RŽ x . is also Yq= Zq= Wq-convex on D. Next we prove that yR Ž x . f int Yq= int Zq= int Wq ,
᭙ x g D.
Otherwise, there is xX g D such that yR Ž xX . [ y Ž yX , zX , wX . y Ž O, z 0 , w 0 . g int Yq= int Zq= int W .
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HUANG YONG-WEI
By definition of the contingent epiderivative,
Ž xX y x 0 , yX , zX , wX . g epi Ž D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . . s T Ž epi Ž F , G, H . , Ž x 0 , y 0 , z 0 , w 0 . . . According to the definition of the contingent cone, the following sequences exist: x n g D, c n g Yq, d n g Zq, e n g Wq, yn g F Ž x n . q c n , z n g GŽ x n . q d n , wn g H Ž x n . q e n , satisfying x n ª x 0 , yn ª y 0 , z n ª z 0 ; n ) 0, such that
n Ž x n y x 0 , yn y y 0 , z n y z 0 , wn y w 0 . ª Ž xX y x 0 , yX , zX , wX . . Thus, nŽ y 0 y yn . ª yyX , nŽ z 0 y z n . ª yzX , nŽ w 0 y wn . ª ywX . Because yyX g int Yq, so ᭚M1 g N such that nŽ y 0 y yn . g int Yq, ᭙ n G M1. Since int Yq is a cone, and since n ) 0, hence y 0 y yn g int Yq ,
᭙ n G M1 .
Ž 4.4.
Because yzX y z 0 g int Zq, so ᭚M2 g N such that nŽ z 0 y z n . y z 0 g int Zq, ᭙ n G M2 . That is nŽŽ1 y 1r n . z 0 y z n . g int Zq, ᭙ n G M2 , or Ž1 y 1r n . z 0 y z n g int Zq, ᭙ n G M2 . In a similar way, ᭚M3 g N such that Ž1 y 1r n . w 0 y wn g int Wq, ᭙ n G M3 . Take M s maxŽ M1 , M2 , M3 .. Whenever n G M, we have n ) k ) 1. Otherwise nŽ y 0 y yn . ª O g int Yq because of yn ª y 0 . This is absurd. Since Ž1 y 1r n . z 0 g yZq, and since int Zqq Zq; int Zq, therefore yz n g int Zq; Zq. So yz n g yŽ G Ž x n . q Zq . l Zq, ᭙ n G M. Since Zq is a convex cone, we get yG Ž x n . l Zq/ ⭋,
᭙ n G M.
Similarly, we have yH Ž x n . l Wq/ ⭋, ᭙ n G M. This means that x n belongs to a feasible set of ŽP.. It follows by Ž4.4. that
Ž y 0 y F Ž x n . y Yq . l int Yq/ ⭋,
᭙ n G M,
since yn g F Ž x n . q Yq. So, Ž y 0 y F Ž x n .. l int Yq/ ⭋, which does not coincide with the supposition that Ž x 0 , y 0 , z 0 , w 0 . is a weak minimizer of ŽP.. Therefore, yR Ž x . f int Yq= int Zq= int Wq ,
᭙ x g D.
By applying Lemma 2.2 of w13x or Theorem 3.1 of w12x, we have that U U U ᭚Ž yU , zU , wU . g Yq = Zq = Wq _Ž O, O, O .4 , such that
² R Ž x . , Ž yU , zU , wU .: G 0,
᭙ x g D.
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319
Hence,
² P Ž x . , Ž yU , zU , wU .: q ² z 0 , zU : q ² w 0 , wU : G 0,
᭙ x g D,
where P Ž x . s DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 .Ž x y x 0 .. Since Ž O, O, O . s P Ž x 0 ., hence ² z 0 , zU : q ² w 0 , wU : G 0. On the other hand, due to z 0 g yZq, w 0 g yWq, consequently, ² z 0 , zU : q ² w 0 , wU : F 0. Therefore, ² z 0 , zU : q ² w 0 , wU : s 0. Thus ² P Ž x ., Ž yU , zU , wU .: G 0, ᭙ x g D, i.e.,
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0,
᭙ x g D.
Next we show yU / O. Since Q 0 is fulfilled, then from Theorem 3.3, there exists y 1 g Y such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. Again by Lemma 3 of w1x, we have
Ž F , G, H . Ž x . y Ž y 0 , z 0 , w 0 . 4 ; D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . 4 qYq= Zq= Wq ,
᭙ x g D.
Thus ² y y y 0 , yU : q ² z, zU : q ² w, wU : G 0, ᭙ Ž y, z, w . g Ž F , G, H . Ž x . , ᭙ x g D.
Ž 4.5.
Suppose that yU s O. Then Ž zU , wU . / 0, and Ž4.5. can be written as
² G Ž x . , zU: q² H Ž x . , wU: G 0,
᭙ x g D.
Ž 4.6.
Thus,
² F Ž x . q Yq , yU: q² G Ž x . q Zq , zU: q² H Ž x . q Wq , wU: G 0, ᭙ x g D.
Ž 4.7.
Since Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq ., then for any x g D, ¨ 1 g GŽ x ., ¨ 2 g H Ž x ., k 1 g Zq, k 2 g Wq, there is ) 0 such that Ž y 1 , O, O . " Ž O, ¨ 1 q k 1 , ¨ 2 q k 2 . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. By Ž4.7., we have ² ¨ 1 q k 1 , zU : q ² ¨ 2 q k 2 , wU : s 0. Observing Ž4.6., we get ² k 1 , zU : q ² k 2 , wU : s 0,
᭙k 1 g Zq , ᭙k 2 g Wq .
Ž 4.8.
Then we also have two cases. One case is zU / 0. Then Ž4.8. contradicts Lemma 3.2. The other case is zU s 0. Then wU / 0. For any ¨ g W, there is 1 ) 0 such that Ž y 0 , O, O . " 1Ž O, O, ¨ . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. By Ž4.7., we obtain ² ¨ , wU : s 0, ᭙ ¨ g W. Thus wU s 0. This is a contradiction. Therefore we have yU / 0.
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COROLLARY 4.1. Let D ; Y be a con¨ ex set, let Ž F, G, H .Ž x . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a weak minimizer of ŽP.. Suppose that there exists y 1 g Y such that Ž y 1 , O, O . g intŽŽ F, G, H .Ž D . q Yq= Zq= Wq .. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 . exists, and TX ŽepiŽ F, G, H ., Ž x 0 , y 0 , z 0 , w 0 .. s X, then there is Ž yU , zU , wU . U U U g Yq = Zq = Wq , with yU / O, such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0,
᭙ x g D;
² z 0 , zU : q ² w 0 , wU : s 0. COROLLARY 4.2. Let D ; Y be a con¨ ex set, let Ž F, G, H .Ž x . be Yq= Zq= Wq-con¨ ex on D, and let Ž x 0 , y 0 , z 0 , w 0 . be a weak minimizer of ŽP.. Suppose that Q 1 is fulfilled. If the contingent epideri¨ ati¨ e DŽ F, G, H .Ž x 0 , y 0 , z 0 , w 0 . exists, and TX ŽepiŽ F, G, H ., Ž x 0 , y 0 , z 0 , w 0 .. s X, then there is U U U Ž yU , zU , wU . g Yq = Zq = Wq , with yU / O, such that
² D Ž F , G, H . Ž x 0 , y 0 , z 0 , w 0 . Ž x y x 0 . , Ž yU , zU , wU .: G 0, U
᭙ x g D;
U
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