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


 Alan Leonard
 1 years ago
 Views:
Transcription
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.
Duality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationQuasi Contraction and Fixed Points
Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa00168, 6 Pages doi:10.5899/2012/jnaa00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationTypes of Degrees in Bipolar Fuzzy Graphs
pplied Mathematical Sciences, Vol. 7, 2013, no. 98, 48574866 HIKRI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2013.37389 Types of Degrees in Bipolar Fuzzy Graphs Basheer hamed Mohideen Department
More informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationMaxMin Representation of Piecewise Linear Functions
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 297302. MaxMin Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department,
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationSOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction
italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationNotes: Chapter 2 Section 2.2: Proof by Induction
Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case  S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example
More information0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup
456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),
More informationA NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION
1 A NEW LOOK AT CONVEX ANALYSIS AND OPTIMIZATION Dimitri Bertsekas M.I.T. FEBRUARY 2003 2 OUTLINE Convexity issues in optimization Historical remarks Our treatment of the subject Three unifying lines of
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More information6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium
6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline PricingCongestion Game Example Existence of a Mixed
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
More informationResearch Article On the Rank of Elliptic Curves in Elementary Cubic Extensions
Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationSection 3 Sequences and Limits, Continued.
Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More informationNumerical Verification of Optimality Conditions in Optimal Control Problems
Numerical Verification of Optimality Conditions in Optimal Control Problems Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der JuliusMaximiliansUniversität Würzburg vorgelegt von
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationDetermination of the normalization level of database schemas through equivalence classes of attributes
Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,
More informationAbsolute Value Programming
Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationSome remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s1366101505080 R E S E A R C H Open Access Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationMixedÀ¾ Ð½Optimization Problem via Lagrange Multiplier Theory
MixedÀ¾ Ð½Optimization Problem via Lagrange Multiplier Theory Jun WuÝ, Sheng ChenÞand Jian ChuÝ ÝNational Laboratory of Industrial Control Technology Institute of Advanced Process Control Zhejiang University,
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationRelatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations
Wang and Agarwal Advances in Difference Equations (2015) 2015:312 DOI 10.1186/s1366201506500 R E S E A R C H Open Access Relatively dense sets, corrected uniformly almost periodic functions on time
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationTECHNICAL NOTE. Proof of a Conjecture by Deutsch, Li, and Swetits on Duality of Optimization Problems1
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 102, No. 3, pp. 697703, SEPTEMBER 1999 TECHNICAL NOTE Proof of a Conjecture by Deutsch, Li, and Swetits on Duality of Optimization Problems1 H. H.
More informationSEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov
Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationScheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates
Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates Donatas Elvikis, Horst W. Hamacher, Marcel T. Kalsch Department of Mathematics, University of Kaiserslautern, Kaiserslautern,
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationTiers, Preference Similarity, and the Limits on Stable Partners
Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationFuzzy Probability Distributions in Bayesian Analysis
Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:
More informationLocal periods and binary partial words: An algorithm
Local periods and binary partial words: An algorithm F. BlanchetSadri and Ajay Chriscoe Department of Mathematical Sciences University of North Carolina P.O. Box 26170 Greensboro, NC 27402 6170, USA Email:
More informationOn the Algebraic Structures of Soft Sets in Logic
Applied Mathematical Sciences, Vol. 8, 2014, no. 38, 18731881 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.43127 On the Algebraic Structures of Soft Sets in Logic Burak Kurt Department
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMath212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.
More informationLow upper bound of ideals, coding into rich Π 0 1 classes
Low upper bound of ideals, coding into rich Π 0 1 classes Antonín Kučera the main part is a joint project with T. Slaman Charles University, Prague September 2007, Chicago The main result There is a low
More informationSome New Classes of Generalized Concave Vector Valued Functions. Riccardo CAMBINI
UNIVERSITÀ DEGLI STUDI DI PISA DIPARTIMENTO DI STATISTICA E MATEMATICA APPLICATA ALL ECONOMIA Some New Classes of Generalized Concave Vector Valued Functions Riccardo CAMBINI This is a preliminary version
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationChapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z
More informationA Note on Di erential Calculus in R n by James Hebda August 2010.
A Note on Di erential Calculus in n by James Hebda August 2010 I. Partial Derivatives o Functions Let : U! be a real valued unction deined in an open neighborhood U o the point a =(a 1,...,a n ) in the
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationModule1. x 1000. y 800.
Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationRecall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:
Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1
More informationSome other convexvalued selection theorems 147 (2) Every lower semicontinuous mapping F : X! IR such that for every x 2 X, F (x) is either convex and
PART B: RESULTS x1. CHARACTERIZATION OF NORMALITY TYPE PROPERTIES 1. Some other convexvalued selection theorems In this section all multivalued mappings are assumed to have convex values in some Banach
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationA Simple Proof of Duality Theorem for MongeKantorovich Problem
A Simple Proof of Duality Theorem for MongeKantorovich Problem Toshio Mikami Hokkaido University December 14, 2004 Abstract We give a simple proof of the duality theorem for the Monge Kantorovich problem
More informationA class of online scheduling algorithms to minimize total completion time
A class of online scheduling algorithms to minimize total completion time X. Lu R.A. Sitters L. Stougie Abstract We consider the problem of scheduling jobs online on a single machine and on identical
More informationMathematical Induction
Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers
More information1 Fixed Point Iteration and Contraction Mapping Theorem
1 Fixed Point Iteration and Contraction Mapping Theorem Notation: For two sets A,B we write A B iff x A = x B. So A A is true. Some people use the notation instead. 1.1 Introduction Consider a function
More informationFINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS
FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROSSHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure
More information