Second order optimality conditions with the envelope like effect in nonsmooth multiobjective mathematical programming i l stability and set valued directional derivatives

J. Math. Anal. Appl. 403 (2013) 695–702 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming I: l-stability and set-valued directional derivatives Phan Quoc Khanh a,∗ , Nguyen Dinh Tuan b a Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Nam b Department of Mathematics and Statistics, University of Economics of Hochiminh City, 59C Nguyen Dinh Chieu, D.3, Hochiminh City, Viet Nam article info Article history: Available online 14 February 2013 Submitted by Heinz Bauschke Keywords: Nonsmooth multiobjective programming Weak solutions Firm solutions Set-valued second-order directional derivatives Strict differentiability l-stability abstract Second-order necessary conditions and sufficient conditions, with the envelope-like effect, for optimality in nonsmooth multiobjective mathematical programming are established. We use set-valued second-order directional derivatives and impose strict differentiability for necessary conditions and l-stability for sufficient conditions, avoiding continuous differentiability. The results improve and sharpen several recent existing ones. Examples are provided to show advantages of our theorems over some known ones in the literature. In Part I, we consider l-stability and second-order set-valued directional derivatives of vector functions. Part II is devoted to second-order necessary optimality conditions and sufficient ones. © 2013 Elsevier Inc. All rights reserved. 1. Introduction and preliminaries In mathematical programming, and more generally in optimization, second-order optimality conditions occupy an important place, since they provide significant additional information to the first-order ones. To meet the diversity of practical applications, the considered optimization problems, and hence the tools and techniques of study, have been becoming more and more complicated. But we can observe, in most of related contributions in the literature, that the core of the results can be roughly stated the same as in the classical result of calculus that the second derivative of objective functions (or Lagrange functions in constrained problems) at minimizers is nonnegative. Kawasaki [14] was the first researcher to reveal that directional derivatives of Lagrange functions may be negative at minimizers, if the directional derivative of the map composed from the objective and constraints lies on a particular part of the boundary of the negative composite cone in the product of the image spaces. He called this phenomenon the envelope-like effect. Kawasaki’s results were developed by several authors in [5,7,20,21], considering always C scalar programs, like in [14]. In multiobjective programming, the first results of this type were given in [12,13] also for smooth cases. For nonsmooth (multiobjective) programming, Gutiérrez–Jiménez–Novo [11] used the set-valued parabolic and Dini second-order directional derivatives to establish second-order optimality conditions with the envelope-like effect. They considered Fréchet differentiable functions whose derivative is continuous or stable at the point of study. However, still many authors ignore the envelope-like effect when investigating second-order optimality conditions. This may lead to unaware mistakes. Furthermore, in the mentioned papers, sometimes it was not made clear ∗ Corresponding author. E-mail addresses: pqkhanh@hcmiu.edu.vn, pqkhanhus@yahoo.com (P.Q. Khanh), ndtuan73@yahoo.com (N.D. Tuan). 0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.12.076 696 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 enough when this effect occurs and when it does not. (Even in the careful study of second-order optimality conditions with the envelope-like effect in [11], there is a confusion with this phenomenon, see Remark (ii) in Part II ([17]).) This observation motivates our first aim of this paper, which is to make the envelope-like effect clear in second-order optimality conditions. On the other hand, a major approach to nonsmooth optimization is to propose and apply suitable generalized derivatives to replace the classical Fréchet and Gateaux derivatives which not exist in establishing optimality conditions. Various kinds of derivatives have been employed, each has advantages in some situations but none is universal. Recently, set-valued derivatives for a single-valued vector function have been used effectively to provide multiplier rules in nonsmooth programs, see, e.g., [4,8,9,11,15,18] (but in [4,8,9,15,18], the envelope-like effect was not attained). Our second aim in this paper is to apply the Hadamard second-order directional derivative (we proposed in [15]) together with the l-stability (developed in [1–3,10]) to obtain new second-order optimality conditions, which improve and sharpen recent existing ones. Namely, since the value of our Hadamard second-order directional derivative at a point is larger than that of both the parabolic and Dini second-order directional derivatives, our necessary optimality conditions are stronger than that in [11]. Furthermore, we relax major assumptions imposed in [11]: replacing continuous differentiability and stability by strict differentiability and l-stability, respectively (shortly, resp.). We consider the following multiobjective mathematical programming problem. Let f : Rn → Rm , g : Rn → Rp and h : Rn → Rr be given. Let C be a closed convex cone in Rm and K a convex set in Rp . The problem under our consideration is (P) f (x), s.t. g (x) ∈ −K , h(x) = 0. If K = Rn+ , the constraint g (x) ∈ −K collapses to the usual inequality constraint. The organization of the paper is as follows. In the rest of this section we recall some preliminary facts, including those concerning locally Lipschitz functions and second-order tangency. In Section 2, the notion of l-stability of scalar and vector functions and some properties are presented. Section is devoted to second-order directional derivatives and properties which will be in use. In Part II ([17]), we establish in Section necessary optimality conditions, with the envelope-like effect, for local weak solutions of (P). Section contains sufficient optimality conditions for local firm solutions. In the forthcoming paper [16], we continue to study problem (P) for the case where the involved maps are between general (infinite dimensional) Banach spaces. Second-order optimality conditions with the envelope-like effect are obtained also for local weak and firm solutions, but in terms of other generalized derivatives, with a higher level of nonsmoothness. Our notations are basically standard. N and R are the sets of natural numbers and real numbers, resp. For a normed space X , X ∗ stands for the topological dual of X ; ⟨., .⟩ is the canonical pairing. ∥.∥ is used for the norm in any normed space and d(y, S ) for the distance from a point y to a set S. Bn (x, r ) = {y ∈ Rn : ∥x − y∥ < r }; Sn = {y ∈ Rn : ∥y∥ = 1}; Sn∗ = {y ∈ (Rn )∗ : ∥y∥ = 1}; L(X , Y ) denotes the space of bounded linear mappings from X into Y , where X and Y are normed spaces. For a cone C ⊂ Rn , C ∗ = {c ∗ ∈ (Rn )∗ : ⟨c ∗ , c ⟩ ≥ 0, ∀c ∈ C } is the polar cone of C . For A ⊂ Rn , riA, intA, clA, bdA, coneA and LinA stand for the relative interior, interior, closure, boundary, cone hull of A and the linear space generated by A, resp. For t > and r ∈ N, o(t r ) designates a point depending on t such that o(t r )/t r → as t → 0+ . Let us recall some definitions and preliminary facts. A map f : Rn → X , where X is a normed space, is called strictly differentiable at x ∈ Rn if it has Fréchet derivative f ′ (x) at x and    1 (f (y + th) − f (y)) − f ′ (x)h sup   = 0.  t y→x,t →0+ h∈Sn lim For locally Lipschitz function f : Rn → Rm , the Clarke generalized Jacobian of f at x is defined by ∂ f (x) = conv{limf ′ (xk ) : xk ∈ Ω , xk → x}, where f is differentiable in Ω , which is dense by the Rademacher theorem. We collect some basic properties of the Clarke generalized Jacobian in the following. Proposition 1.1 ([6]). Let f : Rn → Rm be locally Lipschitz at x. Then, (i) ∂ f (x) is a nonempty, convex, compact subset of L(Rn , Rm ); (ii) ∂ f (x) is a singleton if and only if f is strictly differentiable at x: ∂ f (x) = {f ′ (x)}; (iii) (robustness) ∂ f (x) = {limk→∞ vk : vk ∈ ∂ f (xk ), xk → x}, in other words (as ∂ f (x) is compact), the map ∂ f (.) is upper semicontinuous at x; (iv) (Lebourg′ s mean value theorem) if f is locally Lipschitz in a convex neighborhood U of x and a, b ∈ U, then f (b) − f (a) ∈ conv(∂ f ([a, b])(b − a)) and when m = 1, there is some point c ∈ (a, b) such that f (b) − f (a) ∈ ∂ f (c )(b − a). Now we recall the notions of tangent cones and second-order tangent sets that we will use later. P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 697 Definition 1.2. Let x0 , u ∈ Rn and M ⊂ Rn . (a) The contingent cone of M at x0 is T (M , x0 ) = {v ∈ Rn : ∃tk → 0+ , ∃vk → v, ∀k ∈ N, x0 + tk vk ∈ M }. (b) The interior tangent cone of M at x0 is IT (M , x0 ) = {v ∈ Rn : ∀tk → 0+ , ∀vk → v, ∀k large enough, x0 + tk vk ∈ M }. (c) The second-order contingent set of M at (x0 , u) is  T (M , x0 , u) =  w ∈ Rn : ∃tk → 0+ , ∃wk → w, ∀k ∈ N, x0 + tk u + tk2 wk ∈ M . (d) The asymptotic second-order tangent cone of M at (x0 , u) is   w ∈ Rn : ∃(tk , rk ) → (0+ , 0+ ) : tk /rk → 0, ∃wk → w, ∀k ∈ N, x0 + tk u + tk rk wk ∈ M . T ′′ (M , x0 , u) = (e) The second-order adjacent set of M at (x0 , u) is A ( M , x , u) =   w ∈ Rn : ∀tk → 0+ , ∃wk → w, ∀k ∈ N, x0 + tk u + tk2 wk ∈ M . (f) The second-order interior tangent set of M at (x0 , u) is IT (M , x0 , u) =   w ∈ Rn : ∀tk → 0+ , ∀wk → w, ∀k large enough, x0 + tk u + tk2 wk ∈ M . The following proposition summarizes some basic properties of the above second-order tangent sets (we not give references for well-known facts). Proposition 1.3. Let M ⊂ Rn , x0 ∈ Rn and u ∈ Rn . Then, (i) IT (M , x0 , u) ⊂ A2 (M , x0 , u) ⊂ T (M , x0 , u) ⊂ clcone[cone(M − x0 ) − u]; (ii) if u ̸∈ T (M , x0 ), then T (M , x0 , u) = ∅. If, in addition, M is convex, intM ̸= ∅ and u ∈ T (M , x0 ), then (see [13,19,22]) (iii) intcone(M − x0 ) = IT (intM , x0 ); (iv) if A2 (M , x0 , u) ̸= ∅, then IT (M , x0 , u) = intA2 (M , x0 , u), clIT (M , x0 , u) = A2 (M , x0 , u); (v) if u ∈ cone(M − x0 ), then (a) IT (M , x0 , u) = intcone[cone(M − x0 ) − u]; (b) A2 (M , x0 , u) = clcone[cone(M − x0 ) − u]. 2. l-stable scalar and vector functions Recall that a function h : Rn → Rm is called stable (or calm) at x ∈ Rn if there are a neighborhood U of x and a ϑ > such that, for all y ∈ U, ∥h(y) − h(x)∥ ≤ ϑ∥y − x∥. Definition 2.1 ([1,10]). (i) The lower (resp, upper) directional derivative of a function ϕ : Rn → R at x in direction u is defined by ϕ l (x, u) = lim inf (ϕ(x + tu) − ϕ(x)) t →0+ t   resp, ϕ u (x, u) = lim sup (ϕ(x + tu) − ϕ(x)) . t →0+ t (ii) A function ϕ is called l-stable (resp, u-stable) at x if there exist a neighborhood U of x and a ϑ > such that, for all y ∈ U and u ∈ Sn , |ϕ l (y, u) − ϕ l (x, u)| ≤ ϑ∥y − x∥ (2.1) (resp, |ϕ (y, u) − ϕ (x, u)| ≤ ϑ∥y − x∥). (2.2) u u Some properties of l-stability of ϕ : Rn → R are summarized in the next proposition. 698 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 Proposition 2.2. (i) [1,3] Any l-stable function is locally Lipschitz and strictly differentiable. (ii) [10] ϕ is l-stable at x if and only if ϕ is (Fréchet) differentiable at x and there is a neighborhood U of x such that ϕ is Lipschitz on U, and there is a ϑ > such that ∥ϕ ′ (y) − ϕ ′ (x)∥ ≤ ϑ∥y − x∥ a.e. in U (in the sense of Lebesgue measure). (iii) [10] The notions of l-stability and u-stability are equivalent; moreover the same neighborhood U and constant ϑ applied in the inequality (2.1) or in the inequality (2.2) can also be applied in the other one. Due to Proposition 2.2 (iii), in the sequel we use only the lower directional derivative and l-stability, the upper notions are mentioned only if necessary. The notion of l-stability is extended to vector functions as follows. Definition 2.3 ([2]). The lower (resp, upper) directional derivative of a function Φ : Rn → Rm at x in a direction u with respect to (shortly wrt) ξ ∗ ∈ (Rm )∗ is defined by Φξl ∗ (x, u) = lim inf ⟨ξ ∗ , Φ (x + tu) − Φ (x)⟩ t →0+ t   resp, Φξu∗ (x, u) = lim sup ⟨ξ ∗ , Φ (x + tu) − Φ (x)⟩ . t →0+ t The following mean value property for continuous vector functions will be needed. Proposition 2.4 ([2]). Let Φ : Rn → Rm be continuous on an open subset U ⊂ Rn containing a segment [a, b] and ξ ∗ ∈ (Rm )∗ . Then, there are points γ1 , γ2 ∈ (a, b) such that Φξl ∗ (γ1 , b − a) ≤ ⟨ξ ∗ , Φ (b) − Φ (a)⟩ ≤ Φξl ∗ (γ2 , b − a). ∗ Definition 2.5. Let Φ : Rn → Rm and Γ = C ∗ ∩ Sm . (i) [2] Assume that C ⊂ Rm is a closed, convex and pointed cone with intC ̸= ∅. We say that Φ is l-stable at x in the sense of Bednařík–Pastor if there are a neighborhood U of x and a ϑ > such that, for all y ∈ U, u ∈ Sn , and ξ ∗ ∈ Γ , |Φξl ∗ (y, u) − Φξl ∗ (x, u)| ≤ ϑ∥y − x∥. (ii) [10] Φ is said to be l-stable at x in the sense of Ginchev if, for any ξ ∗ ∈ (Rm )∗ , the scalar function Φξ ∗ (.) := ⟨ξ ∗ , Φ (.)⟩ is l-stable at x. Of course, if a scalar or vector function f has a Fréchet derivative f ′ which is stable (i.e., calm), then f is l-stable. Note that in the previous papers (e.g., [2,10]) dealing with l-stability, this notion was defined and applied only for the case of Euclidean spaces. Here, we consider Banach spaces. It is worth noting that, in many applications, e.g., in economics, Lagrange multipliers, being elements of the dual spaces involved in the problem under consideration, are prices. Hence, the dual spaces cannot coincide with the primal ones as for the Euclidean case. Several properties of l-stable vector functions taken from [2,10], which are valid also for Banach spaces, are collected below. Proposition 2.6. (i) [10] Φ : Rn → Rm is l-stable at x in the sense of Ginchev if and only if there exist a neighborhood U of x and a ϑ > such that, for all y ∈ U, u ∈ Sn , and ξ ∗ ∈ (Rm )∗ , |Φξl ∗ (y, u) − Φξl ∗ (x, u)| ≤ ϑ∥ξ ∗ ∥∥y − x∥. (ii) [10] Φ : Rn → Rm is l-stable at x in the sense of Ginchev if and only if it is Fréchet differentiable at x and there exist an open neighborhood U of x and a ϑ > such that Φ is Lipschitz on U, and, for almost every y ∈ U, ∥Φ ′ (y) − Φ ′ (x)∥ ≤ ϑ∥y − x∥. (iii) [2,10] If function Φ : Rn → Rm is l-stable at x in the sense of Bednařík–Pastor or Ginchev, then Φ is locally Lipschitz at x and strictly differentiable at x. Now we prove that the above two definitions of l-stability for vector functions are equivalent in a sense. Proposition 2.7. If Φ : Rn → Rm is l-stable at x in the sense of Ginchev, then the inequality in the definition of l-stability of Bednařík–Pastor holds. If C is pointed and intC ̸= ∅, the two definitions are equivalent. P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 699 Proof. It follows from Proposition 2.6 (i) that Φ is l-stable at x in the sense of Ginchev if and only if there are a neighborhood ∗ U of x and a ϑ > such that, for all y ∈ U , u ∈ Sn , and ξ ∗ ∈ Sm , |Φξl ∗ (y, u) − Φξl ∗ (x, u)| ≤ ϑ∥y − x∥. ∗ Hence, this inequality holds for ξ ∗ ∈ Γ ⊂ Sm as required by Bednařík–Pastor. m ∗ Conversely, suppose C is pointed, intC ̸= ∅ and the last inequality holds for ξ ∗ ∈ Γ . Because LinΓ = every r(i R ) , for ∗ ∗ ∗ ∗ α ξ , where i ∈ {1, 2, . . . , m}, there exist ξi,1 , . . . , ξi,ri ∈ Γ and αi,1 , . . . , αi,ri ∈ R with ri = 1, . . . , m such that ei = i , j i,j j=1 e∗i ∈ (Rm )∗ is defined by ⟨e∗i , x⟩ = xi , for x = (x1 , x2 , . . . , xm ). Let M = maxi,j |αi,j | and Φi be the ith component of Φ . Observe that, since Φ is l-stable at x in the sense of Bednařík–Pastor, Φel ∗ (x, u) = Φeu∗ (x, u) = ⟨e∗i , Φ ′ (x)u⟩. For every y ∈ U , u ∈ Sn , and i ∈ {1, 2, . . . , m}, one has that i Φil (y, u) − Φil i (y, u) − Φel ∗ (x, u) i   ′ ∗ = lim inf ei , (Φ (y + tu) − Φ (y)) − Φ (x)u (x, u) = Φel ∗ i t t →0+ ri = lim inf  t →0+   αi,j ξi∗,j , (Φ (y + tu) − Φ (y)) − Φ ′ (x)u t j =1 ri ≥−  |αi,j ||Φξl ∗ (y, u) − Φξl ∗ (x, u)| ≥ −ϑ1 ∥y − x∥, i,j j =1 i,j where ϑ1 = mM ϑ . Now, by the equivalence of l-stability and u-stability, one has further Φil (y, u) − Φil (x, u) ≤ Φeu∗ (y, u) − Φeu∗ (x, u) i i   = lim sup e∗i , (Φ (y + tu) − Φ (y)) − Φ ′ (x)u t t →0+ = lim sup t →0+ ≤ ri  ri    αi,j ξi∗,j , (Φ (y + tu) − Φ (y)) − Φ ′ (x)u t j=1 |αi,j ||Φξu∗ (y, u) − Φξu∗ (x, u)| ≤ ϑ1 ∥y − x∥. i,j j =1 i,j The obtained two inequalities say that Φi is l-stable at x in the sense of Ginchev for all i = 1, . . . , m. Hence, using the technique given in the proof of Theorem 3.3 in [10], also the vector function Φ is l-stable at x in the sense of Ginchev. By virtue of this proposition, from now on we mention only one notion of l-stability, namely that of Ginchev. 3. Second-order set-valued directional derivatives Let us recall the upper limit in the sense of Painlevé–Kuratowski of a set-valued mapping Φ : Rn ⇒ Rm Limsupu→u Φ (u) = {y ∈ Rm : ∃uk → u, ∃yk ∈ Φ (uk ) such that yk → y}. In this paper, we are concerned with the following three kinds of set-valued derivatives of a single-valued vector function. Definition 3.1. Let h : Rn → Rm be Fréchet differentiable at x0 ∈ Rn and u, w ∈ Rn . (i) [15] The Hadamard second-order directional derivative of h at x0 in direction u is D2 h(x0 , u) = Limsupv→u,t →0+ h(x0 + t v) − h(x0 ) − th′ (x0 )u t /2 . (ii) [15] The Dini second-order directional derivative of h at x0 in direction u is d2 h(x0 , u) = Limsupt →0+ h(x0 + tu) − h(x0 ) − th′ (x0 )u t /2 . (iii) [11] The parabolic second-order directional derivative of h at x0 in direction (u, w) is p D2 h(x0 , u, w) = Limsupv→w,t →0+ h(x0 + tu + 21 t v) − h(x0 ) − th′ (x0 )u t /2 . 700 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 p Note that D2 h(x0 , u, w) is known in many works also as the contingent derivative of the set-valued map x → {h(x)} at (x0 , h(x0 )) in direction (u, w). It is clear that d2 h(x0 , u) ⊂ D2 h(x0 , u), p D2 h(x0 , u, w) ⊂ D2 h(x0 , u). To obtain some more relations between these derivatives we need the following. Lemma 3.2. Let h : Rn → Rm be l-stable at x0 ∈ Rn . Then, there is ϑ > such that, for all a, b near x0 , there exists γ ∈ (a, b) satisfying ∥h(b) − h(a) − h′ (x0 )(b − a)∥ ≤ ϑ∥b − a∥∥γ − x0 ∥. ∗ Proof. By the Hahn–Banach theorem, there exists ξ ∗ ∈ Sm such that ∥h(b) − h(a) − h′ (x0 )(b − a)∥ = ⟨ξ ∗ , h(b) − h(a) − h′ (x0 )(b − a)⟩. Proposition 2.4 yields γ ∈ (a, b) fulfilling ⟨ξ ∗ , h(b) − h(a)⟩ ≤ hlξ ∗ (γ , b − a). Therefore, for ϑ being the l-stability constant of h, ⟨ξ ∗ , h(b) − h(a) − h′ (x0 )(b − a)⟩ ≤ hlξ ∗ (γ , b − a) − hlξ ∗ (x0 , b − a) ≤ ϑ∥γ − x0 ∥∥b − a∥. Hence, ∥h(b) − h(a) − h′ (x0 )(b − a)∥ ≤ ϑ∥b − a∥∥γ − x0 ∥. The following relation improves Proposition 2.2 of [15]. Proposition 3.3. If h : Rn → Rm is l-stable at x0 ∈ Rn with h′ (x0 ) = 0, then, for all u ∈ Rn , d2 h(x0 , u) = D2 h(x0 , u). Proof. It suffices to check that D2 h(x0 , u) ⊂ d2 h(x0 , u) for all u ∈ Rn . Let y ∈ D2 h(x0 , u), i.e., there are tk → 0+ and uk → u such that h(x0 + tk uk ) − h(x0 ) − tk h′ (x0 )u tk2 /2 → y. We have that P := h(x0 + tk uk ) − h(x0 ) − tk h′ (x0 )u /2 h(x0 + tk uk ) − h(x0 + tk u) = . tk2 /2 tk2 − h(x0 + tk u) − h(x0 ) − tk h′ (x0 )u tk2 /2 Lemma 3.2 with a = x0 + tk u and b = x0 + tk uk yields, for all k ∈ N large enough, γk ∈ (x0 + tk u, x0 + tk uk ) such that ϑ tk ∥uk − u∥∥γk − x0 ∥ tk2 /2 ϑ tk ∥uk − u∥tk (∥uk ∥ + ∥u∥) ≤ tk2 /2 = 2ϑ∥uk − u∥(∥uk ∥ + ∥u∥) → 0, ∥P ∥ ≤ which implies that y ∈ d2 h(x0 , u). Proposition 3.4. Let h : Rn → Rm be l-stable at x0 ∈ Rn and u, w ∈ Rn . (i) If wk := (xk − x0 − tk u)/ 21 tk2 → w with tk → 0+ , then there is y ∈ Dp2 h(x0 , u, w) such that (for a subsequence) yk := h(xk ) − h(x0 ) − tk h′ (x0 )u tk2 /2 → y. (ii) Dp2 h(x0 , u, w) is nonempty and compact. (iii) If wk := (xk − x0 − tk u)/ 12 tk rk → w with (tk , rk ) → (0+ , 0+ ) and tk /rk → 0, then yk := h(xk ) − h(x0 ) − tk h′ (x0 )u tk rk /2 → h′ (x0 )w. (3.1) P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 701 Proof. (i) Suppose wk := (xk − x0 − tk u)/ 12 tk2 → w as tk → 0+ . Applying Lemma 3.2 to a = x0 and b = xk , we see that, for all k ∈ N large enough, there are γk ∈ (x0 , xk ) such that ∥h(xk ) − h(x0 ) − h′ (x0 )(xk − x0 )∥ ≤ ϑ∥xk − x0 ∥∥γk − x0 ∥ ≤ ϑ∥xk − x0 ∥2 , (3.2) which implies that 2        h(xk ) − h(x0 ) − tk h′ (x0 )u ′    − h (x0 )wk  ≤ 2ϑ u + tk wk   .  2 t /2 (3.3) k p Hence, {yk } is bounded, and therefore, there exists a subsequence converging to some y ∈ D2 h(x0 , u, w). p p p (ii) Part (i) shows that D2 h(x0 , u, w) ̸= ∅. Since D2 h(x0 , u, w) is closed, we prove that D2 h(x0 , u, w) is bounded. Let p + y ∈ D2 h(x0 , u, w). Then, yk , defined by (3.1) for some (tk , wk ) → (0 , w), tends to y. By the argument in (i), we have (3.3) p and, passing it to limit gives ∥y − h′ (x0 )w∥ ≤ 2ϑ∥u∥2 , i.e., D2 h(x0 , u, w) is bounded. (iii) Suppose wk := (xk − x0 − tk u)/ 21 tk rk → w as (tk , rk ) → (0+ , 0+ ) with tk /rk → 0. According to (i), for all k ∈ N large enough, we have (3.2) for some γk ∈ (x0 , xk ). Therefore, yk → h′ (x0 )w since    h(xk ) − h(x0 ) − tk h′ (x0 )u  tk ′   − h ( x )w k  ≤ 2ϑ  tk rk /2 rk  2   u + rk wk  → 0.   p Proposition 3.4(i), (ii) sharpens Proposition of [11]. Furthermore, part (i) says, more than D2 h(x0 , u, w) being nonempty, that any yk defined by (3.1) has a subsequence tending to some point of this derivative. Part (iii) of Proposition 3.4 extends Lemma of [11]. The following fact improves Proposition 3(ii) of [11]. Proposition 3.5. If h : Rn → Rm is strictly differentiable at x0 ∈ Rn , then, for all u, w ∈ Rn , p D2 h(x0 , u, w) = h′ (x0 )w + d2 h(x0 , u). Proof. For (tk , wk ) → (0+ , w), set yk = = h(x0 + tk u + 21 tk2 wk ) − h(x0 ) − tk h′ (x0 )u tk2 /2 h(x0 + tk u + 12 tk2 wk ) − h(x0 + tk u) tk2 /2 + h(x0 + tk u) − h(x0 ) − tk h′ (x0 )u tk2 /2 := hˆ k + yˆ k . Using Lebourg’s mean value theorem in Proposition 1.1 (iv) for h and [a, b] = [x0 + tk u, x0 + tk u + t k wk ], we have hˆ k ∈ conv(∂ h([a, b])wk ). The robustness of ∂ h implies that hˆ k → h′ (x0 )w as k → ∞. p Now, if y ∈ D2 h(x0 , u, w), then there exists (tk , wk ) → (0+ , w) such that yk → y. Hence, yˆ k = yk −hˆ k → y−h′ (x0 )w := yˆ , i.e., yˆ ∈ d2 h(x0 , u). Conversely, if we have a yˆ ∈ d2 h(x0 , u), then tk → 0+ exists such that yˆ k → yˆ , where yˆ k is defined at the beginning of p the proof. For wk ≡ w and hˆ k defined above, we have y ∈ D2 h(x0 , u, w) since yk := yˆ k + hˆ k → yˆ + h′ (x0 )w := y. We have the following direct implication of Propositions 3.3 and 3.5. Corollary 3.6. Let h : Rn → Rm and x0 , u ∈ Rn . (i) If h is strictly differentiable at x0 and h′ (x0 ) = 0, then, for w ∈ Rn , p d2 h(x0 , u) = D2 h(x0 , u, w). (ii) If h is l-stable at x0 and h′ (x0 ) = 0, then, for w ∈ Rn , p D2 h(x0 , u) = d2 h(x0 , u) = D2 h(x0 , u, w). The following example says that the condition h′ (x0 ) = in Proposition 3.3 is essential, but strict differentiability and l-stability are only sufficient conditions in Propositions 3.3 and 3.5 and Corollary 3.6. 702 P.Q. Khanh, N.D. Tuan / J. Math. Anal. Appl. 403 (2013) 695–702 Example 3.1. (a) Let h : R2 → R be defined by h(x1 , x2 ) = 21 x21 + x2 , x0 = (0, 0), u = (1, 0), and w = (w1 , w2 ) ∈ R2 . Then, h is l-stable (and so strictly differentiable) at x0 , h′ (x0 ) ̸= 0. Direct computations give d2 h(x0 , u) = {1} ̸= D2 h(x0 , u) = R, p D2 h(x0 , u, w) = {1 + w2 } = h′ (x0 )w + d2 h(x0 , u). (b) Let u = 1, x0 = 0, w ∈ R, and h(x) = x2 sin (and so, not l-stable) at x0 . However, x if x ̸= 0, h(0) = 0. 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Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. . solutions Set-valued second-order directional derivatives Strict differentiability l-stability a b s t r a c t Second-order necessary conditions and sufficient conditions, with the envelope-like effect, for. N.D. Tuan, Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming II: Optimality conditions, J. Math. Anal. Appl. Online first 2013,. optimality conditions and sufficient ones. © 2013 Elsevier Inc. All rights reserved. 1. Introduction and preliminaries In mathematical programming, and more generally in optimization, second-order