Research Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization


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1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID , 15 pages doi: /2010/ Research Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization X. K. Sun and S. J. Li College of Mathematics and Science, Chongqing University, Chongqing , China Correspondence should be addressed to X. K. Sun, Received 29 March 2010; Accepted 3 August 2010 Academic Editor: Jong Kim Copyright q 2010 X. K. Sun and S. J. Li. 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. By virtue of higherorder adjacent derivative of setvalued maps, relationships between higherorder adjacent derivative of a setvalued map and its profile map are discussed. Some results concerning stability analysis are obtained in parametrized vector optimization. 1. Introduction Research on stability and sensitivity analysis is not only theoretically interesting but also practically important in optimization theory. A number of useful results have been obtained in scalar optimization see 1, 2. Usually, by stability, we mean the qualitative analysis, which is the study of various continuity properties of the perturbation or marginal function or map of a family of parametrized optimization problems. On the other hand, by sensitivity, we mean the quantitative analysis, which is the study of derivatives of the perturbation function. Some authors have investigated the sensitivity of vector optimization problems. In 3, Tanino studied some results concerning the behavior of the perturbation map by using the concept of contingent derivative of setvalued maps for general multiobjective optimization problems. In 4, Shi introduced a weaker notion of setvalued derivative TPderivative and investigated the behavior of contingent derivative for the setvalued perturbation maps in a nonconvex vector optimization problem. Later on, Shi also established sensitivity analysis for a convex vector optimization problem see 5. In 6, Kuk et al. investigated the relationships between the contingent derivatives of the perturbation maps i.e., perturbation map, proper perturbation map, and weak perturbation map and those of feasible set map in the objective space by virtue of contingent derivative, TPderivative and Dini derivative. Considering convex vector optimization problems, they also investigated the behavior of the above three kinds of perturbation maps under some convexity assumptions see 7.
2 2 Journal of Inequalities and Applications On the other hand, some interesting results have been proved for stability analysis in vector optimization problems. In 8, Tanino studied some qualitative results concerning the behavior of the perturbation map in convex vector optimization. In 9, Li investigated the continuity and the closedness of contingent derivative of the marginal map in multiobjective optimization. In 10, Xiang and Yin investigated some continuity properties of the mapping which associates the set of efficient solutions to the objective function by virtue of the additive weight method of vector optimization problems and the method of essential solutions. To the best of our knowledge, there is no paper to deal with the stability of higherorder adjacent derivative for weak perturbation maps in vector optimization problems. Motivated by the work reported in 3 9, in this paper, by higherorder adjacent derivative of setvalued maps, we first discuss some relationships between higherorder adjacent derivative of a setvalued map and its profile map. Then, by virtue of the relationships, we investigate the stability of higherorder adjacent derivative of the perturbation maps. The rest of this paper is organized as follows. In Section 2, we recall some basic definitions. In Section 3, after recalling the concept of higherorder adjacent derivative of setvalued maps, we provide some relationships between the higherorder adjacent derivative of a setvalued map and its profile map. In Section 4, we discuss some stability results of higherorder adjacent derivative for perturbation maps in parametrized vector optimization. 2. Preliminaries Throughout this paper, let X and Y be two finite dimensional spaces, and let K Y be a pointed closed convex cone with a nonempty interior int K, where K is said to be pointed if K K {0}. LetF : X Y be a setvalued map. The domain and the graph of F are defined by Dom F {x X : F x / } and Graph F { x, y X Y : y F x,x Dom F }, respectively. The socalled profile map F K : X Y is defined by F K x : F x K, for all x Dom F. At first, let us recall some important definitions. Definition 2.1 see 11. LetQ be a nonempty subset of Y. An elements ŷ Q is said to be a minimal point resp. weakly minimal point of Q if Q ŷ K {0} resp., Q ŷ int K. The set of all minimal points resp., weakly minimal point of Q is denoted by Min K Q resp., WMin K Q. Definition 2.2 see 12. AbaseforK is a nonempty convex subset B of Kwith 0 / B such that every k K, k / 0 has a unique representation k αb, where b B and α>0. Definition 2.3 see 13. The weak domination property is said to hold for a subset H of Y if H WMin K H int K {0}. Definition 2.4 see 14. LetF be a setvalued map from X to Y. i F is said to be lower semicontinuous l.s.c at x X if for any generalized sequence {x n } with x n x and y F x, there exists a generalized sequence {y n } with y n F x n such that y n y. ii F is said to be upper semicontinuous u.s.c at x X if for any neighborhood N F x of F x, there exists a neighborhood N x of x such that F x N F x, for all x N x.
3 Journal of Inequalities and Applications 3 iii F is said to be closed at x X if for any generalized sequence x n,y n Graph F, x n,y n x, y, it yields x, y Graph F. We say that F is l.s.c resp., u.s.c, closed on X if it is l.s.c resp., u.s.c, closed at each x X. F is said to be continuous on X if it is both l.s.c and u.s.c on X. Definition 2.5 see 14. F is said to be Lipschitz around x X if there exist a real number M>0 and a neighborhood N x of x such that F x 1 F x 2 M x 1 x 2 B Y, x 1,x 2 N x, 2.1 where B Y denotes the closed unit ball of the origin in Y. Definition 2.6 see 14. F is said to be uniformly compact near x X if there exists a neighborhood N x of x such that x N x F x is a compact set. 3. HigherOrder Adjacent Derivatives of SetValued Maps In this section, we recall the concept of higherorder adjacent derivative of setvalued maps and provide some basic properties which are necessary in the following section. Throughout this paper, let m be an integer number and m>1. Definition 3.1 see 15. Letx C X and u 1,...,u m 1 be elements of X. The set T b m C x, u 1,...,u m 1 is called the mthorder adjacent set of C at x, u 1,...,u m 1,ifandonly if, for any x T b m C x, u 1,...,u m 1, for any sequence {h n } R \{0} with h n 0, there exists a sequence {x n } X with x n x such that x h n u 1 h 2 nu 2 h m 1 n u m 1 h m n x n C, n. 3.1 Definition 3.2 see 15. Let x, y Graph F and u i,v i X Y, i 1, 2,...,m 1. The mthorder adjacent derivative D b m F x, y, u 1,v 1,...,u m 1,v m 1 of F at x, y for vectors u 1,v 1,..., u m 1,v m 1 is the setvalued map from X to Y defined by Graph D b m F ) ) x, y, u 1,v 1,...,u m 1,v m 1 TGraph F b m ) x, y, u1,v 1,...,u m 1,v m Proposition 3.3. Let x, y Graph F and u i,v i X Y, i 1, 2,...,m 1. Then, for any x Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1, D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x K D b m F K x, y, u 1,v 1,...,u m 1,v m Proof. The proof follows on the lines of Proposition 2.1 in 3 by replacing contingent derivative by mthorder adjacent derivative. Note that the converse inclusion of 3.3 may not hold. The following example explains the case where we only take m 2, 3.
4 4 Journal of Inequalities and Applications Example 3.4. Let X Y R and K R,letF : X Y be defined by {0} if x 0, F x { } 1,x 3 if x> Let x, y 0, 0 Graph F and u 1,v 1 u 2,v 2 1, 0. For any x>0, we have D b 2 F x, y, u 1,v 1 ) x {0}, D b 2 F K x, y, u 1,v 1 ) x R, D b 3 F x, y, u 1,v 1,u 2,v 2 ) x {1}, D b 3 F K x, y, u 1,v 1,u 2,v 2 ) x R. 3.5 Thus, for any x>0, we have D b 2 F K x, y, u 1,v 1 ) x / D b 2 F x, y, u 1,v 1 ) x K, D b 3 F K x, y, u 1,v 1,u 2,v 2 ) x / D b 3 F x, y, u 1,v 1,u 2,v 2 ) x K. 3.6 Proposition 3.5. Let x, y Graph F and u i,v i X Y, i 1, 2,...,m 1. Assume that K has a compact base. Then, for any x Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1, WMin K D b m x, ) F K) y, u1,v 1,...,u m 1,v m 1 x D b m F ) x, y, u 1,v 1,...,u m 1,v m 1 x. 3.7 where K is a closed convex cone contained in int K {0}. Proof. If WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x, the inclusion holds trivially. Thus, we suppose that WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x /. Let y 0 WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x. Then, y 0 D b m F K) x, y, u1,v 1,...,u m 1,v m Since K int K {0}, WMin K D b m x, ) F K) y, u1,v 1,...,u m 1,v m 1 x ) 3.9 x, Min K Db m ) F K y, u1,v 1,...,u m 1,v m 1 x, then it follows that y 0 Min K Db m F K) x, y, u1,v 1,...,u m 1,v m
5 Journal of Inequalities and Applications 5 From 3.8 and the definition of mthorder adjacent derivative, we have that for any sequence {h n } R \{0} with h n 0, there exist sequences { x n,y n } with x n,y n x, y 0 and { k n } K such that n v m 1 h m n y n k n F x h n u 1 h m 1 n u m 1 h m n x n ) Since K is a closed convex cone contained in int K {0}, K has a compact base. It is clear that B K is a compact base for K, where B is a compact base for K. In this proposition, we assume that B is a compact base of K. Since k n K, there exist α n > 0andb n B such that k n α n b n. Since B is compact, we may assume without loss of generality that b n b B. Now, we show that α n /h m n 0. Suppose that α n /h m n 0. Then, for some ε>0, we may assume, without loss of generality, that α n /h m n ε, for all n. Letk n εh m n /α n k n K. Then, we have k n k n K By 3.11 and 3.12, weobtainthat ) n v m 1 h m n y n k n F x h n u 1 h m 1 n u m 1 h m n x n K From 3.13 and k n /h m n ε/α n k n εb n εb / 0, we have y 0 εb D b m F K) x, y, u1,v 1,...,u m 1,v m 1 ) x, 3.14 which contradicts Therefore, α n /h m n 0andy n k n /h m n y 0. Thus, it follows from 3.11 that y 0 D b m F x, y, u 1,v 1,...,u m 1,v m 1 x, and the proof is complete. Remark 3.6. The inclusion of WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 ) x D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x 3.15 may not hold under the assumptions of Proposition 3.5. The following example explains the case where we only take m 2, 3. Example 3.7. Let X R and Y R 2,letK R 2 and F : X Y be defined by F x { y R 2 : y x 3,x 3)}. 3.16
6 6 Journal of Inequalities and Applications Suppose that x, y 0, 0, 0 Graph F, u 1,v 1 u 2,v 2 1, 0, 0. Then, for any x X, D b 2 F x, y, u 1,v 1 ) x { 0, 0 }, D b 3 F ) x, y, u 1,v 1,u 2,v 2 x { 1, 1 }, D b 2 F K ) { y1 ) } x, y, u 1,v 1 x,y 2 R 2 : y 1 0,y 2 0, D b 3 F K ) { y1 ) } x, y, u 1,v 1,u 2,v 2 x,y 2 R 2 : y 1 1,y Naturally, we have WMin K D b 2 F K ) { y1 ) } x, y, u 1,v 1 x,y 2 R 2 : y 1 y 2 0,y 1 0,y 2 0, WMin K D b 3 F K ) { y1 ) } x, y, u 1,v 1,u 2,v 2 x,y 2 R 2 : y 1 1,y 2 1 { y1 ) },y 2 R 2 : y 1 1,y Thus, for any x X, WMin K D b 2 F K x, y, u 1,v 1 ) x / D b 2 F x, y, u 1,v 1 ) x, WMin K D b 3 F K x, y, u 1,v 1,u 2,v 2 ) x / D b 3 F x, y, u 1,v 1,u 2,v Proposition 3.8. Let x, y Graph F, and u i,v i X Y, i 1, 2,...,m 1, and let K has a compact base. Suppose that P x : D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x fulfills the weak domination property for any x Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1. Then, for any x Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1, WMin K D b m F K) x, y, u1,v 1,...,u m 1,v m 1 ) x WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x, 3.20 where K is a closed convex cone contained in int K {0}. Proof. Let y 0 WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x. Then, y 0 D b m F K) x, y, u1,v 1,...,u m 1,v m By Proposition 3.5, we also have y 0 D b m F x, y, u 1,v 1,...,u m 1,v m 1 x.
7 Journal of Inequalities and Applications 7 Suppose that y 0 / WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. Then, there exists y D b m F x, y, u 1,v 1,...,u m 1,v m 1 x such that y 0 y int K From y D b m F x, y, u 1,v 1,...,u m 1,v m 1 x and Proposition 3.3, we have y D b m F K) x, y, u1,v 1,...,u m 1,v m So, by 3.21, 3.22,and 3.23, y 0 / WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x, which leads to a contradiction. Thus, y 0 WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. Conversely, let y 0 WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. Then, y 0 D b m F ) x, y, u 1,v 1,...,u m 1,v m 1 x D b m x, ) F K) y, u1,v 1,...,u m 1,v m 1 x Suppose that y 0 / WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x. Then, there exists y D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x such that y 0 y k int K Since P x fulfills the weak domination property for any x Dom D b m F x, y, u 1,v 1,..., u m 1,v m 1, there exists k int K {0} such that y k WMin K D b m F K) x, y, u1,v 1,...,u m 1,v m From 3.25 and 3.26, we have y 0 k k WMin K D b m F K) x, y, u1,v 1,...,u m 1,v m It follows from Proposition 3.5 and 3.27 that y 0 k k D b m F x, y,u 1,v 1,...,u m 1,v m 1 ) x, 3.28 which contradicts y 0 WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. Thus, y 0 WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x, and the proof is complete. Obviously, Example 3.4 can also show that the weak domination property of P x is essential for Proposition 3.8.
8 8 Journal of Inequalities and Applications Remark 3.9. From Example 3.7, the equality of WMin K D b m F K x, y, u 1,v 1,...,u m 1,v m 1 ) x WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x 3.29 may still not hold under the assumptions of Proposition 3.8. Proposition Let x, y Graph F and u i,v i X Y, i 1, 2,...,m 1. Suppose that F is Lipschitz at x. Then, D b m F x, y, u 1,v 1,...,u m 1,v m 1 is continuous on Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1. Proof. Since F is Lipschitz at x, there exist a real number M>0 and a neighborhood N x of x such that F x 1 F x 2 M x 1 x 2 B Y, x 1,x 2 N x First, we prove that D b m F x, y, u 1,v 1,...,u m 1,v m 1 is l.s.c. at x Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1. Indeed, for any ŷ D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. Fromthe definition of mthorder adjacent derivative, we have that for any sequence {h n } R \{0} with h n 0, there exists a sequence { x n, ŷ n } with x n, ŷ n x, ŷ such that n v m 1 h m n ŷ n F x h n u 1 h m 1 n u m 1 h m n x n ) Take any x X and x n x. Obviously, x h n u 1 h m 1 n u m 1 h m n x n, x h n u 1 u m 1 h m n x n N x, for any n sufficiently large. Therefore, by 3.30, we have h m 1 n ) F x h n u 1 h m 1 n u m 1 h m n x n ) F x h n u 1 h m 1 n u m 1 h m n x n Mh m n x n x n B Y So, with 3.31, there exists b n B Y such that n v m 1 h m ) n ŷn M x n x n b n F x h n u 1 h m 1 n u m 1 h m n x n ) We may assume, without loss of generality, that b n b B Y.Thus,by 3.33, ŷ M x x b D b m F x, y, u 1,v 1,...,u m 1,v m It follows from 3.34 that for any sequence {x k } with x k x, ŷ D b m F x, y, u 1,v 1,..., u m 1,v m 1 x, there exists a sequence {y k } with y k : ŷ M x x k b D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) xk. 3.35
9 Journal of Inequalities and Applications 9 Obviously, y k ŷ. Hence, D b m F x, y, u 1,v 1,...,u m 1,v m 1 is l.s.c. on Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1. We will prove that D b m F x, y, u 1,v 1,...,u m 1,v m 1 is u.s.c. on x Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1. In fact, for any ε>0, we consider the neighborhood x ε/m B X of x. Letx x ε/m B X and y D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. From the definition of D b m F x, y, u 1,v 1,...,u m 1,v m 1 x, we have that for any sequence {h n } R \{0} with h n 0, there exists a sequence { x n,y n } with x n,y n x, y such that n v m 1 h m n y n F x h n u 1 h m 1 n u m 1 h m n x n ) Take any x n x. Obviously, x h n u 1 h m 1 n u m 1 h m n x n, x h n u 1 h m 1 n u m 1 h m n x n N x, for any n sufficiently large. Therefore, by 3.30, we have ) ) F x h n u 1 h m 1 n u m 1 h m n x n F x h n u 1 h m 1 n u m 1 h m n x n Mh m n x n x n B Y Similar to the proof of l.s.c., there exists b B Y such that y M x x b D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x Thus, y D b m F x, y, u 1,v 1,...,u m 1,v m 1 x εb Y. Hence, D b m F x, y, u 1,v 1,...,u m 1,v m 1 is u.s.c. on Dom D b m F x, y, u 1,v 1,...,u m 1,v m 1, and the proof is complete. 4. Continuity of HigherOrder Adjacent Derivative for Weak Perturbation Map In this section, we consider a family of parametrized vector optimization problems. Let F be a setvalued map from U toy, where U is the Banach space of perturbation parameter vectors, Y is the objective space, and F is considered as the feasible set map in the objective space. In the optimization problem corresponding to each parameter valued x, our aim is to find the set of weakly minimal points of the feasible objective valued set F x. Hence, we define another setvalued map S from U to Y by S x WMin K F x, for any x U. 4.1 The setvalued map S is called the weak perturbation map. Throughout this section, we suppose that K is a closed convex cone contained in int K {0}. Definition 4.1 see 11. F is said to be Kminicomplete by S near x if F x S x K, for any x N x, where N x is a neighborhood of x.
10 10 Journal of Inequalities and Applications Remark 4.2. Since S x F x,thekminicompleteness of F by S near x implies that S x K F x K, for any x N x. 4.2 Hence, if F is Kminicomplete by S near x, then, for any y S x D b m F K x, y, u 1,v 1,...,u m 1,v m 1 ) D b m S K x, y, u 1,v 1,...,u m 1,v m 1 ). 4.3 The following lemma palys a crucial role in this paper. Lemma 4.3. Let x, y Graph S and u i,v i U Y, i 1, 2,...,m 1, and let K have a compact base. Suppose that the following conditions are satisfied: i P x : D b m F K x, y, u 1,v 1,...,u m 1,v m 1 x fulfills the weak domination property for any x Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1 ; ii F is Lipschitz at x; iii F is Kminicomplete by S near x. Then, for any x Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1, D b m S x, y, u 1,v 1,...,u m 1,v m 1 ) x WMinK D b m F x, y, u 1,v 1,...,u m 1,v m Proof. We first prove that WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x D b m S x, y, u 1,v 1,...,u m 1,v m In fact, from Proposition 3.5, Proposition 3.8, and the Kminicompleteness of F by S near x, we have WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x WMin K D b m F K) x, y, u1,v 1,...,u m 1,v m 1 ) x WMin K D b m S K) x, y, u1,v 1,...,u m 1,v m 1 ) x 4.6 D b m S x, y, u 1,v 1,...,u m 1,v m 1 Thus, result 4.5 holds. Now, we prove that D b m S x, y, u 1,v 1,...,u m 1,v m 1 ) x WMinK D b m F x, y, u 1,v 1,...,u m 1,v m 1 4.7
11 Journal of Inequalities and Applications 11 In fact, assume that y D b m S x, y, u 1,v 1,...,u m 1,v m 1 x. Then, for any sequence {h n } R \{0} with h n 0, there exists a sequence { x n,y n } with x n,y n x, y such that ) n v m 1 h m n y n S x h n u 1 h m 1 n u m 1 h m n x n F x h n u 1 h m 1 n u m 1 h m n x n ). 4.8 Suppose that y / WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. Then, there exists ŷ D b m F x, y, u 1,v 1,...,u m 1,v m 1 x such that y ŷ int K. Thus, for the preceding sequence {h n }, there exists a sequence { x n, ŷ n } with x n, ŷ n x, y such that n v m 1 h m n ŷ n F x h n u 1 h m 1 n u m 1 h m n x n ). 4.9 Obviously, x h n u 1 h m 1 n u m 1 h m n x n, x h n u 1 h m 1 n u m 1 h m n x n N x, for any n sufficiently large. Therefore, by ii, we have ) ) F x h n u 1 h m 1 n u m 1 h m n x n F x h n u 1 h m 1 n u m 1 h m n x n Mh m n x n x n B Y So, with 4.9, there exists b n B Y such that n v m 1 h m ) n ŷn M x n x n b n F x h n u 1 h m 1 n u m 1 h m n x n ) Since y n ŷ n M x n x n b n y ŷ and y ŷ int K, y n ŷ n M x n x n b n int K, for n sufficiently large. Then, we have n v m 1 h m n y n y h n v 1 h m 1 n v m 1 h m ) n ŷn ) M x n x n b n int K, 4.12 which contradicts 4.8. Then, y WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 x. This completes the proof. The following example shows that the Lemma 4.3, where we only take m 2, 3. Kminicompleteness of F is essential in Example 4.4 F is not Kminicomplete by S near x. LetU R, Y R 2 and K R 2,andlet F : U Y be defined by { y1 ) } { y1 F x,y 2 R 2 ) : y 1 0,y 2 0,y 2 R 2 : y 2 > } y
12 12 Journal of Inequalities and Applications Then, for any x U, { y1 ) } F x K F x, S x,y 2 R 2 : y 1 0,y Suppose that x, y 0, 0, 0 Graph S, u 1,v 1 u 2,v 2 1, 0, 0. Then, F is Lipschitz at x, and for any x U, D b 2 x, ) F K) y, u1,v 1 x D b 3 x, ) F K) y, u1,v 1,u 2,v 2 x { y1 ) },y 2 R 2 : y 1 0,y 2 0 { y1 ),y 2 R 2 : y 2 } y fulfills the weak domination property. We also have D b 2 F ) x, y, u 1,v 1 x D b 3 F ) x, y, u 1,v 1,u 2,v 2 x { y1 ) },y 2 R 2 : y 1 0,y 2 0 { y1 ),y 2 R 2 : y 2 } y On the other hand, D b 2 S x, y, u 1,v 1 ) x D b 3 S x, y, u 1,v 1,u 2,v 2 ) x S x, WMin K D b 2 F ) x, y, u 1,v 1 x WMinK D b 3 F ) x, y, u 1,v 1,u 2,v 2 x { y1 ) },y 2 R 2 : y 1 0,y 2 0 { y1 ) },y 2 R 2 : y 2 y 1,y 1 < Thus, for any x X, D b 2 S x, y, u 1,v 1 ) x / WMin K D b 2 F x, y, u 1,v 1 ) x, D b 3 S x, y, u 1,v 1,u 2,v 2 ) x / WMin K D b 3 F x, y, u 1,v 1,u 2,v Theorem 4.5. Let x, y Graph S and u i,v i U Y,i 1, 2,...,m 1. Then, D b m S x, y, u 1,v 1,...,u m 1,v m 1 x is closed on Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1. Proof. From the definition of D b m S x, y, u 1,v 1,...,u m 1,v m 1 x, we have that Graph D b m S x, y, u 1,v 1,...,u m 1,v m 1 ) ) T b m Graph S x, y, u1,v 1,...,u m 1,v m 1 ). 4.19
13 Journal of Inequalities and Applications 13 Since T b m Graph S x, y, u 1,v 1,...,u m 1,v m 1 is closed set, D b m S x, y, u 1,v 1,...,u m 1,v m 1 is closed on Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1, and the proof is complete. Theorem 4.6. Let x, y Graph S and u i,v i U Y, i 1, 2,...,m 1.IfY is a compact space, then D b m S x, y, u 1,v 1,...,u m 1,v m 1 is u.s.c. on Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1. Proof. Since Y is a compact space, it follows from Corollary 9 of Chapter 3 in 14 and Theorem 4.5 that D b m S x, y, u 1,v 1,...,u m 1,v m 1 is u.s.c. on Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1. Thus, the proof is complete. Theorem 4.7. Let x Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1. Suppose that D b m F x, y, u 1,v 1,...,u m 1,v m 1 x is a compact set and the assumptions of Lemma 4.3 are satisfied. Then, D b m S x, y, u 1,v 1,...,u m 1,v m 1 is u.s.c. at x. Proof. It follows from Lemma 4.3 and Theorem 4.5 that WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 is closed. By Proposition 3.10, we have that D b m F x, y, u 1,v 1,..., u m 1,v m 1 is u.s.c. at x. Since D b m F x, y, u 1,v 1,...,u m 1,v m 1 x is a compact set, it follows from Theorem 8 of Chapter 3 in 14 that D b m S ) x, y, u 1,v 1,...,u m 1,v m 1 WMinK D b m F ) x, y, u 1,v 1,...,u m 1,v m 1 [WMin K D b m F ) ] x, y, u 1,v 1,...,u m 1,v m 1 D b m F ) x, y, u 1,v 1,...,u m 1,v m is u.s.c. at x, and the proof is complete. Now, we give an example to illustrate Theorem 4.7, where we also take m 2, 3. Example 4.8. Let U 0, 1, Y R 2,andK R 2,andletF : U Y be defined by F x { y1,y 2 ) R 2 :0 y 1 x 3, 0 y 2 x 3} Then, for any x U, { y1 ) } { y1 S x,y 2 R 2 :0 y 1 x 3 ),y 2 0,y 2 R 2 : y 1 0, 0 y 2 x 3} Suppose that x, y 0, 0, 0 Graph S, x 1/3, u 1,v 1 u 2,v 2 1, 0, 0 and K { y 1,y 2 R 2 : 1/4 y 2 y 1 4y 2 }. Obviously, K has a compact base, F is Lipschitz at x, and F is Kminicomplete by S near x. By direct calculating, for any x U, D b 2 x, ) F K) y, u1,v 1 x K, D b 3 x, ) { y1 ) } F K) y, u1,v 1,u 2,v 2 x,y 2 R 2 :0 y 1 4y 2 1, 0 y 2 4y
14 14 Journal of Inequalities and Applications fulfill the weak domination property, which is a strong property for a setvalued map. We also have D b 2 F x, y, u 1,v 1 ) x { 0, 0 }, D b 2 S ) x, y, u 1,v 1 x { 0, 0 }, D b 3 F ) { y1 ) } x, y, u 1,v 1,u 2,v 2 x,y 2 R 2 :0 y 1 1, 0 y 2 1, D b 3 S ) { y1 ) } x, y, u 1,v 1,u 2,v 2 x,y 2 R 2 :0 y 1 1,y 2 0 { y1 ) },y 2 R 2 : y 1 0, 0 y Thus, the conditions of Theorem 4.7 are satisfied. Obviously, both D b 2 S x, y, u 1,v 1 x and D b 3 S x, y, u 1,v 1,u 2,v 2 x are u.s.c at x. Theorem 4.9. Let x Dom D b m S x, y, u 1,v 1,...,u m 1,v m 1. Suppose that D b m F x, y, u 1,v 1,...,u m 1,v m 1 is uniformly compact near x and the assumptions of Lemma 4.3 are satisfied. Then, D b m S x, y, u 1,v 1,...,u m 1,v m 1 is l.s.c. at x. Proof. By Lemma 4.3,itsuffices to prove that WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 is l.s.c. at x.letx n x and ŷ WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x By Proposition 3.10, we have that D b m F x, y, u 1,v 1,...,u m 1,v m 1 is l.s.c. at x. Then, there exists a sequence {y n } with y n D b m F x, y, u 1,v 1,...,u m 1, v m 1 x n such that y n ŷ. Since K int K {0}, WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x Min K Db m F x, y, u 1,v 1,...,u m 1,v m Then, for any sequence {y n} with y n have WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 x n,we y n Min K Db m F x, y,u 1,v 1,...,u m 1,v m 1 ) xn, 4.27 then it follows that y n y n K Since D b m F x, y, u 1,v 1,...,u m 1,v m 1 is uniformly compact near x, we may assume, without loss of generality, that y n y. It follows from the closedness of D b m F x, y, u 1,v 1,...,u m 1,v m 1 that y D b m F x, y, u 1,v 1,...,u m 1,v m 1 ) x. 4.29
15 Journal of Inequalities and Applications 15 From 4.28 and K is closed, we have ŷ y K int K {0}. Then, it follows from 4.25 and 4.29 that y ŷ. Thus,WMin K D b m F x, y, u 1,v 1,...,u m 1,v m 1 is l.s.c. at x, and the proof is complete. It is easy to see that Example 4.8 can also illustrate Theorem 4.9. Acknowledgments The authors would like to thank the anonymous referees for their valuable comments and suggestions which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China Grant no and Chongqing University Postgraduates Science and Innovation Fund Grant no B1A References 1 J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer, New York, NY, USA, A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, vol. 165 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, vol. 56, no. 3, pp , D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, Journal of Optimization Theory and Applications, vol. 70, no. 2, pp , D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, vol. 77, no. 1, pp , H. Kuk, T. Tanino, and M. Tanaka, Sensitivity analysis in vector optimization, Journal of Optimization Theory and Applications, vol. 89, no. 3, pp , H. Kuk, T. Tanino, and M. Tanaka, Sensitivity analysis in parametrized convex vector optimization, Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp , T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, vol. 26, no. 3, pp , Shengjie Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, vol. 11, no. 2, pp , S. W. Xiang and W. S. Yin, Stability results for efficient solutions of vector optimization problems, Journal of Optimization Theory and Applications, vol. 134, no. 3, pp , Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of Multiobjective Optimization, vol. 176 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, R. B. Holmes, Geometric Functional Analysis and Its Applications, Springer, New York, NY, USA, D.T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, J.P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, J.P. Aubin and H. Frankowska, SetValued Analysis, vol. 2 of Systems & Control: Foundations & Applications, Birkhäuser, Boston, Mass, USA, 1990.
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