In the paper, we give some remarks on [1]. Then, we modify main results concerning the sum rule of second-order contingent derivatives for set-valued maps and its application to the sensitivity analysis of generalized perturbation maps. The obtained results are new and better than those in [1]. Some examples are proposed to illustrate our results.
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 The second-order contingent derivative of generalized perturbation maps Nguyen Le Hoang Anh University of Science, VNU– HCMC (Received on 5th December 2016, accepted on 28 th November 2017) ABSTRACT In the paper, we give some remarks on [1] Then, generalized perturbation maps The obtained results we modify main results concerning the sum rule of are new and better than those in [1] Some examples second-order contingent derivatives for set-valued are proposed to illustrate our results maps and its application to the sensitivity analysis of Key words: second-order proto-differentiability, second-order semi-differentiability, metric subregularity, generalized perturbation map, set-valued map INTRODUCTION In [1], the second-order proto-differentiability and second-order semi-differentiability for set-valued maps were firsthy discussed and applied to sum rules of two set-valued maps Then, the authors established secondorder sensitivity analysis of generalized perturbation maps as an application of sum rules The semidifferentiability plays an essential role in all main results in [1] In the paper, we give some remarks on the Proposition and Theorem in [1] On the other hand, a new result is proposed to avoid the semidifferentiability by using a weaker hypothesis of the proto-differentiability The layout of this paper is as follows Section is devoted to several concepts needed in the sequel Our main remarks and modified results are given in Section PRELIMINARIES Throughout the paper, let X and Y be normed spaces For a set-valued map F : X 2Y , the domain, image, and graph of F are defined, respectively (resp for short), by dom( F ) : x X | F ( x) , im( F ) : y Y | y F ( X ) , gr ( F ) : ( x, y) X Y | y F ( x) Definition 2.1 ([2, 3]) Let S X , x cl (S ) and w X , where cl ( S ) denotes the closure of S (i) The contingent cone and the adjacent cone of S at x are defined by, resp, T (S , x) : u X | tn 0 , un u, x tnun S , T b (S , x) : u X | tn 0 , un u, x tnun S Trang 203 Science & Technology Development, Vol 5, No.T20- 2017 (ii) The second-order contingent cone and the second-order adjacent cone of S at x in the direction w are defined by, resp, t2 T ( S , x, w) : u X | tn 0 , un u, x tn w n un S , t2 T b (2) ( S , x, w) : u X | tn 0 , un u, x tn w n un S Remark 2.1 From the Observation in [4], we obtain the equivalent formulae of Definition 2.1(ii) as follows n , n : n , n , n 2, n T ( S , x, w) : u X , xn S : n ( xn x) w, n ( n ( xn x) w) u T 2( b ) n , n : n , n , n 2, n ( S , x, w) : u X xn S : n ( xn x) w, n ( n ( xn x) w) u Definition 2.2 ([2, 3]) Let F : X 2Y , ( x, y) gr ( F ) and (w, r ) X Y (i) The contingent derivative (the adjacent derivative) of F at ( x, y) is a set-valued map DF ( x, y) : X 2Y ( D F ( x, y) : X 2Y , resp) such that b gr ( DF ( x, y)) : T ( gr ( F ),( x, y)) ( gr ( Db F ( x, y)) : T b ( gr ( F ), ( x, y)) , resp) (ii) The second-order contingent derivative (the second-order adjacent derivative) of F at ( x, y) in the direction ( w, r ) is a set-valued map D2 F ( x, y, w, r ) : X 2Y ( D2(b) F ( x, y, w, r ) : X 2Y , resp) such that gr ( D2 F ( x, y, w, r )) : T ( gr ( F ), ( x, y), (w, r )) ( gr ( D2(b) F ( x, y, w, r )) : T 2(b) ( gr ( F ), ( x, y), (w, r )) , resp) Remark 2.2 From Definition 2.1 and Remark 2.1 is follous D F ( x, y, w, r )(u) = v Y | tn , un u, vn v, y tn r tn2 t F x tn w n un 2 n , n : n , n , n 2, n = v Y ( xn , yn ) gr ( F ) : n ( xn x) w, n ( n ( xn x) w) u, , n ( yn y ) r , n ( n ( yn y ) r ) v b (2) D tn2 tn2 v Y | t , u u , v v , y t r v F x t w un F ( x, y, w, r )(u) = n n n n n n 2 Trang 204 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 n , n : n , n , n 2, n = v Y ( xn , yn ) gr ( F ) : n ( xn x) w, n ( n ( xn x) w) u, n ( yn y ) r , n ( n ( yn y ) r ) v Definition 2.3 ([5]) Let F : X 2Y , ( x, y) gr ( F ) and (w, r ) X Y The second-order lower Dini derivative of F at ( x, y) in direction ( w, r ) is a set-valued map Dl2 F ( x, y, w, r ) : X 2Y that is defined by D F ( x, y, w, r )(u) = v Y | tn , un u, vn v, y tn r l tn2 t F x tn w n un 2 Remark 2.3 (i) By the proof similar to that of Observation in [4], we get n , n : n , n , n 2, n Dl2 F ( x, y, w, r )(u) = v Y xn dom( F ) : n ( xn x) w, n ( n ( xn x) w) u, yn F ( xn ) : n ( yn y ) r , n ( n ( yn y ) r ) v (ii) It is obvious to see that D F ( x, y, w, r )(u) D l b (2) F ( x, y, w, r )(u) D F ( x, y, w, r )(u) Definition 2.4 ([6, 7]) Let ( x, y) gr ( F ) and (w, r ) X Y F: X 2 , Y (i) The map F is said to be second-order protodifferentiable at x relative to y in the direction ( w, r ) if D2 F ( x, y, w, r ) Db(2) F ( x, y, w, r ) (ii) The map F is said to be second-order semidifferentiable at x relative to y in the direction ( w, r ) if D2 F ( x, y, w, r ) Dl2 F ( x, y, w, r ) It is easy to see that if F is second-order semidifferentiable then F is second-order protodifferentiable Definition 2.5 ([1]) Let F : X 2Y , ( x, y) gr ( F ) and (w, r ) X Y The map F is said to be second- order lower semi-differentiable at ( x, y) in the direction ( w, r ) if for any n , n , xn dom( F ) n , n , with n 2, n n ( xn x) w and n (n ( xn x) w) u for some u X , there exists a subsequence n yn y r i i and such y y r yni F xni n i ni ni that is convergent Remark 2.4 By Remark 2.3(i), if F is second-order semi-differentiable then F is second-order lower semidifferentiable This assertion can be also implied immediately by Proposition in [1] Recall that a set-valued map F : X 2Y is called to be metric regular at ( x, y) gr ( F ) if there are , r such that for all u BX ( x, r ), v BY ( y, r ) , Trang 205 Science & Technology Development, Vol 5, No.T20- 2017 d u, F 1 (v) d v, F (u) , (ii) In the proof of the Proposition (see page 248 of [1]), the authors implied that v v D G x, y, w, r (u ) from the second-order lower (1) where BX ( x, r ) denotes the open ball in X centered at x with radius r semi-differentiability of G This assertion is right if yn G xn Since By fixing v y in (1), we get a weaker definition named by metric subregularity The metric (sub)regularity plays an important role in variational analysis and has been applied to many topics of optimization, see [2, 8-10] and the references therein Inspired of the above definition, we propose the following concept yn yn yn G xn G xn G xn G xn G xn and G is a set-valued map, it is not sure that yn belongs to G xn Thus, the conclusion of the Proposition 3.1 may be not true, see Example 2.1 in [11] Hence, the Proposition 3.1 should be presented as follows Definition 2.6 Let F : X 2Y , ( x, y) gr ( F ) and S X The map F is said to be metric subregularity at ( x, y) gr ( F ) with respect to S if there are , r such that for all u BX ( x, r ) S , v BY ( y, r ) , d u, F 1 ( y) S d y, F (u) Proposition in the direction w, r Then, for all D2 G x, y, w, r (u) D2 G x, y, w, r (u) D2 G( x, y, w, r)(u), w, r X Y , (2) where G : G G , y : y y and r : r r w, r X Y When G is a single-valued map, (2) become an equality If, additionally, G is second-order proto- w, r Then, the set-valued differentiable at x relative to y in the direction w, r map G : X defined by G : G G is secondorder proto-differentiable at x relative to y : y y in w, r , uX , Y the direction G, G : X 2Y , and w, r , r X Y Y x relative to y in the direction and G is second-order lower semi-differentiable at x, y Let Suppose that G is second-order semi-differentiable at G : X 2Y is second-order semi-differentiable at x relative to y G( x) in the direction 3.2 x, y gr G , x, y gr G MAIN RESULTS Firstly, we recall a main proposition in [1] as follows Proposition 3.1 (Proposition in [1]) Suppose that G : X 2Y is second-order proto-differentiable at x relative to y G( x) in the direction , then G is second-order proto-differentiable at x relative to y in the direction ( w, r ) where r : r r , and for all Proof It is similar to the proof of Proposition in [1] uX , D2G( x, y, w, r )(u) D G x, y, w, r (u) D G x, y, w, r (u) Remark 3.1 (i) From Remark 2.4, the assumption on the second-order lower semi-differentiability of G at x, y in the direction w, r is superfluous As an application of Proposition 3.1 to sensitivity analysis of generalized perturbation maps, Theorem in [1] should be stated by Theorem 3.1 Let X , Y , Z be normed spaces, defined by G : X Z 2Y be G( x, z) : y D | z F ( x, y) K ( y) , where D is a closed Trang 206 convex subset in Y, F : X Y 2Z , TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 K : D 2Z , y, z gr ( K ) , ( x, y), z z gr ( F ) and z z in the direcction w, r , q q Then, for all u X w, r, q q X ( D y) Z , Suppose that F is second-order semi-differentiable at ( x, y) relative to v T (D, y, r) | p D F x, y, z z, w, r, q q (u, v) D K y, z, r, q (v) 2 D2G( x, z, y, w, q, r )(u, p) (3) When F is a single-valued map, (3) is an equality If, additionally, K is second-order proto-differentiable at y relative to z in the direction r , q , then G is second-order proto-differentiable at ( x, z ) relative to y in the direction ( w, q, r ) Proof The reader is referred to Theorem in [1] A natural question arises: which conditions ensure that (3.1) becomes an equality when all maps are setvalued? To get the answer, we recall a concept of the TP-derivative (see [12]) of a set-valued map F : X 2Y at ( x, y) gr ( F ) as follows DTP F ( x, y)(u) : v Y | tn 0, un , (u, v) : x tnun x, y tnvn F x tnun The following concept is necessary for our next result Definition 3.1 Let F : X 2Y , ( x, y) gr ( F ) and (w, r ) X Y The asymptotic second-order TP-derivative of F at ( x, y) in the direction ( w, r ) is a set-valued map DTP F x, y, w, r : X 2Y defined by th tn , hn 0, un , (u, v) : x tn w n n un x, DTP F ( x, y, w, r )(u ) : v Y t h t h n n n n y tn r F x tn w un 2 It is obvious to see that DTP F ( x, y, 0, 0)(u) DTP F ( x, y)(u) for all u X By virtue of the asymptotic secondorder TP-derivative, we obtain the converse inclusion of (3.1) for set-valued maps as follows Proposition 3.3 Let G, G : X 2Y , x, y gr G , x, y gr G finite dimensional and DTP G x, y, w, r (0) w, r, r X Y Y Suppose that and D G x, y, w, r (0) 0 TP (4) Then, for all u X , D2G( x, y, w, r )(u) D G x, y, w, r (u ) D G x, y, w, r (u ) , where G : G G , y : y y and r : r r Proof Let v D2G( x, y, w, r )(u) , then there exist tn 0 , un , (u, v) such that y tn r Y is tn2 t2 t2 t2 G x tn w n un G x tn w n un G x t n w n un 2 Trang 207 Science & Technology Development, Vol 5, No.T20- 2017 t2 t2 Thus, there are yn G x tn w n un and yn G x tn w n un such that , where : (5) yn y t n r y y tn r , : n We now prove that the sequence has a convergent subsequence (1/ 2)tn (1/ 2)tn2 Suppose to the contrary, i.e., By setting zn : / , then zn (taking a subsequence if necessary) has a limit point z with z Moreover, y tn r tn tn z n tn tn yn G x t n w u v n n Let hn : tn , we get z DTP G x, y, w, r (0) It follows from (5) that vn , which implies / z On the other hand, tn tn v tn tn n y tn r yn G x t n w v 2 n Thus, z DTP G x, y, w, r (0) , which contradicts (4) converges to v , then v D2 G x, y, w, r (u) From (5), v v gets v v D2 G x, y, w, r (u ) , 3.4 G, G : X , Y Without loss of generality, we assume that one Proposition Hence i.e., u v n n Let Y be finite dimensional, x, y gr G , x, y gr G , w, r, r X Y Y Suppose that and G is second-order semi-differentiable at x relative to y in the direction w, r and (4) holds Then, (2) becomes an equality If, v D2 G x, y, w, r (u ) D G x, y, w, r (u ) additionally, G is second-order proto-differentiable at x relative to y in the direction w, r , then G is Note that (3.3) is only a sufficient condition (not necessary condition) for the converse inclusion of (2), see Example in [1] It follows from the Proposition 3.3 that the Proposition 3.2 and the Theorem 3.1 can be modified for set-valued maps as follows second-order proto-differentiable at x relative to y in the direction w, r Theorem 3.2 Let Y be finite dimensional, defined by G : X Z 2Y be G( x, z) : y D | z F ( x, y) K ( y) , where D is a closed Trang 208 convex subset in Y, F : X Y 2Z , TAÏP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 w, r, q q X ( D y) Z Suppose that K : D 2Z , y, z gr ( K ) , x, y , z z gr F second-order semi-differentiable at ( x, y) relative to recent papers for calculus of several kinds of generalized derivatives, see [13-16] In this paper, we propose another hypothesis to obtain (2) without the semi-differentiability as follows zz Proposition and in the w, r, q q direcction F is and TP then G G is second-order proto-differentiable at x relative to y in the direction w, r is second-order proto- and the map g : X Y X g , , , : is differentiable at ( x, z ) relative to y in the direction ( w, q, r ) x, y, x, y, X By taking (w, r ) (0, 0) , the Proposition 3.4 and the Theorem 3.2 were reduced to Proposition 2.1 and the Theorem 3.1 in [11], respectively metric defined by subregular at with respect to gr G gr G Then, (3.1) holds Proof and there exist v D2 G x, y, w, r (u) Let v D2 G x, y, w, r (u) , In [1], the second semi-differentiability is employed to get (3.1) Although it is a quite strict condition, this concept (or relative versions) is used in y tn r and w, r, r X Y Y Suppose that Then, (3) is an equality If, additionally, K is secondorder proto-differentiable at y relative to z in the r, q , G, G : X 2Y , Let D K y, z, r, q (0) 0x., y gr G , x, y gr G DTP F x, y, z z, w, r , q q (0,0) direction 3.5 then tn 0 , un , u, v such that tn2 t2 G x tn w n un 2 Since G is second-order proto-differentiable at x relative to y in the direction w, r , with tn above, there are u , v u, v such that n n y tn r tn2 t2 G x tn w n un 2 According to the metric subregularity of g , there exist , such that for every B x, y , B x, y , gr G gr G , d u , v , u , v , g gr G gr G d , g u , v , u , v X Y u1 , v1 , u2 , v2 X Y 1 1 2 X X 1 2 For n large enough, we have tn2 tn2 tn2 tn2 x tn w un , y tn r , x tn w un , y tn r BX Y 2 2 x, y , B x , y , X Y Thus, it follows from (3.5) that there exists xn , yn , xn , yn gr G gr G with xn xn for all n such that Trang 209 (6) Science & Technology Development, Vol 5, No.T20- 2017 x, y, x, y t w, r, w, r t2 u , v , u , v x , y , x , y t2 n n n n n n n n n n n un un , which implies tn2 tn2 y t r v y y t r yn n n n n 2 y tn r tn2 t2 yn y tn r n yn 2 x, y, x, y tn w, r , w, r tn2 un , , un , xn , yn , xn , yn tn2 un un and x tn w tn2 t2 un xn x, y, x, y tn w, r , w, r n un , , un , xn , yn , xn , yn 2 t2 n un un Thus, y n yn y y t n r r 1/ 2 t v n n 2 un un (7) and xn x tn w un un un 1/ tn2 Setting : y n yn y y t n r r 1/ t n ,u n : xn x tn w , 1/ tn2 then v v, un u (let n in (7) and (8) and y tn r tn2 t2 yn yn G xn G xn G x tn w n un , 2 where y : y y and r : r r Hence, v v D2G( x, y, w, r )(u) Taking w, r , r 0, 0, , the Proposition 3.5 reduces to the following result Trang 210 (8) TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ T5- 2017 Corollary 3.1 Let G, G : X 2Y , x, y gr G , x, y gr G Suppose that G is proto-differentiable at x relative to y and the map g : X Y X defined as in Proposition 3.5 is metric subregular at x, y, x, y, X with respect to gr G gr G Then, DG x, y (u) DG x, y (u) D G G x, y y (u ) Corollary 3.1 can be used to replace the semi-differentiability of G in Proposition 2.1 in [11] Our condition on the metric subregularity in the Proposition 3.5 is very different from the semi-differentiability condition However, the following example shows a case where the Corollary 3.1 works, while the Proposition 3.2 does not be defined by Example 3.1 Let G, G : 1 , x n , G ( x) G ( x) : n x , otherwise Then, we have 1 , x n , G G ( x) : n 2 x , otherwise By calculating, one gets DG 0,0 (u) Db G 0,0 (u) u , Dl G 0,0 (0) , which implies that G is proto-differentiable at relative to 0, but it is not semi-differentiable at relative to Thus, Proposition 3.2 cannot be employed in this example However, the metric subregularity of Corollary 3.1 is fulfilled , we need to show that there exists such that for all Indeed, let u, v ', u, v ' B gr G gr G , d u, v ', u, v ' , g gr G gr G d 0, g u, v ', u, v ' 0,0 , B 0,0 , 1 Since u, v ', u, v ' gr G gr G , we get u v ' and u v ' Thus, it is enough to find such that inf x , y , y GG ( x ) On the other hand, we have inf x , y , y GG ( x ) Taking x u x v ' y u x v ' y u u (9) u x v ' y u x v ' y 2inf u x u x x uu , then we get Trang 211 Science & Technology Development, Vol 5, No.T20- 2017 u u u u 2inf u x u x u u x 2 u u Thus, (9) is true for every Hence, by Corollary 3.1, we get DG(0, 0)(u) DG(0, 0)(u) D G G (0, 0)(u) CONCLUSION In the paper, we propose remarks for some results in [1] Then, by virtue of the proto-differentiability, a weaker hypothesis than the semi-differentiability introduced in [1], we obtain a new result on second- order sensitivity analysis of generalized perturbation maps Acknowledgment: The author acknowledges financial support from University of Science, VNU-HCM (under grant number T2016-01) Đạo hàm contingent cấp hai ánh xạ nhiễu suy rộng Nguyễn Lê Hoàng Anh Trường Đại học Khoa học Tự nhiên, ĐHQG -HCM TĨM TẮT Trong báo này, chúng tơi đưa số nhận xét kết [1] Sau đó, chúng tơi phát triển kết liên 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superfluous As an application of Proposition 3.1 to sensitivity analysis of generalized perturbation maps, Theorem in [1] should be stated by Theorem... x, y)) : T b ( gr ( F ), ( x, y)) , resp) (ii) The second-order contingent derivative (the second-order adjacent derivative) of F at ( x, y) in the direction ( w, r ) is a set-valued map D2 F