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Xu et al Journal of Inequalities and Applications (2017) 2017:7 DOI 10.1186/s13660-016-1275-x RESEARCH Open Access Second-order lower radial tangent derivatives and applications to set-valued optimization Bihang Xu1 , Zhenhua Peng2,3 and Yihong Xu3* * Correspondence: xuyihong@ncu.edu.cn Department of Mathematics, Nanchang University, Nanchang, 330031, China Full list of author information is available at the end of the article Abstract We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated MSC: 46G05; 90C29; 90C46 Keywords: Henig efficiency; radial tangent derivative; set-valued optimization; optimality condition Introduction In recent years, first-order optimality conditions in the set-valued optimization have attracted a great deal of attention, and various derivative-like notions have been utilized to express these optimality conditions For example, Gong et al [] introduced the concept of radial tangent cone and presented several kinds of necessary and sufficient conditions for some proper efficiencies, Kasimbeyli [] gave necessary and sufficient optimality conditions based on the concept of the radial epiderivatives At the same time, second-order optimality conditions and higher-order optimality conditions for vector optimization problems have been extensively studied in the literature (see [–]) Jahn et al [] proposed second-order epiderivatives for set-valued maps, and by using these concepts they gave second-order necessary optimality conditions and a sufficient optimality condition in set optimization Khan and Isac [] proposed the concept of a second-order composed contingent derivative for set-valued maps, using which they established second-order optimality conditions in set-valued optimization With a second-order composed contingent derivative, Zhu et al [] established second-order Karush-Kuhn-Tucker necessary and sufficient optimality conditions for a set-valued optimization problem However, in [, , –, ], in the expressions of first-order and higher-order optimality conditions, the tan© The Author(s) 2017 This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 gent derivatives of the objective function and the constraint function are not separated, and thus the properties of the derivatives of the objective function are not easily obtained from those of the constraint function On the other hand, some efficient points exhibit certain abnormal properties To eliminate such anomalous efficient points, various concepts of proper efficiency have been introduced [–] Henig [] introduced the concept of Henig efficiency, which is very important for the study of set-valued optimization [, , , ] In this paper, we introduce a new class of lower radial tangent cones and two new kinds of second-order tangent sets, using which we introduce four new kinds of second-order tangent derivatives We discuss the properties of these second-order tangent derivatives, using which we establish second-order necessary optimality conditions for a point pair to be a Henig efficient element of a set-valued optimization problem Basic concepts Throughout the paper, let X, Y , and Z be three real normed linear spaces, X , Y , and Z denote the original points of X, Y , and Z, respectively Let M be a nonempty subset of Y As usual, we denote the interior, closure, and cone hull of M by int M, cl M, and cone M, respectively The cone hull of M is defined by cone M = {λm : λ ≥ , m ∈ M} Let C and D be closed convex pointed cones in Y and Z, respectively A nonempty convex subset B ⊂ C is called a base of C if ∈/ cl B and C = cone B Denote the closed unit ball of Y by U Suppose that C has a base B Let δ := inf{ b : b ∈ B} and Cε (B) := cone(εU + B) for all < ε < δ It is clear that δ > and Cε (B) is a pointed convex cone for all < ε < δ (see []) Let F : X → Y be a set-valued map The domain, graph, and epigraph of F are defined respectively by dom F := x ∈ X : F(x) = ∅ , graph F := (x, y) ∈ X × Y : y ∈ F(x) , epi F := (x, y) ∈ X × Y : y ∈ F(x) + C Definition . (See []) Let A be a nonempty subset of X, and let xˆ ∈ cl A The radial tangent cone of A at xˆ , denoted by R(A, xˆ ), is given by R(A, xˆ ) := u ∈ X : ∃tn > and xn ∈ A such that tn (xn – xˆ ) → u Remark . Equation (.) is equivalent to R(A, xˆ ) = {u ∈ X : ∃λn > and un → u such that xˆ + λn un ∈ A, ∀n ∈ N}, where N denotes the set of positive integers (.) Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 Definition . (See []) Let A be a nonempty subset of X, and let xˆ ∈ cl A The contingent cone of A at xˆ , denoted by T(A, xˆ ), is given by T(A, xˆ ) := u ∈ X : ∃tn → + and un → u such that xˆ + tn un ∈ A, ∀n ∈ N (.) Remark . (See []) Equation (.) is equivalent to T(A, xˆ ) := u ∈ X : ∃λn → +∞ and xn ∈ A such that xn → xˆ and λn (xn – xˆ ) → u Definition . (See []) Let A be a nonempty subset of X, and let xˆ ∈ cl A The secondorder contingent set of A at xˆ in the direction w, denoted by T (A, xˆ , w), is given by T (A, xˆ , w) := v ∈ X : ∃tn → + and → v such that xˆ + tn w + tn ∈ A Definition . (See [, ]) Let F : X → Y be a set-valued map, (ˆx, yˆ ) ∈ graph F, and ˆ vˆ ) ∈ X × Y The second-order composed contingent derivative of F at (ˆx, yˆ ) in the di(u, ˆ vˆ ) is the set-valued map D F(ˆx, yˆ , u, ˆ vˆ ) : X → Y defined by rection (u, ˆ vˆ ) = T T graph F, (ˆx, yˆ ) , (u, ˆ vˆ ) graph D F(ˆx, yˆ , u, ˆ vˆ ) ∈ Definition . (See []) Let F : X → Y be a set-valued map, (ˆx, yˆ ) ∈ graph F, and (u, ˆ vˆ ) is the X × Y The second-order contingent derivative of F at (ˆx, yˆ ) in the direction (u, Y ˆ vˆ ) : X → defined by set-valued map D F(ˆx, yˆ , u, ˆ vˆ ) ˆ vˆ )(x) = y ∈ Y : (x, y) ∈ T graph F, (ˆx, yˆ ), (u, D F(ˆx, yˆ , u, In the following, we introduce a new class of lower radial tangent cones and two new kinds of second-order tangent sets Definition . Let Q be a nonempty subset of X × Y , and let (ˆx, yˆ ) ∈ cl Q The lower radial tangent cone of Q at (ˆx, yˆ ) is defined by Rl Q, (ˆx, yˆ ) := (u, v) ∈ X × Y : ∀tn > , ∀un → u, ∃vn → v such that (ˆx + tn un , yˆ + tn ) ∈ Q Definition . Let Q be a nonempty subset of X × Y , and let (ˆx, yˆ ) ∈ cl Q The secondˆ vˆ ), denoted by Rl (Q, (ˆx, yˆ ), order lower radial tangent set of Q at (ˆx, yˆ ) in the direction (u, ˆ vˆ )), is given by (u, ˆ vˆ ) := (u, v) ∈ X × Y : ∀tn > , ∀un → u, ∃vn → v Rl Q, (ˆx, yˆ ), (u, such that xˆ + tn uˆ + tn un , yˆ + tn vˆ + tn ∈ Q Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 Definition . Let A be a nonempty subset of X, and let xˆ ∈ cl A The second-order radial tangent set of A at xˆ in the direction w, denoted by R (A, xˆ , w), is given by R (A, xˆ , w) := v ∈ X : ∃tn > and → v such that xˆ + tn w + tn ∈ A Remark . Let ∅ = Q ⊂ X × Y , (ˆx, yˆ ) ∈ cl Q Then (i) Rl (Q, (ˆx, yˆ )) ⊂ T(Q, (ˆx, yˆ )) ⊂ R(Q, (ˆx, yˆ )); ˆ vˆ )) ⊂ T (Q, (ˆx, yˆ ), (u, ˆ vˆ )) ⊂ R (Q, (ˆx, yˆ ), (u, ˆ vˆ )) (ii) Rl (Q, (ˆx, yˆ ), (u, However, none of the inverse inclusions is necessarily true, as is shown in the following example Example . Let R be the set of real numbers, X = Y = R, Q = {(– n , n ) : n = , , } ∪ ˆ vˆ ) = (, ) A direct calculation gives {(x, y) : x ≥ , y ≥ } ∪ {(–, –)}, and (ˆx, yˆ ) = (u, Rl (Q, (, ), (, )) = {(x, y) : x > , y ≥ }, T (Q, (, ), (, )) = {(x, y) : x ≥ , y ≥ } ∪ {(x, ) : x < }, and R (Q, (, ), (, )) = {(x, y) : x ≥ , y ≥ } ∪ {(x, ) : x < } ∪ {(x, x) : x < } ∪ ∞ n= {λ(– n , n ) : λ > } The second-order lower radial tangent derivative In this section, by virtue of the radial tangent cone, the second-order radial tangent set, the lower radial tangent cone, and the second-order lower radial tangent set, we introduce the concepts of the second-order radial composed tangent derivative, the second-order radial tangent derivative, the second-order lower radial composed tangent derivative, and the second-order lower radial tangent derivative for a set-valued map Furthermore, we discuss some important properties of the second-order lower radial composed tangent derivative and the second-order lower radial tangent derivative ˆ vˆ ) ∈ X × Y Definition . Let F : X → Y be a set-valued map, (ˆx, yˆ ) ∈ graph F, and (u, ˆ vˆ ) The second-order radial composed tangent derivative of F at (ˆx, yˆ ) in the direction (u, Y ˆ vˆ ) : X → defined by is the set-valued map R F(ˆx, yˆ , u, ˆ vˆ ) = R R epi F, (ˆx, yˆ ) , (u, ˆ vˆ ) graph R F(ˆx, yˆ , u, ˆ vˆ )) = ∅, then F is said to be second-order radial composed derivIf R(R(epi F, (ˆx, yˆ )), (u, ˆ vˆ ) or that the second-order radial composed tangent able at (ˆx, yˆ ) in the direction (u, ˆ vˆ ) exists derivative of F at (ˆx, yˆ ) in the direction (u, ˆ vˆ ) ∈ X × Y Definition . Let F : X → Y be a set-valued map, (ˆx, yˆ ) ∈ graph F, and (u, ˆ vˆ ) is the setThe second-order radial tangent derivative of F at (ˆx, yˆ ) in the direction (u, ˆ vˆ ) : X → Y defined by valued map R F(ˆx, yˆ , u, ˆ vˆ ) ˆ vˆ ) = R epi F, (ˆx, yˆ ), (u, graph R F(ˆx, yˆ , u, ˆ vˆ )) = ∅, then F is called second-order radial derivable at (ˆx, yˆ ) in the If R (epi F, (ˆx, yˆ ), (u, ˆ vˆ ) or that the second-order radial tangent derivative of F at (ˆx, yˆ ) in the direcdirection (u, ˆ vˆ ) exists tion (u, Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 ˆ vˆ ) ∈ X × Y Definition . Let F : X → Y be a set-valued map, (ˆx, yˆ ) ∈ graph F, and (u, The second-order lower radial composed tangent derivative of F at (ˆx, yˆ ) in the direction ˆ vˆ ) : X → Y defined by ˆ vˆ ) is the set-valued map Rl F(ˆx, yˆ , u, (u, ˆ vˆ ) ˆ vˆ ) = Rl Rl epi F, (ˆx, yˆ ) , (u, graph Rl F(ˆx, yˆ , u, ˆ vˆ )) = ∅, then F is said to be second-order lower radial composed If Rl (Rl (epi F, (ˆx, yˆ )), (u, ˆ vˆ ) or that the second-order lower radial composed derivable at (ˆx, yˆ ) in the direction (u, ˆ vˆ ) exists tangent derivative of F at (ˆx, yˆ ) in the direction (u, ˆ vˆ ) ∈ X × Y Definition . Let F : X → Y be a set-valued map, (ˆx, yˆ ) ∈ graph F, and (u, ˆ vˆ ) is the The second-order lower radial tangent derivative of F at (ˆx, yˆ ) in the direction (u, Y ˆ vˆ ) : X → defined by set-valued map Rl F(ˆx, yˆ , u, ˆ vˆ ) ˆ vˆ ) = Rl epi F, (ˆx, yˆ ), (u, graph Rl F(ˆx, yˆ , u, ˆ vˆ )) = ∅, then F is called second-order lower radial derivable at (ˆx, yˆ ) If Rl (epi F, (ˆx, yˆ ), (u, ˆ vˆ ) or that the second-order lower radial tangent derivative of F at (ˆx, yˆ ) in the direction (u, ˆ vˆ ) exists in the direction (u, Proposition . Suppose that E ⊂ X and the second-order lower radial composed tangent ˆ vˆ ) exists Then derivative of F : X → Y at (ˆx, yˆ ) ∈ graph F in the direction (u, ˆ vˆ ) R R(E, xˆ ), uˆ Rl F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ ˆ vˆ )(R(R(E, xˆ ), u)) ˆ Then there exists u ∈ R(R(E, xˆ ), u) ˆ such that Proof Let v ∈ Rl F(ˆx, yˆ , u, ˆ vˆ )(u) v ∈ Rl F(ˆx, yˆ , u, Thus, ˆ vˆ ) ˆ vˆ ) = Rl Rl epi F, (ˆx, yˆ ) , (u, (u, v) ∈ graph Rl F(ˆx, yˆ , u, (.) ˆ it follows that there exist sequences tn > and un → u such that From u ∈ R(R(E, xˆ ), u) uˆ + tn un ∈ R(E, xˆ ) Therefore, there exist sequences tnk > and ukn → uˆ + tn un such that xˆ + tnk ukn ∈ E For such tn and un , it follows from (.) that there exists a sequence → v such that (uˆ + tn un , vˆ + tn ) ∈ Rl epi F, (ˆx, yˆ ) Then, for the same tnk and ukn , there exists a sequence vkn → vˆ + tn such that xˆ + tnk ukn , yˆ + tnk vkn ∈ epi F, Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 and, consequently, yˆ + tnk vkn ∈ F xˆ + tnk ukn + C Thus, vkn ∈ F xˆ + tnk ukn + C – yˆ , tnk and, consequently, vkn ∈ cone F(E) + C – yˆ Since vkn → vˆ + tn as k → ∞, we obtain vˆ + tn ∈ clcone F(E) + C – yˆ Thus, ∈ clcone F(E) + C – yˆ – vˆ , tn and, consequently, ∈ cone clcone F(E) + C – yˆ – vˆ Taking n → ∞, we get v ∈ clcone clcone F(E) + C – yˆ – vˆ So, ˆ vˆ ) R R(E, xˆ ), uˆ Rl F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ Proposition . Suppose that E ⊂ X and the second-order lower radial tangent derivative ˆ vˆ ) exists Then of F : X → Y at (ˆx, yˆ ) ∈ graph F in the direction (u, ˆ vˆ ) R (E, xˆ , u) ˆ ⊂ clcone cone F(E) + C – yˆ – vˆ Rl F(ˆx, yˆ , u, ˆ vˆ )(R (E, xˆ , u)) ˆ Then there exists u ∈ R (E, xˆ , u) ˆ such that Proof Let v ∈ Rl F(ˆx, yˆ , u, ˆ vˆ )(u) v ∈ Rl F(ˆx, yˆ , u, Thus, ˆ vˆ ) ˆ vˆ ) = Rl epi F, (ˆx, yˆ ), (u, (u, v) ∈ graph Rl F(ˆx, yˆ , u, (.) Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 ˆ it follows that there exist sequences tn > and un → u such that From u ∈ R (E, xˆ , u) xˆ + tn uˆ + tn un ∈ E For such tn and un , it follows from (.) that there exists a sequence → v such that xˆ + tn uˆ + tn un , yˆ + tn vˆ + tn ∈ epi F Then yˆ + tn vˆ + tn ∈ F xˆ + tn uˆ + tn un + C, and, consequently, vˆ + tn ∈ F xˆ + tn uˆ + tn un + C – yˆ tn Thus, vˆ + tn ∈ cone F(E) + C – yˆ Hence, ∈ cone F(E) + C – yˆ – vˆ tn Therefore, ∈ cone cone F(E) + C – yˆ – vˆ Taking n → ∞, we get v ∈ clcone cone F(E) + C – yˆ – vˆ So, ˆ vˆ ) R (E, xˆ , u) ˆ ⊂ clcone cone F(E) + C – yˆ – vˆ Rl F(ˆx, yˆ , u, ˆ vˆ ) or R F(ˆx, yˆ , u, ˆ vˆ ) for Rl F(ˆx, yˆ , u, ˆ vˆ ) in ProposiRemark . If we substitute D F(ˆx, yˆ , u, tion ., then none of the inclusions ˆ vˆ ) R R(E, xˆ ), uˆ D F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ ˆ vˆ ) R R(E, xˆ ), uˆ R F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ and Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 ˆ vˆ ) or R F(ˆx, yˆ , u, ˆ vˆ ) for Rl F(ˆx, yˆ , u, ˆ vˆ ) in is necessarily true If we substitute D F(ˆx, yˆ , u, Proposition ., then none of the inclusions ˆ vˆ ) R (E, xˆ , u) ˆ ⊂ clcone cone F(E) + C – yˆ – vˆ D F(ˆx, yˆ , u, and ˆ vˆ ) R (E, xˆ , u) ˆ ⊂ clcone cone F(E) + C – yˆ – vˆ R F(ˆx, yˆ , u, is necessarily true, as is shown in the following example Example . Let R be the set of real numbers, X = Y = R, C = {t : t ≥ }, and E = {x : x ≥ } Define the set-valued map F : X → Y by F(x) = {y : y ≥ } if x ≥ , √ {y : y ≥ x} otherwise ˆ vˆ ) = (, –) A direct calculation gives (i) Let (ˆx, yˆ ) = (, ), (u, R(E, ) = R R(E, ), = [, +∞), T epi F, (, ) = R epi F, (, ) = (x, y) : x > , y ≥ ∪ (x, y) : x ≤ , y ∈ R , T T epi F, (, ) , (, –) = (x, y) : x ≤ , y ∈ R , R R epi F, (, ) , (, –) = (x, y) : x ≤ , y ∈ R ∪ (x, y) : x > , y ≥ , D F(, , , –)(x) = R, x ≤ , ∅, x > , R F(, , , –)(x) = R, x ≤ , {y : y ≥ }, x > , Rl epi F, (, ) = (x, y) : x ∈ R, y ≥ , Rl Rl epi F, (, ) , (, –) = ∅, Rl F(, , , –)(x) = ∅, x ∈ R Consequently, D F(, , , –) R R(E, ), = R F(, , , –) R R(E, ), Rl F(, , , –) R R(E, ), = ∅, = R, clcone clcone F(E) + C – yˆ – vˆ = [, +∞) Then, the inclusion of Proposition . is true However, ˆ vˆ ) R R(E, xˆ ), uˆ D F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page of 19 and ˆ vˆ ) R R(E, xˆ ), uˆ R F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ ˆ vˆ ) = (, ) A direct calculation gives (ii) Let (ˆx, yˆ ) = (, ), (u, R R(E, ), = R (E, , ) = R(E, ) = [, +∞), T T epi F, (, ) , (, ) = R R epi F, (, ) , (, ) = T epi F, (, ), (, ) = R epi F, (, ), (, ) = T epi F, (, ) = R epi F, (, ) = (x, y) : x > , y ≥ ∪ (x, y) : x ≤ , y ∈ R , D F(, , , )(x) = R F(, , , )(x) = R, x ≤ , {y : y ≥ }, x > , D F(, , , )(x) = R F(, , , )(x) = R, x ≤ , {y : y ≥ }, x > , Rl Rl epi F, (, ) , (, ) = Rl epi F, (, ), (, ) = Rl epi F, (, ) = (x, y) : x ∈ R, y ≥ , Rl F(, , , )(x) = Rl F(, , , )(x) = [, +∞), x ∈ R Consequently, D F(, , , ) R R(E, ), = R F(, , , ) R R(E, ), = D F(, , , ) R (E, , ) = R F(, , , ) R (E, , ) = R, Rl F(, , , ) R R(E, ), = Rl F(, , , ) R (E, , ) = [, +∞), clcone clcone F(E) + C – yˆ – vˆ = clcone cone F(E) + C – yˆ – vˆ = [, +∞) Then, the inclusions of Propositions . and . are true However, ˆ vˆ ) R R(E, xˆ ), uˆ D F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ , ˆ vˆ ) R R(E, xˆ ), uˆ R F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ , ˆ vˆ ) R (E, xˆ , u) ˆ ⊂ clcone cone F(E) + C – yˆ – vˆ , D F(ˆx, yˆ , u, and ˆ vˆ ) R (E, xˆ , u) ˆ ⊂ clcone cone F(E) + C – yˆ – vˆ R F(ˆx, yˆ , u, Second-order necessary optimality conditions Let F : X → Y , G : X → Z , and (F, G) : X → Y ×Z be defined by (F, G)(x) = F(x) × G(x) Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 10 of 19 Consider the following optimization problem with set-valued maps: (VP) F(x), s.t G(x) ∩ (–D) = ∅, x ∈ X ˆ that is, Eˆ = {x ∈ X : G(x) ∩ (–D) = ∅} The feasible set of (VP) is denoted by E, ˆ yˆ ∈ F(ˆx) A pair (ˆx, yˆ ) is called a Henig efficient Definition . (See [, , ]) Let xˆ ∈ E, element of (VP) if there exists ε ∈ (, δ) such that ˆ – yˆ ∩ – intcone(εU + B) = ∅, F(E) ˆ = where δ := inf{ b : b ∈ B}, F(E) x∈Eˆ F(x), and U is the closed unit ball of Y Definition . (See []) The interior tangent cone IT(S, y¯ ) of S at y¯ is the set of all y ∈ Y such that for any tn → + and yn → y, we have y¯ + tn yn ∈ S Remark . (See []) If S ⊂ Y is convex, y¯ ∈ S, and int S = ∅, then IT(S, y¯ ) = IT(int S, y¯ ) = intcone(S – y¯ ) Theorem . Suppose that (ˆx, yˆ ) is a Henig efficient element of (VP), zˆ ∈ G(ˆx) ∩ (–D), ˆ vˆ , w) ˆ ∈ X × (–C) × (–D), F is second-order lower radial composed derivable at (ˆx, yˆ ) (u, ˆ vˆ ), and G is second-order radial composed derivable at (ˆx, zˆ ) in the diin the direction (u, ˆ w) ˆ Then there exists εˆ ∈ (, δ) such that rection (u, ˆ vˆ )(x), R G(ˆx, zˆ , u, ˆ w)(x) ˆ Rl F(ˆx, yˆ , u, ∩ – intcone(ˆε U + B) × (– int D) = ∅ (.) ˆ vˆ ) ∩ dom R G(ˆx, zˆ , u, ˆ w) ˆ for all x ∈ dom Rl F(ˆx, yˆ , u, Proof Since (ˆx, yˆ ) is a Henig efficient element of (VP), there exists a number ε ∈ (, δ) such that ˆ – yˆ ∩ – intcone(ε U + B) = ∅ F(E) (.) On the contrary, suppose that (.) does not hold Then there exist x¯ ∈ dom Rl F(ˆx, yˆ , ˆ vˆ ) ∩ dom R G(ˆx, zˆ , u, ˆ w), ˆ y¯ ∈ Rl F(ˆx, yˆ , u, ˆ vˆ )(¯x), and z¯ ∈ R G(ˆx, zˆ , u, ˆ w)(¯ ˆ x) such that u, y¯ ∈ – intcone(ε U + B) (.) z¯ ∈ – int D (.) and ˆ w)(¯ ˆ x) it follows that From z¯ ∈ R G(ˆx, zˆ , u, ˆ w) ˆ = R R epi G, (ˆx, zˆ ) , (u, ˆ w) ˆ (¯x, z¯ ) ∈ graph R G(ˆx, zˆ , u, Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 11 of 19 Hence, there exist tn > and (un , wn ) ∈ R(epi G, (ˆx, zˆ )) such that ˆ w) ˆ → (¯x, z¯ ) tn (un , wn ) – (u, (.) From (.) it follows that there exists N ∈ N such that ˆ ∈ – int D, tn (wn – w) ∀n > N Since – int D is a cone, we obtain ˆ ∈ – int D, wn – w ∀n > N ˆ ∈ –D and –D is a convex cone, it follows that Since w wn ∈ – int D – D = – int D, ∀n > N (.) Since (un , wn ) ∈ R(epi G, (ˆx, zˆ )), there exist sequences tnk > and (xkn , znk ) ∈ epi G such that tnk xkn , znk – (ˆx, zˆ ) → (un , wn ), k → +∞ (.) It follows from (.) that there exists K (n) ∈ N such that tnk znk – zˆ ∈ – int D, ∀n > N , ∀k > K (n) Since – int D is a cone, we obtain znk – zˆ ∈ – int D, ∀n > N , ∀k > K (n) Since zˆ ∈ –D and –D is a convex cone, it follows that znk ∈ – int D – D = – int D, ∀n > N , ∀k > K (n) Since (xkn , znk ) ∈ epi G, we obtain znk ∈ G(xkn ) + D Hence, there exists z¯ nk ∈ G(xkn ) such that znk ∈ z¯ nk + D Consequently, z¯ nk ∈ znk – D ⊂ – int D – D = – int D ˆ It follows from (.) that tnk (xkn – xˆ ) → un as Thus, G(xkn ) ∩ (–D) = ∅, that is, xkn ∈ E ˆ xˆ ) It follows from (.) that tn (un – u) ˆ → x¯ , and hence, k → ∞, and hence, un ∈ R(E, ˆ ˆ By Proposition ., since y¯ ∈ Rl F(ˆx, yˆ , u, ˆ vˆ )(¯x), we conclude that x¯ ∈ R(R(E, xˆ ), u) ˆ xˆ ), uˆ ˆ vˆ ) R R(E, y¯ ∈ Rl F(ˆx, yˆ , u, ˆ + C – yˆ – vˆ ⊂ clcone clcone F(E) From (.) it follows that ˆ + C – yˆ – vˆ ∩ – intcone(ε U + B) = ∅ clcone clcone F(E) Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 12 of 19 Since – intcone(ε U + B) is open, we obtain ˆ + C – yˆ – vˆ ∩ – intcone(ε U + B) = ∅ cone clcone F(E) Since cone(ε U + B) is a pointed cone, it follows that ˆ + C – yˆ – vˆ ∩ – intcone(ε U + B) = ∅, clcone F(E) and thus, ˆ + C – yˆ ∩ vˆ – intcone(ε U + B) = ∅ clcone F(E) It follows from vˆ ∈ –C ⊂ – cone(ε U + B) that vˆ – int cone(ε U + B) ⊂ – cone(ε U + B) – int cone(ε U + B) ⊂ – intcone(ε U + B) Consequently, ˆ + C – yˆ ∩ – intcone(ε U + B) = ∅ clcone F(E) In the similar way, we conclude that ˆ + C – yˆ ∩ – intcone(ε U + B) = ∅ F(E) Since C ⊂ cone(ε U + B) and cone(ε U + B) is a point cone, we obtain ˆ – yˆ ∩ – intcone(ε U + B) = ∅ F(E) This is a contradiction to (.) The proof is completed Corollary . Suppose that (ˆx, yˆ ) is a Henig efficient element of (VP), zˆ ∈ G(ˆx) ∩ (–D), ˆ vˆ , w) ˆ ∈ X × (–C) × (–D), F is second-order lower radial composed derivable at (ˆx, yˆ ) in (u, ˆ vˆ ), and G is second-order lower radial composed derivable at (ˆx, zˆ ) in the the direction (u, ˆ w) ˆ Then there exists a number εˆ ∈ (, δ) such that direction (u, ˆ vˆ )(x), Rl G(ˆx, zˆ , u, ˆ w)(x) ˆ Rl F(ˆx, yˆ , u, ∩ – intcone(ˆε U + B) × (– int D) = ∅ ˆ vˆ ) ∩ dom Rl G(ˆx, zˆ , u, ˆ w) ˆ for all x ∈ dom Rl F(ˆx, yˆ , u, Proof The proof follows directly from Theorem . and Remark .(ii) Corollary . Suppose that (ˆx, yˆ ) is a Henig efficient element of (VP), zˆ ∈ G(ˆx) ∩ (–D), ˆ vˆ , w) ˆ ∈ X × (–C) × (–D), C has a convex base B, F is second-order lower radial com(u, ˆ vˆ ), and G is second-order lower radial composed posed derivable at (ˆx, yˆ ) in the direction (u, ˆ w) ˆ Then there exists a number εˆ ∈ (, δ) such that derivable at (ˆx, zˆ ) in the direction (u, ˆ vˆ )(x), Rl G(ˆx, zˆ , u, ˆ w)(x) ˆ ∩ IT – intcone(ˆε U + B), –ˆv × (– int D) = ∅ Rl F(ˆx, yˆ , u, ˆ vˆ ) ∩ dom Rl G(ˆx, zˆ , u, ˆ w) ˆ for all x ∈ dom Rl F(ˆx, yˆ , u, Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 13 of 19 Proof IT – intcone(ˆεU + B), –ˆv = intcone – intcone(ˆεU + B) + vˆ ⊂ –C – intcone(ˆεU + B) ⊂ – cone(ˆεU + B) – intcone(ˆεU + B) ⊂ – intcone(ˆεU + B) We provide the following example to explain Theorem . and Corollaries . and . Example . Let R be the set of real numbers, X = Y = Z = R, C = D = {t : t ≥ }, B = {} Define the set-valued maps F : X → Y and G : X → Z by F(x) = G(x) = {y : y ≥ } if x ≥ , {y : y ≥ x } otherwise ˆ vˆ , w) ˆ = (, , ) ∈ X × (–C) × (–D), ε = A direct calculation gives Let (ˆx, yˆ ) = (, ), (u, zˆ ∈ G() ∩ (–D) = {}, Rl epi F, (, ) = Rl epi G, (, ) = (x, y) : x > , y ≥ , R epi G, (, ) = (x, y) : x ∈ R, y ≥ , Rl Rl epi F, (, ) , (, ) = Rl Rl epi G, (, ) , (, ) = (x, y) : x > , y ≥ , R R epi G, (, ) , (, ) = (x, y) : x ∈ R, y ≥ , Rl F(, , , )(x) = Rl G(, , , )(x) = R G(, , , )(x) = [, +∞), {y : y ≥ } if x > , ∅ otherwise, x ∈ R, IT – intcone(εU + B), –ˆv = – intcone(εU + B) = (–∞, ) Then, the inclusions of Theorem . and Corollaries . and . are true Theorem . Suppose that (ˆx, yˆ ) is a Henig efficient element of (VP), zˆ ∈ G(ˆx) ∩ (–D), ˆ vˆ , w) ˆ ∈ X × (–C) × (–D), F is second-order lower radial derivable at (ˆx, yˆ ) in the di(u, ˆ w) ˆ Then ˆ vˆ ), and G is second-order radial derivable at (ˆx, zˆ ) in the direction (u, rection (u, there exists a number εˆ ∈ (, δ) such that ˆ vˆ )(x), R G(ˆx, zˆ , u, ˆ w)(x) ˆ Rl F(ˆx, yˆ , u, ∩ – intcone(ˆεU + B) × (– int D) = ∅ (.) ˆ vˆ ) ∩ dom R G(ˆx, zˆ , u, ˆ w) ˆ for all x ∈ dom Rl F(ˆx, yˆ , u, Proof On the contrary, suppose that (.) does not hold Then, for any ε ∈ (, δ), there exist ˆ vˆ ) ∩ dom R G(ˆx, zˆ , u, ˆ w), ˆ y¯ ∈ Rl F(ˆx, yˆ , u, ˆ vˆ )(¯x), and z¯ ∈ R G(ˆx, zˆ , u, ˆ w)(¯ ˆ x) x¯ ∈ dom Rl F(ˆx, yˆ , u, such that y¯ ∈ – intcone(εU + B) (.) Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 14 of 19 and z¯ ∈ – int D (.) ˆ w)(¯ ˆ x) it follows that From z¯ ∈ R G(ˆx, zˆ , u, ˆ w) ˆ = R epi G, (ˆx, zˆ ), (u, ˆ w) ˆ (¯x, z¯ ) ∈ graph R G(ˆx, zˆ , u, Hence, there exist tn > , xn → x¯ , and zn → z¯ such that ˆ + tn zn ∈ epi G xˆ + tn uˆ + tn xn , zˆ + tn w Thus, ˆ + tn zn ∈ G xˆ + tn uˆ + tn xn + D zˆ + tn w (.) The set of positive integers is denoted by N From (.) and zn → z¯ it follows that there exists N ∈ N such that zn ∈ – int D, ∀n > N Since – int D and –D are convex cones, we obtain ˆ + tn zn ∈ –D – D – int D = – int D, zˆ + tn w ∀n > N (.) It follows from (.) that there exists z˜ n ∈ G(ˆx + tn uˆ + tn xn ) such that ˆ + tn zn ∈ {˜zn } + D zˆ + tn w Since (.) and D is a convex cone, we obtain ˆ + tn zn – D ⊂ – int D – D = – int D ⊂ –D z˜ n ∈ zˆ + tn w Thus, G xˆ + tn uˆ + tn xn ∩ (–D) = ∅, that is, ˆ xˆ + tn uˆ + tn xn ∈ E ˆ xˆ , u) ˆ By Proposition . and y¯ ∈ From tn > and xn → x¯ it follows that x¯ ∈ R (E, ˆ vˆ )(¯x) we obtain Rl F(ˆx, yˆ , u, ˆ xˆ , u) ˆ + C – yˆ – vˆ ˆ vˆ ) R (E, ˆ ⊂ clcone cone F(E) y¯ ∈ Rl F(ˆx, yˆ , u, Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 15 of 19 It follows from (.) that ˆ + C – yˆ – vˆ ∩ – intcone(εU + B) = ∅ clcone cone F(E) Since – intcone(εU + B) is open, we obtain ˆˆ + C – yˆ – vˆ ∩ – intcone(εU + B) = ∅ cone cone F(E) Since cone(εU + B) is a pointed cone, it follows that ˆ + C – yˆ – vˆ ∩ – intcone(εU + B) = ∅, cone F(E) and thus, ˆ + C – yˆ ∩ vˆ – intcone(εU + B) = ∅ cone F(E) It follows from vˆ ∈ –C ⊂ – cone(εU + B) that vˆ – int cone(εU + B) ⊂ – cone(εU + B) – int cone(εU + B) ⊂ – intcone(εU + B) Consequently, ˆ + C – yˆ ∩ – intcone(εU + B) = ∅ cone F(E) In a similar way, we conclude that ˆ + C – yˆ ∩ – intcone(εU + B) = ∅ F(E) Since C ⊂ cone(εU + B) and cone(εU + B) is a pointed cone, we obtain ˆ – yˆ ∩ – intcone(εU + B) = ∅ F(E) This is a contradiction to the assumption that (ˆx, yˆ ) is a Henig minimizer of (VP) Corollary . Suppose that (ˆx, yˆ ) is a Henig efficient element of (VP), zˆ ∈ G(ˆx) ∩ (–D), ˆ vˆ , w) ˆ ∈ X × (–C) × (–D), F is second-order lower radial derivable at (ˆx, yˆ ) in the direction (u, ˆ vˆ ), and G is second-order lower radial derivable at (ˆx, zˆ ) in the direction (u, ˆ w) ˆ Then (u, there exists a number εˆ ∈ (, δ) such that ˆ vˆ )(x), Rl G(ˆx, zˆ , u, ˆ w)(x) ˆ ∩ Rl F(ˆx, yˆ , u, – intcone(ˆεU + B) × (– int D) = ∅ ˆ vˆ ) ∩ dom Rl G(ˆx, zˆ , u, ˆ w) ˆ for all x ∈ dom Rl F(ˆx, yˆ , u, Proof The proof follows immediately from Theorem . and Remark .(ii) Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 16 of 19 Corollary . Suppose that (ˆx, yˆ ) is a Henig efficient element of (VP), zˆ ∈ G(ˆx) ∩ (–D), ˆ vˆ , w) ˆ ∈ X × (–C) × (–D), B is a base of C, F is second-order lower radial derivable at (u, ˆ vˆ ), and G is second-order lower radial derivable at (ˆx, zˆ ) in the (ˆx, yˆ ) in the direction (u, ˆ w) ˆ Then there exists a number εˆ ∈ (, δ) such that direction (u, ˆ vˆ )(x), Rl G(ˆx, zˆ , u, ˆ w)(x) ˆ Rl F(ˆx, yˆ , u, ∩ IT – intcone(ˆεU + B), –ˆv × (– int D) = ∅ ˆ vˆ ) ∩ dom Rl G(ˆx, zˆ , u, ˆ w) ˆ for all x ∈ dom Rl F(ˆx, yˆ , u, Proof It is similar to the proof of Corollary . We give the following example to illustrate Theorem . and Corollaries . and . Example . Let R be the set of real numbers, X = Y = Z = R, C = D = {t : t ≥ }, and B = {} Define the set-valued maps F : X → Y and G : X → Z by F(x) = {y : y ≥ }, x ∈ R, G(x) = {y : y ≥ x}, x ∈ R ˆ vˆ , w) ˆ = (–, , –), and ε = A direct calculation gives Let (ˆx, yˆ ) = (, ), (u, zˆ ∈ G() ∩ (–D) = {}, Rl epi F, (, ), (–, ) = (x, y) : x ∈ R, y ≥ , R epi G, (, ), (–, –) = (x, y) : x ∈ R, y ≥ x , Rl epi G, (, ), (–, –) = (x, y) : x ∈ R, y ≥ x , Rl F(, , –, )(x) = {y : y ≥ }, x ∈ R, Rl G(, , –, –)(x) = R G(, , –, –)(x) = {y : y ≥ x}, x ∈ R, IT – intcone(εU + B), –ˆv = – intcone(εU + B) = (–∞, ) Then, the inclusions of Theorem . and Corollaries . and . are true Let us recall that the upper (inferior) limit in the sense of Painlevé-Kuratowski of a ¯ ∃yn ∈ set-valued map : X → Y is defined as lim supu→u¯ (u) := {y ∈ Y : ∃un → u, ¯ ∃yn ∈ (un ) such that (un ) such that yn → y} and lim infu→u¯ (u) := {y ∈ Y : ∀un → u, yn → y} If f : X → Y is Fréchet differentiable at xˆ ∈ X, its Fréchet derivative is denoted by f (ˆx) The profile map of F is the set-valued map F+ : X → Y defined by F+ (x) = F(x) + C, x ∈ dom F In what follows, we consider vector optimization Let f : X → Y , g : X → Z Consider the following vector optimization: (P) f (x), s.t g(x) ∈ –D, x ∈ X Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 17 of 19 Similarly to Definition . in [], we introduce the following second-order generalized lower (upper) directional derivative for vector-valued functions ˆ x ∈ X The parabolic Definition . Let f : X → Y be Fréchet differentiable at xˆ , and u, ˆ x) is second-order generalized lower directional derivative of xˆ in the direction (u, ˜ l f (ˆx, u)(x) ˆ D := lim inf f (ˆx + t uˆ + t x ) – f (ˆx) – tf (ˆx)uˆ t t>,x →x Remark . When the set-valued map F becomes to a vector-valued function f , which is ˆ we have Fréchet differentiable at xˆ , letting vˆ := f (ˆx)u, ˜ l f+ (ˆx, u)(x) ˆ vˆ )(x) = D ˆ = lim inf Rl F(ˆx, yˆ , u, f+ (ˆx + t uˆ + t x ) – f+ (ˆx) – tf (ˆx)uˆ t t>,x →x ˆ x ∈ X The parabolic Definition . Let f : X → Y be Fréchet differentiable at xˆ , and u, ˆ x) is second-order generalized upper directional derivative of xˆ in the direction (u, ˜ f (ˆx, u)(x) ˆ D := lim sup f (ˆx + t uˆ + t x ) – f (ˆx) – tf (ˆx)uˆ t t>,x →x Remark . When the set-valued map F becomes to a vector-valued function f , which is ˆ we have Fréchet differentiable at xˆ , letting vˆ := f (ˆx)u, ˜ f+ (ˆx, u)(x) ˆ vˆ )(x) = D ˆ = lim sup R F(ˆx, yˆ , u, t>,x →x f+ (ˆx + t uˆ + t x ) – f+ (ˆx) – tf (ˆx)uˆ t Corollary . Suppose that (ˆx, yˆ ) is a Henig efficient element of (P) and g(ˆx) ∈ –D Then there exists a number εˆ ∈ (, δ) such that ˜ g+ (ˆx, u)(x) ˜ l f+ (ˆx, u)(x), ˆ ˆ D ∩ D – intcone(ˆεU + B) × (– int D) = ∅ ˜ f+ (ˆx, u) ˜ g+ (ˆx, u) ˆ ∩ dom D ˆ for any x ∈ dom D l Proof The proof follows immediately from Theorem . and Remarks . and . Conclusions In this paper, we introduced some new kinds of lower radial tangent cone, second-order lower radial tangent set, and second-order radial tangent set By virtue of these concepts, second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for a set-valued map are introduced Compared with the secondˆ vˆ ) introduced in [, ], the secondorder composed contingent derivative D F(ˆx, yˆ , u, ˆ vˆ ), second-order radial composed tangent derivaorder contingent derivative D F(ˆx, yˆ , u, ˆ vˆ ), and second-order radial tangent derivative R F(ˆx, yˆ , u, ˆ vˆ ), second-order tive R F(ˆx, yˆ , u, ˆ vˆ ), and second-order lower radial lower radial composed tangent derivative Rl F(ˆx, yˆ , u, ˆ vˆ ) have nice properties: tangent derivative Rl F(ˆx, yˆ , u, ˆ vˆ ) R R(E, xˆ ), uˆ Rl F(ˆx, yˆ , u, ⊂ clcone clcone F(E) + C – yˆ – vˆ Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 18 of 19 and ˆ vˆ ) R (E, xˆ , u) ˆ ⊂ clcone cone F(E) + C – yˆ – vˆ , Rl F(ˆx, yˆ , u, which are demonstrated in Propositions . and . Just applying these properties, we established second-order necessary optimality conditions for a point pair to be a Henig efficient element of a set-valued optimization problem where the second-order tangent derivatives of the objective function and constraint function are separated Competing interests The authors declare that they have no competing interests Authors’ contributions All authors contributed to each part of this work equally, and they all read and approved the final manuscript Author details School of Information Engineering, Nanchang University, Nanchang, 330031, China School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China Department of Mathematics, Nanchang University, Nanchang, 330031, China Authors’ information Yihong Xu (1969-), Professor, Doctor, the major field of interest is the set-valued optimization Acknowledgements This research was supported by the National Natural Science Foundation of China Grant 11461044 and the Natural Science Foundation of Jiangxi Province (20151BAB201027) Received: 14 October 2016 Accepted: December 2016 References Gong, XH, Dong, HB, Wang, SY: Optimality conditions for proper efficient solutions of vector set-valued optimization J Math Anal Appl 284, 332-350 (2003) Kasimbeyli, R: Radial epiderivatives and set-valued optimization Optimization 58, 521-534 (2009) Jahn, J, Khan, AA, Zeilinger, P: Second-order optimality conditions in set optimization J Optim Theory Appl 125, 331-347 (2005) Zhu, SK, Li, SJ, Teo, KL: Second-order Karush-Kuhn-Tucker optimality conditions for set-valued optimization J Glob Optim 58, 673-692 (2014) Ning, E, Song, W, Zhang, Y: Second order sufficient optimality conditions in vector optimization J Glob Optim 54, 537-549 (2012) Xu, YH, Li, M, Peng, ZH: A note on ‘Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality’ [Positivity 18, 449-473(2014)] Positivity 20, 295-298 (2016) Khan, AA, Tammer, C: Second-order optimality conditions in set-valued optimization via asymptotic derivatives Optimization 62, 743-758 (2013) Li, SJ, Teo, KL, Yang, XQ: Higher-order optimality conditions for set-valued optimization J Optim Theory Appl 137, 533-553 (2008) Li, SJ, Zhu, SK, Teo, KL: New generalized second-order contingent epiderivatives and set-valued optimization problems J Optim Theory Appl 152, 587-604 (2012) 10 Jiménez, B, Novo, V: Second-order necessary conditions in set constrained differentiable vector optimization Math Methods Oper Res 58, 299-317 (2003) 11 Anh, NLH, Khanh, PQ: Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives J Glob Optim 56, 519-536 (2013) 12 Anh, NLH, Khanh, PQ, Tung, LT: Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization Nonlinear Anal 74, 7365-7379 (2011) 13 Li, SJ, Chen, CR: Higher order optimality conditions for Henig efficient solutions in set-valued optimization J Math Anal Appl 323, 1184-1200 (2006) 14 Wang, QL, Li, SJ: Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency Numer Funct Anal Optim 30, 849-869 (2009) 15 Khan, AA, Isac, G: Second-order optimality conditions in set-valued optimization by a new tangential derivative Acta Math Vietnam 34, 81-90 (2009) 16 Aghezzaf, B, Hachimi, M: Second-order optimality conditions in multiobjective optimization problems J Optim Theory Appl 102, 37-50 (1999) 17 Xu, YH, Peng, ZH: Higher-order sensitivity analysis in set-valued optimization under Henig efficiency J Ind Manag Optim (2016) doi:10.3934/jimo.2016019 18 Peng, ZH, Xu, YH: New second-order tangent epiderivatives and applications to set-valued optimization J Optim Theory Appl (2016) doi:10.1007/s10957-016-1011-1 19 Henig, MI: Proper efficiency with respect to cones J Optim Theory Appl 36(3), 387-407 (1982) Xu et al Journal of Inequalities and Applications (2017) 2017:7 Page 19 of 19 20 Zheng, XY: Proper efficiency in locally convex topological vector spaces J Optim Theory Appl 94(2), 469-486 (1997) 21 Borwein, JM, Zhuang, D: Super efficiency in vector optimization Trans Am Math Soc 338(1), 105-122 (1993) 22 Benson, HP: An improved definition of proper efficiency for vector maximization with respect to cones J Math Anal Appl 71, 232-241 (1979) 23 Qiu, QS, Yang, XM: Connectedness of Henig weakly efficient solution set for set-valued optimization problems J Optim Theory Appl 152, 439-449 (2012) 24 Aubin, JP, Frankowska, H: Set-Valued Analysis Birkhäuser, Boston (1990)
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