Optimality Conditions of Vector Set Valued Optimization Problem Involving Relative Interior

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Volume 2011, Article ID 183297,15pages doi:10.1155/2011/183297

Research Article

Optimality Conditions of Vector Set-Valued

Optimization Problem Involving Relative Interior

Zhiang Zhou

1, 2

1_{College of Sciences, Shanghai University, Shanghai 200444, China}

2_{Department of Applied Mathematics, Chongqing University of Technology, Chongqing 400054, China}

Correspondence should be addressed to Zhiang Zhou,zhi [email protected]

Received 26 October 2010; Revised 25 December 2010; Accepted 22 January 2011

Academic Editor: Sin E. Takahasi

Copyrightq2011 Zhiang Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Firstly, a generalized weak convexlike set-valued map involving the relative interior is introduced in separated locally convex spaces. Secondly, a separation property is established. Finally, some optimality conditions, including the generalized Kuhn-Tucker condition and scalarization theorem, are obtained.

1. Introduction

In mathematical programming, set-valued optimization is a very important topic. Since the 1980s, many authors have paid attention to it. Some international journals such as Set-Valued and Variational Analysis original name: Set-Valued Analysis were also established. Theories and applications are widely developed. Rong and Wu 1, Li 2,

and Yang3and Yang4introduced cone convexlikeness, subconvexlikeness, generalized subconvexlikeness, and nearly subconvexlikeness, respectively. In these generalized convex set-valued maps, it is clear that nearly subconvexlikeness is the weakest. We find that, in the above-mentioned papers, the convex cone has a nonempty topological interior. However, it is possible that the topological interior of the convex cone is empty. For instance, if

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convex programming. However, we find that only a few papers11,12are about set-valued optimization involving the relative interior. In this paper, we will further study set-valued optimization problems involving relative interior.

This paper is organized as follows. In Section 2, we give some preliminaries. In

Section 3, a kind of generalized weak convexlike set-valued map involving relative interior is introduced, and a separation property is established. In Section 4, some optimality conditions, including the generalized Kuhn-Tucker condition and scalarization theorem, are obtained.

2. Preliminaries

LetX,Y, andZbe three separated locally convex spaces, and let 0 denote the zero element for every space. LetK be a nonempty subset ofY. The generated cone ofK is defined as cone K {λa|a∈K, λ≥0}. A coneK⊆Y is said to be pointed ifK∩−K {0}. A cone

K⊆Y is said to be nontrivial ifK /{0}andK /Y.

Let Y∗ and Z∗ stand for the topological dual space of Y and Z, respectively. From now on, letCandDbe nontrivial pointed closed-convex cones inYandZ, respectively. The topological dual coneCand strict topological dual coneCiofCare defined as

C_y∗_∈_Y∗_|_{y, y}∗ ₀_, _∀_y_∈_C_,

Ciy∗∈Y∗|y, y∗>0, ∀y∈C\ {0},

2.1

where y, y∗ denotes the value of the linear continuous functional y∗ at the pointy. The meanings ofDandDiare similar.

LetKbe a nonempty subset ofY. We denote by clK, intK, and aﬀKthe closed hull, topological interior, and aﬃne hull ofK, respectively.

Definition 2.1see11,13. LetKbe a subset ofY. The relative interior ofKis the set

riKx∈K|there existsU, a neighborhood of x, such that U∩aﬀK⊆K. 2.2

Now, we give some basic properties about the relative interior.

Lemma 2.2. LetKbe a subset ofY. Letk0∈K,k∈riK,α∈R, andλ∈0,1. Then, aαriKriαK;

bifKis convex, then

1−λk0λk∈riK. 2.3

Proof. aSinceαaﬀKaﬀαK, it is clear thatαriKriαK; bsincek∈riK, there existsV, a neighborhood of 0, such that

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By2.4, we have

λkλV∩λaﬀK⊆λK. 2.5

It follows from2.5that

1−λk0λkλV

∩1−λk0λaﬀK⊆1−λk0λK. 2.6

It is clear that

1−λk0λaﬀKaﬀK. 2.7

SinceKis convex, we have

1−λk0λK⊆K. 2.8

By2.6,2.7, and2.8, we obtain

1−λk0λkλV

∩aﬀK⊆K, 2.9

which implies that

1−λk0λk∈riK. 2.10

Remark 2.3. ByLemma 2.2, ifKis a convex cone, then riK∪ {0}is a convex cone.

Lemma 2.4. IfKis a convex cone ofY, then

KriK⊆riK. 2.11

Proof. If riKφ, it is clear that the conclusion holds. If riK /φ, we have

KriK2

1 2K

1

2riK ⊆2 riKri 2KriK, 2.12

whereLemma 2.2bis used in the first inclusion relation andLemma 2.2ais used in the second equality.

Lemma 2.5see14, 15. Let W be a linear topological space and w∗ be a linear functional on W.w∗is continuous if and only ifH{w | w, w∗ 0, w∈W}is closed. IfHis not closed,H is dense inW.

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Lemma 2.6see11. LetK ⊆Y be a closed-convex set withriK /φ. If0∈/riK, then there exists y∗_∈_Y∗_{\ {}₀_}_{such that}_{k, y}∗_≥₀_{for each}_k_∈_K_.

Remark 2.7. The following example will show that the closeness ofK cannot be deleted in

Lemma 2.6.

Example 2.8. Let Y be an infinite-dimensional normed space and k∗ be a non-continuous linear functional onY.Kis defined as

K{k| k, k∗ ₁_{, k}_∈_Y_}_. _2.13

Since aﬀK K, it is clear that 0 ∈/ riK K. ByLemma 2.5,Kis not closed and clK Y. Therefore, for anyy∗∈Y∗\ {0}, y∗cannot separate 0 andK.

Remark 2.9. Example 2.8shows that, even ifK is a convex subset of Y, the expression that riclK riKdoes not hold generally.

3. Separation Property

From now on, we suppose that riC /φ and riD /φ. LetAbe a nonempty subset ofX and

F:A → 2Y be a set-valued map onA. WriteFA ∪x∈AFx.

Definition 3.1see1. LetAbe a nonempty subset ofX. A set-valued mapF :A → 2Y is calledC-convexlike onAif the setFA Cis convex.

In 2, 3, 16, 17, when intC /φ,C-subconvexlike map and generalized C -subconvexlike map were introduced, respectively. The following two definitions are generalizations ofC-subconvexlike map and generalizedC-subconvexlike map, respectively.

Definition 3.2see12. LetAbe a nonempty subset ofX. A set-valued mapF :A → 2Y is calledC-weak convexlike onAif the setFA riCis convex.

Definition 3.3see12. LetAbe a nonempty subset ofX. A set-valued mapF :A → 2Y is called generalizedC-weak convexlike onAif the set coneFA riCis convex.

Remark 3.4. By12, Theorems 3.1 and 3.2, we have the following implications:

C-convexlikeness⇒C-weak convexlikeness⇒generalizedC-weak convexlikeness.

However, the following two examples show that the converse of the above implications is not generally true.

Example 3.5. LetX Y R2_,_C _{_y

1,0 |y1 ≥ 0}, andA {1,0,0,2}. The set-valued mapF :A → 2Y is defined as follows:

F1,0 y1, y2

|1< y1 ≤2, 0≤y2≤1

∪ {1,0, 1,1}, F0,2 y1, y2

|1< y1 ≤2, 1≤y2≤2

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It is clear that FA riCis convex and FA C is not convex. Therefore,F isC-weak convexlike onA. However,Fis notC-convexlike onA.

Example 3.6. LetX Y R2_,_C _{_y

1,0 |y1 ≥ 0}, andA {1,0,0,2}. The set-valued mapF :A → 2Y is defined as follows:

F1,0 y1, y2

|y1≥0, 1≤y2≤ −y12

, F0,2 y1, y2

|y1≥1, 0≤y2≤ −y12

. 3.2

It is clear that coneFA riC is convex and FA riC is not convex. Therefore, F is generalizedC-weak convexlike onA. However,Fis notC-weak convexlike onA.

Now, we consider the following two systems.

System 1: There existsx0∈Asuch thatFx0∩−riC/φ.

System 2: There existsy∗ ∈C\ {0}such thaty, y∗ ≥0, for ally∈FA. Theorem 3.7. LetAbe a nonempty subset ofX.

iSuppose thatF :A → 2Y is generalizedC-weak convexlike onAandriclconeFA

riC riconeFA riC_/φ. If System 1 has no solution, then System 2 has solution.

iiIfy∗∈Ciis a solution of System 2, then System 1 has no solution.

Proof. iFirstly, we assert that 0∈/coneFA riC. Otherwise, there existx0∈A, α≥0 such that 0∈αFx0 riC.

Case 1. Ifα0, then 0∈riC. Thus, there existsU, a neighborhood of 0, such that

U∩aﬀC⊆C. 3.3

Without loss of generality, we suppose thatUis symmetric. It follows from3.3that

U∩−aﬀC⊆−C. 3.4

It is clear that affCis a linear subspace ofY. Therefore, affC−affC. By3.4, we have

U∩aﬀC⊆−C. 3.5

By3.3and3.5, we obtain

U∩aﬀC⊆C∩−C. 3.6

Since C is nontrivial, there exists c ∈ C\ {0}. By the absorption of U, there exists λ, a suﬃciently small positive number, such that

λc∈U∩aﬀC⊆C∩−C, 3.7

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Case 2. Ifα > 0, there existsy0 ∈ Fx0such that−y0 ∈ 1/αriC ⊆ riC, which contradicts

Fx∩−riC φ, for allx∈A.

Therefore, our assertion is true. Thus, we obtain

0∈/riclconeFA riC. 3.8

SinceFis generalizedC-weak convexlike onA, clconeFAriCis a closed-convex set. By

Lemma 2.6, there existsy∗∈Y∗\ {0}such that

y, y∗_≥₀_, _∀_y_∈_clcone_F_A _ri_C_. _3.9

So,

αFx c, y∗_≥₀_, _∀_x_∈_{A, c}_∈_ri_{C, α}_≥₀_. _3.10

Lettingα0 in3.10, we obtain

c, y∗_≥₀_, _∀_c_∈_ri_C. _3.11

We assert thaty∗ ∈ C. Otherwise, there existsc ∈ Csuch thatc, y∗ < 0, hence,

θc_{, y}∗ _<₀_, _{for all}_{θ >}_{0. By}_{Lemma 2.4}_{, we have}

θc_c_∈_ri_C, _∀_c_∈_ri_C. _3.12

It follows from3.11that

θc_{c, y}∗_≥₀_, _∀_{θ >}₀_{, c}_∈_ri_C. _3.13

Thus, we obtain

θc_{, y}∗_{c, y}∗_≥₀_, _∀_{θ >}₀_{, c}_∈_ri_C. _3.14

On the other hand,3.14does not hold whenθ >−c, y∗ /c, y∗ ≥0. Therefore,c, y∗ ≥ 0, for allc∈C, that is,y∗∈C.

Lettingα1 in3.10, we have

Fx c, y∗_≥₀_, _∀_x_∈_{A, c}_∈_ri_C. _3.15

Takingc0 ∈riC, λn>0, limn→ ∞λn0, we have

Fx λnc0, y∗

≥0, ∀x∈A, n∈N. 3.16

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iiSincey∗∈Ciis a solution of System 2, we have

y, y∗_≥₀_, _∀_y_∈_F_A_. _3.17

Now, we suppose that System 1 has solution. Then, there existsx0 ∈ Asuch that Fx0∩ −riC/φ. Thus, there existsy0 ∈Fx0such that−y0 ∈ riC. It is clear that−y0/0. So, we have

y0, y∗

<0, 3.18

which contradicts3.17.

Remark 3.8. If Y Rn, by 5, Theorems 6.2 and 6.3, the condition that riclconeFA

riC riconeFA riC/φholds automatically. However, byRemark 2.9, it is possible that, the condition that riclconeFA riC riconeFA riC/φ does not hold. Therefore, our assumption is reasonable.

4. Optimality Conditions

LetF :A → 2Y andG: A → 2Zbe two set-valued maps fromAtoY andZ, respectively. Now, we consider the following vector optimization problem of set-valued maps:

min Fx

s.t. −Gx∩D /φ. VP

The feasible set ofVPis defined by

Sx∈A| −Gx∩D /φ. 4.1

Now, we define

WMinFS, C y0∈FS|y0−y /∈riC, ∀y∈FS

, PMinFS, C y0∈FS|−C∩cl

coneFS C−y0

{0}. 4.2

Definition 4.1. A pointx0is called a weakly eﬃcient solution ofVPifx0 ∈ SandFx0∩

WMinFS, C/φ. A point pairx0, y0is called a weak minimizer ofVPify0 ∈Fx0∩

WMinFS, C.

Definition 4.2. A pointx0 is called a Benson properly eﬃcient solution ofVPifx0 ∈Sand

Fx0∩PMinFS, C/φ. A point pairx0, y0is called a Benson proper minimizer ofVP ify0∈Fx0∩PMinFS, C.

LetIx Fx×Gx, for allx ∈A. It is clear thatIis a set-valued map fromAto

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coneC×D. The topological dual space ofY×ZisY∗×Z∗, and the topological dual cone of

C×DisC×D.

ByDefinition 3.3, we say that the set-valued mapI :A → 2Y×Zis generalizedC×D -weak convexlike onAif coneIA riC×Dis a convex set ofY ×Z.

Theorem 4.3. LetriclconeI∗A riC×D riconeI∗A riC×D_/φ. Suppose that the following conditions hold:

i x0, y0is a weak minimizer of VP;

iiI∗_x_{is generalized}_C_×_D_{-weak convexlike on}_A_{, where}_I∗_x_F_x₋_y

0×Gx.

Then, there existsy∗, z∗∈C×Dwithy∗, z∗/ 0,0such that

inf

x∈A

Fx, y∗_G_x_{, z}∗ _y

0, y∗

,

infGx0, z∗ 0.

4.3

Proof. According toDefinition 4.1, we have

y0−FS

∩riCφ. 4.4

It is clear thatI∗x Ix−y0,0, for allx∈A. We assert that

−I∗_x_∩_ri_C_×_D _φ, _∀_x_∈_A. _4.5

Otherwise, there existsx∈Asuch that

−I∗_x_∩_ri_C_×_D_/_φ. _4.6

It is easy to check that riC×D riC×riD. Therefore,

−I∗_x_∩_ri_C_×_ri_D_/_φ. _4.7

By4.7, we obtain

y0−Fx

∩riC /φ, 4.8

−Gx∩riD /φ. 4.9

It follows from4.9thatx∈S. Thus, by4.8, we have

y0−FS

∩riC /φ, 4.10

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ByTheorem 3.7, there existsy∗, z∗∈C×Dwithy∗, z∗/ 0,0such that

I∗_x_,_y∗_{, z}∗_≥₀_, _∀_x_∈_A. _4.11

That is,

Fx, y∗_G_x_{, z}∗_≥_y

0, y∗

, ∀x∈A. 4.12

Sincex0 ∈ S, there existsp ∈ Gx0such that−p ∈ D. Becausez∗ ∈ D, we obtain

p, z∗_≤_{0. On the other hand, taking}_x_x

0in4.12, we get

y0, y∗

p, z∗_≥_y

0, y∗

. 4.13

It follows thatp, z∗ ≥0. So,p, z∗ 0. Thus, we have

y0, y∗

∈Fx0, y∗

Gx0, z∗ . 4.14

Therefore, it follows from4.12and4.14that

inf

x∈A

Fx, y∗_G_x_{, z}∗ _y

0, y∗

. 4.15

Finally, taking againxx0in4.12, we obtain

y0, y∗

Gx0, z∗ ≥

y0, y∗

. 4.16

So,Gx0, z∗ ≥0. We have shown that there existsp∈Gx0such thatp, z∗ 0. Thus, we have

infGx0, z∗ 0. 4.17

The following example will be used to illustrateTheorem 4.3.

Example 4.4. LetX Y Z R2_{, C} _D _{_y

1,0 | y1 ≥ 0}, andA {1,0,1,2}. The set-valued mapF:A → 2Y is defined as follows:

F1,0 y1, y2

|y11, 0≤y2 ≤1

, F1,2 y1, y2

|y1>1, 0≤y2≤ −y12

. 4.18

The set-valued mapG:A → 2Y is defined as follows:

G1,0 y1, y2

|y1≤0, 0≤y2≤y11

, G1,2 y1, y2

|y1≥ −1, y11≤y2≤1

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Letx0 1,0andy0 1,0∈Fx0. It is clear that all conditions ofTheorem 4.3are satisfied. Therefore, there existy∗:y1, y2, y∗ y1y2andz∗:y1, y2, z∗ −y1y2such that

inf

x∈A

Fx, y∗_G_x_{, z}∗ _y

0, y∗

,

infGx0, z∗ 0.

4.20

Remark 4.5. Theorem 4.3generalizes Theorem 3.1 of2and Theorem 4.2 of3.

Theorem 4.6. Suppose that the following conditions hold:

ix0∈S;

iithere existy0 ∈Fx0andy∗, z∗∈Ci×Dsuch that

inf

x∈A

Fx, y∗_G_x_{, z}∗ _≥_y

0, y∗

. 4.21

Then,x0is a weakly eﬃcient solution of VP.

Proof. By conditionii, we have

Fx−y0, y∗

Gx, z∗_≥₀_, _∀_x_∈_A. _4.22

Suppose to the contrary thatx0 is not a weakly eﬃcient solution ofVP. Then, there exists

x_∈_S_{such that}_y

0−Fx∩riC /φ. Therefore, there existst∈Fx such thaty0−t∈riC⊆

C\ {0}. Thus, we obtain

t−y0, y∗

<0. 4.23

Sincex∈S, there existsq∈Gxsuch that−q∈D. Hence,

q, z∗_≤₀_. _4.24

Adding4.23to4.24, we have

t−y0, y∗

q, z∗_<₀_, _4.25

which contradicts4.22. Therefore,x0is a weakly eﬃcient solution ofVP.

Example 4.7. LetX Y Z R2_{, C} _D _{_y

1,0 | y1 ≥ 0}, andA {1,0,1,2}. The set-valued mapF:A → 2Y is defined as follows:

F1,0 y1, y2

|y1≥1, y1≤y2≤2

, F1,2 y1, y2

|y1≤2, 1≤y2≤y1

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The set-valued mapG:A → 2Y is defined as follows:

G1,0 y1, y2

| −1≤y1≤0, y20

, G1,2 y1, y2

| −1≤y1≤0, 0≤y2≤1

. 4.27

Letx0 1,0, y0 1,1∈Fx0,y1, y2, y∗ y1y2, andy1, y2, z∗ −y1. It is clear that all conditions ofTheorem 4.6are satisfied. Therefore,1,0is a weakly eﬃcient solution ofVP.

Remark 4.8. Theorem 4.6generalizes2, Theorem 3.3.

Now, we consider the following scalar optimization problemVP_ϕofVP:

min Fx, ϕ

s.t. x∈S, VPϕ

whereϕ∈Y∗\ {0}.

Definition 4.9. Ifx0∈S, y0∈Fx0and

y0, ϕ

≤y, ϕ, ∀y∈FS, 4.28

thenx0andx0, y0are called a minimal solution and a minimizer ofVPϕ, respectively.

Lemma 4.10see18. LetU1, U2⊂Y be two closed-convex cones such thatU1∩U2{0}. IfU2

is pointed and locally compact, then−U₁∩U₂i/φ.

Lemma 4.11. IfV is a subset ofY, then

iclconeVriC clconeV riC,

iiclconeVriC clconeVC.

Proof. iIfV φ, it is obvious that

clconeVriC clconeVriC. 4.29

IfV /φ, there existsc∈riC. It is clear that

λc∈coneVriC, ∀λ∈0,∞. 4.30

Lettingλ → 0 in4.30, we have

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Now, we will show that

coneVriC⊆coneVriC∪ {0}. 4.32

Lety∈coneVriC.

Case 1. Ify0, theny∈coneVriC∪ {0}.

Case 2. Ify /0, there existα >0, v∈V, andc∈riCsuch that

yαvc αvαc∈coneVriC⊆coneV riC∪ {0}. 4.33

Therefore,4.32holds. SinceYis separated, by4.31and4.32, we obtain

clconeVriC⊆clconeVriC∪ {0} clconeVriC∪cl{0} clconeVriC∪ {0} clconeVriC.

4.34

That is,

clconeVriC⊆clconeVriC. 4.35

Using the technique of Lemma 2.1 in19, we easily obtain

coneVriC⊆clconeVriC. 4.36

So,

clconeV riC⊆clconeV riC. 4.37

By4.35and4.37, we have

clconeVriC clconeVriC. 4.38

iiIt is obvious that

clconeV riC⊆clconeVC. 4.39

We will show that

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It is clear that4.40holds ifV φ. Now, we suppose thatV /φ. Lety ∈coneV C, then there existλ≥0, v∈V, andc∈Csuch that

yλvc. 4.41

Since riC /φ, there existsc0∈riC. It follows fromLemma 2.4that

λ

αc0yλ

1

αc0cv ∈coneV riC, ∀α >0. 4.42

Lettingα → ∞in4.42, we have

y∈clconeV riC, 4.43

which implies that4.40holds. By4.40, we obtain

clconeV C⊆clconeVriC. 4.44

By4.39and4.44, we have

clconeV riC clconeVC. 4.45

Theorem 4.12. Suppose that the following conditions hold:

iC⊆Yis locally compact;

ii x0, y0is a Benson proper minimizer of VP; iiiF−y0is generalizedC-weak convexlike onS.

Then, there existsϕ∈Cisuch thatx0, y0is a minimizer ofVP_ϕ.

Proof. By conditionii, we have

−C∩clconeFS C−y0

{0}. 4.46

ByLemma 4.11and conditioniii, we obtain that clconeFS C−y0is a closed-convex cone. Thus, conditions ofLemma 4.10are satisfied. Therefore, there existsϕ∈Cisuch that

ϕ∈clconeFS C−y0

. 4.47

SinceFS−y0⊆clconeFS C−y0, we obtain

y−y0, ϕ

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That is,

y, ϕ≥y0, ϕ

, ∀y∈FS. 4.49

So,x0, y0is a minimizer ofVP_ϕ.

Example 4.13. LetX Y Z R2, C D {y1,0 |y1 ≥0}, andA {1,0,1,2}. The set-valued mapF:A → 2Y is defined as follows:

F1,0 y1, y2

|y1≥1,2≤y2≤ −y14

∪ {1,1}, F1,2 y1, y2

|y1≥2,1≤y2≤ −y14

. 4.50

The set-valued mapG:A → 2Zis defined as follows:

G1,0 y1, y2

|y1≤0, 0≤y2≤y11

, G1,2 y1, y2

|y1≥ −1, y11≤y2≤1

. 4.51

Let x0 1,0, y0 1,1 ∈ Fx0. Thus, all conditions of Theorem 4.12 are satisfied. Therefore, there existsϕ:y1, y2, ϕ y1y2such thatx0, y0is a minimizer ofVPϕ.

Remark 4.14. Theorem 4.12generalizes Theorem 4.2 of16and the necessity of Theorem 4.1 of17.

In this paper, our results improve some results in the literature, and our results are very useful to form Lagrange multipliers rule and establish duality theory.

Acknowledgments

This paper was supported by the National Nature Science Foundation of China Grant 10831009. The author would like to express his thanks to his supervisor Professor X. M. Yang for guidance and the referees for valuable comments and suggestions.

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