Zhang et al Journal of Inequalities and Applications (2015) 2015:361 DOI 10.1186/s13660-015-0882-2 RESEARCH Open Access A note on the optimality condition for a bilevel programming Jie Zhang1,2* , Huan Wang1 and Yue Sun1 * Correspondence: zhangjie04212001@163.com School of Mathematics, Liaoning Normal University, Huanghe Road 850, Dalian, 116029, China Research Center of Information and Control, Dalian University of Technology, Hongling Road, Dalian, 116024, China Abstract The equality type Mordukhovich coderivative rule for a solution mapping to a second-order cone constrained parametric variational inequality is derived under the constraint nondegenerate condition, which improves the result published recently The rule established is then applied to deriving a necessary and sufficient local optimality condition for a bilevel programming with a second-order cone constrained lower level problem MSC: 90C33; 90C46 Keywords: coderivative; second-order cone; parametric variational inequality; constraint nondegenerate condition; bilevel programming Introduction In this paper, we focus on the following bilevel programming (BP): f (x, y) () s.t y ∈ S(x), where f (·, ·) : n × m → is continuously differentiable and S(x) is the optimal solution set of the following problem: ψ(x, y) () s.t A(x)y + b ∈ Kp with ψ(·, ·) : n × m → being a continuously differentiable convex mapping, b ∈ p , A(·) : n → p×m , and Kp ⊆ p being the second-order cone (SOC), also called the Lorentz cone, defined by Kp := z = (z , z ) ∈ × p– : z ≥ z , where · stands for the Euclidean norm If p = , then Kp is the set of nonnegative reals + , in this case, problem () is the bilevel programming studied by [] and [] If the lower level problem of BP is replaced by its KKT condition, this is a mathematical program with a second-order cone complementarity problem among the constraints [] © 2015 Zhang et al 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 Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Since for fixed x ∈ can be rewritten as S(x) := y ∈ m n , problem () is a convex problem, the solution mapping S(·) in () : F(x, y), y – y ≥ , ∀y ∈ (x) , where F(x, y) := ∇y ψ(x, y) and (x) = y ∈ m Page of 12 : n ⇒ : A(x)y + b ∈ Kp m () is a convex-valued multifunction defined by () For fixed x, S(x) denotes the solution set of a variational inequality problem, which has been intensively studied by [–] To establish the necessary and sufficient local optimality condition for bilevel programming (), a crucial step is to compute generalized differentiation for the solution mapping S(·) defined by () The generalized differentiation in our study is Mordukhovich’s coderivative [], which plays an important role in characterizations of metric regularity and openness properties of set-valued mappings; see [] and the references therein Mordukhovich and Outrata [] has established upper estimations of the coderivatives for the solution mapping () with Kp being a closed convex set under appropriate calmness assumptions and constraint qualifications However, the equality type calculus rules of the coderivatives of a solution mapping S () are not mentioned Recently, Zhang et al [] has established equality type calculus rules of the coderivatives of a solution mapping S () under the constraint nondegenerate condition and applied the results obtained to deriving necessary and sufficient condition of the Lipschitz-like property [] of the solution mapping S () In this paper, the equality type representation of the coderivative of a solution mapping S () is established under conditions weaker than [], Theorem ., and it then is used to obtain a necessary and sufficient local optimality conditions for the bilevel programming () This is done on the basis of an exact description of the coderivative of the normal cone operator onto the second-order cone This paper is organized as follows: Section gives preliminaries needed throughout the paper In Section , the main results are established, i.e., the equality type calculus rule of the coderivatives of a solution mapping S () is established and then used to derive the optimality condition of bilevel programming () Some examples are provided Preliminaries Throughout this paper we use the following notations For an extended real-valued function ϕ : n → ∪ {±∞}, ∇ϕ(x) denotes its the gradient of ϕ at x For a continuously differentiable mapping φ : n → m , J φ(x) denotes the Jacobian of φ at x We use Bn , · and + to stand for the closed unit ball in n , the Euclidean norm and the nonnegative reals, respectively [|x|] = {tx : t ∈ }, S⊥ = {η ∈ n : η, x = , ∀x ∈ S}, Sp(S) = + (S – S) and lin(C) denote the linear space generated by vector x, the orthogonal complement of the set S ⊆ n , the linear space generated by S and the linearity subspace of the convex cone C, respectively Given a closed set ⊂ n and a point x¯ ∈ , the Mordukhovich limiting normal cone to at x¯ is defined by N (¯x) := lim sup N (x), x→¯x Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page of 12 see for instance [] and [], where the cone N (¯x) := x∗ ∈ n lim sup x→¯x x∗ , x – x¯ ≤ x – x¯ is called the regular normal cone to at x¯ with ‘lim sup’ being the outer limit of a setvalued mapping or the upper limit of a real-valued function; see [] It follows from the definition that N (¯x) ⊆ N (¯x) If the above inclusion becomes equality, we say that is normally regular at x¯ (or Clarke regular by []) According to [], Theorem ., each convex set is normally regular at all its points For set-valued maps, the definition of the coderivative was introduced by Mordukhovich in [] based on the Mordukhovich limiting normal cone Definition . Consider a mapping S : n ⇒ m and a point x¯ ∈ dom S The coderivative ¯ : m ⇒ n defined by of S at x¯ for any u¯ ∈ S(¯x) is the mapping D∗ S(¯x, u) ¯ ¯ D∗ S(¯x, u)(y) = v : (v, –y) ∈ Ngph S (¯x, u) ¯ is simplified to D∗ S(¯x) when S is single-valued at x¯ , S(¯x) = {u} ¯ The notation D∗ S(¯x, u) Similarly, and with the same provision for simplified notation, the regular coderivative ¯ : m ⇒ n is defined by D∗ S(¯x, u) ¯ ¯ D∗ S(¯x, u)(y) = v : (v, –y) ∈ Ngph S (¯x, u) Next we give the following proposition to show the description of the coderivative of some special set-valued mappings Proposition . ([], Proposition .) For any (¯x, y¯ ) ∈ gph NKp , let z¯ = x¯ + y¯ () In the case when x¯ = , y¯ = , we have ∗ ∗ D NKp (¯x, y¯ ) y ∗ ∗ = D NKp (¯x, y¯ ) y = ⎧ ⎪ ⎨ [|η|] + ⎪ ⎩ ∅, z¯ – z¯ z¯ + z¯ z¯ T where η = (, – z¯ )T () In the case when z¯ ∈ int(Kp )– , we have D∗ NKp (¯x, y¯ ) y∗ = D∗ NKp (¯x, y¯ ) y∗ = , y∗ = , ∅, y∗ = p () In the case when z¯ ∈ int Kp , we have D∗ NKp (¯x, y¯ ) y∗ = D∗ NKp (¯x, y¯ ) y∗ = {}p for any y∗ ∈ p We need the following stability notations; see [] z¯ T y∗ z¯ ∗ –y , ηT y∗ = , otherwise, Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page of 12 Definition . Consider the multifunction F : m ⇒ n (a) (Lipschitz-like property) We say F has Lipschitz-like property at (¯y, x¯ ) ∈ gph F, if there exist some κ > and some neighborhoods U of x¯ and V of y¯ such that F y ∩ U ⊂ F(y) + κ y – y Bn for all y, y ∈ V (b) (Calmness) We say F is calm at (¯y, x¯ ) ∈ gph F if there exist some k > and some neighborhoods U of x¯ and V of y¯ such that d x, F(¯y) ≤ k y – y¯ for all y ∈ V , x ∈ F(y) ∩ U We know from the definition that the calmness property is weaker than the Lipschitz-like property As shown in [], Theorem ., F has Lipschitz-like property at (¯y, x¯ ) ∈ gph F if and only if the coderivative condition D∗ F(¯y, x¯ )() = {}, () see [], Proposition . This condition is the famous Mordukhovich criterion [], Theorem . Under the calmness condition, when the constraint set is structured, the normal cones can be estimated or calculated Proposition . ([], Theorem .) Assume the multifunction M : n ⇒ n , defined by M(q) := z ∈ Z : G(z) + q ∈ K for closed sets Z ⊆ n and K ⊆ gph M Then one has n and a C mapping G : NM() (¯z) ⊆ J G(¯z)T NK G(¯z) + NZ (¯z) n → n , is calm at (, z¯ ) ∈ () We know from [], Theorem . that NM() (¯z) ⊇ J G(¯z)T NK G(¯z) + NZ (¯z) Thus if, in addition, Z is normally regular at z¯ and K is normally regular at G(¯z), then C is normally regular at z¯ and inclusion in () becomes equality We know from [], Theorem . that M defined in Proposition . is Lipschitz-like around (, z¯ ) ∈ gph M if the following constraint qualification holds: ∈ J G(¯z)T η + NZ (¯z), η ∈ NK (G(¯z)) ⇒ η = () Main results In this section, we provide conditions ensuring the equality type calculus rule of the coderivatives of a solution mapping S (), which is an improvement of [], Theorem . Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page of 12 The result obtained is used to derive a necessary and sufficient condition for the bilevel programming () We know from the definition of normal cone in convex analysis that the solution mapping S () can be rewritten as S(x) = y ∈ m : ∈ F(x, y) + N (x) (y) , where N (x) (y) denotes the normal cone of (x) at y For a parameter x¯ ∈ ing Slater constraint qualification (SCQ) is satisfied at x¯ : ∃¯y ∈ m such that x) (¯x) (y) = A(¯ T , if the follow- A(¯x)¯y + b ∈ int Kp , () then, by [], Theorem ., we can compute the normal cone N obtain N n at y ∈ (¯x) (y) (¯x) and NKp A(¯x)y + b () We need the following conditions, which are popularly used conditions in SOCP Definition . Let x¯ ∈ n , y¯ ∈ (¯x) and v¯ ∈ N (¯x) (¯y) (a) We say that the constraint nondegenerate condition (CN C ) holds true at y¯ for x¯ , if A(¯x) m + lin TKp A(¯x)¯y + b p = () (b) We say the strict complementarity (SC ) condition holds at (¯x, y¯ , v¯ ), if λ ∈ ri NKp A(¯x)¯y + b for all λ satisfying λ ∈ NKp (A(¯x)¯y + b) and A(¯x)T λ = v¯ We introduce the Lagrangian mapping L : n × m × p → m defined by L(x, y, λ) := F(x, y) + A(x)T λ () and the Lagrangian multiplier mapping (x, y) := λ ∈ p : n × m ⇒ p defined by | L(x, y, λ) = In [], Theorem ., an equality type representation of the coderivative of a solution mapping S () has been established under some constraint qualifications, we cite it as a lemma Lemma . Assume the SCQ () holds for x¯ and the multifunction P : m × p defined by P(γ , q) := (x, y, λ) ∈ n × m × p | L(x, y, λ) + γ = ∩ M(q) m × p ⇒ n × () Zhang et al Journal of Inequalities and Applications (2015) 2015:361 ¯ with λ¯ ∈ is calm at the points (, , x¯ , y¯ , λ) p n+m+p M: ⇒ is defined by M(q) := (x, y, λ) ∈ n × m × p Page of 12 (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b), where the multifunction q+ A(x)y + b ∈ gph NKp λ () Then: (a) In the case when z¯ > z¯ , where z¯ := A(¯x)¯y + b, we have for any λ¯ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b), ¯ T u + Jx A(¯x)¯y Jx L(¯x, y¯ , λ) D∗ S(¯x, y¯ ) y∗ = T w| ∈ y∗ + Jy L(¯x, y¯ , λ¯ ) u + A(¯x)T w, T w ∈ D∗ NKp A(¯x)¯y + b, λ¯ A(¯x)u () holds for all y∗ ∈ m ¯ for any (b) In the case when z¯ = z¯ , if the mapping M(·) () is calm at (, x¯ , y¯ , λ) ¯λ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b), CN C () holds at y¯ for x¯ and SC condition holds at (¯x, y¯ , –F(¯x, y¯ )) Then the equality () holds for any λ¯ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b) Under conditions weaker than the ones in (b) of Theorem . in [], we obtain the same equality type coderivative rule as follows Theorem . Assume: (a) SCQ () holds for x¯ and P(γ , q) () is calm at the points (, , x¯ , y¯ , λ¯ ) with λ¯ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b) (b) CN C () holds at y¯ for x¯ and SC condition holds at (¯x, y¯ , –F(¯x, y¯ )) Then in the case when z¯ = z¯ , the equality () holds for any λ¯ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b) Proof According to Lemma .(b), we need to show under conditions (a) and (b) that the mapping M(·) () is calm at (, x¯ , y¯ , λ¯ ) for any λ¯ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b) We know ¯ is ensured by the Lipschitz-like from Definition . that the calmness of M(·) at (, x¯ , y¯ , λ) ¯ property of M(·) at (, x¯ , y¯ , λ), which holds under the condition = J (A(x)y)T |(x,y)=(¯x,¯y) η, η ∈ D∗ NKp (A(¯x)¯y + b, λ¯ )() ⇒ η = () Indeed, notice that, by the Mordukhovich criterion (), we only need to verify ¯ D∗ M(, x¯ , y¯ , λ)() = {} () under condition () Let y∗ ∈ D∗ M(, x¯ , y¯ , λ¯ )(), by Definition ., we have y∗ ⎛ ⎞ ⎜ x¯ ⎟ ⎜ ⎟ ∈ Ngph M ⎜ ⎟ ⎝ y¯ ⎠ λ¯ () Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page of 12 Since gph M = (q, x, y, λ) ∈ p × n × m × p q+ A(x)y + b ∈ gph NKp , λ we know from Proposition . that if the condition = J(q,x,y,λ) q + ξ A(x)y+b λ ∈ Ngph NKp A(¯xλ¯)¯y+b T |(q,x,y,λ)=(,¯x,¯y,λ) ¯ ξ, ⇒ ξ = () holds, then ⎛ ⎞ ⎜ x¯ ⎟ A(x)y + b ⎜ ⎟ Ngph M ⎜ ⎟ ⊆ J(q,x,y,λ) q + ⎝ y¯ ⎠ λ ¯λ T ¯ (q,x,y,λ)=(,¯x,¯y,λ) Ngph NKp A(¯x)¯y + b () λ¯ Notice that J(q,x,y,λ) A(x)y + b q+ λ T = ¯ (q,x,y,λ)=(,¯x,¯y,λ) Ip Ip J (A(x)y)|(x,y)=(¯x,¯y) , Ip we have (), then () holds and hence by (), we have y∗ Ip ∈ Ip J (A(x)y)|(x,y)=(¯x,¯y) Ip T Ngph NKp A(¯x)¯y + b , λ¯ which, by () and Definition ., means that y∗ = Therefore () holds Next we show CN C condition implies () In the case when z¯ = z¯ = , CN C condition means that A(¯x) m = p , which is equivalent to = A(¯x)T η ⇒ η= and hence condition () holds In the case when z¯ = z¯ = , we proceed in the proof in two main steps Step Taking the orthogonal complements on both sides of (), the CN C condition can be rewritten as A(¯x) m ⊥ ∩ lin TKp A(¯x)¯y + b ⊥ = {} () We know from [], Proposition . that lin TKp A(¯x)¯y + b ⊥ = Sp NKp A(¯x)¯y + b , which, by (), means that the CN C condition is equivalent to Sp NKp A(¯x)¯y + b ∩ Ker A(¯x)T = {} () Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page of 12 Step We next show Sp NKp A(¯x)¯y + b = D∗ NKp A(¯x)¯y + b, λ¯ () () Since λ¯ ∈ NKp (A(¯x)¯y + b), we have λ¯ = k(–¯z , z¯ ) with k ∈ know from Proposition . that D∗ NKp A(¯x)¯y + b, λ¯ () = –, ¯ T (¯z + λ) ¯ (¯z + λ) = –, (k + )¯zT (k + )¯z + , where z¯ = A(¯x)¯y + b Then we T T = –, z¯ T z¯ T , which, by z¯ = z¯ , means that () holds Combining with () and (), the CN C condition is equivalent to = A(¯x)T η, η ∈ D∗ NKp (A(¯x)¯y + b, λ¯ )() ⇒ η = , which implies () We complete the proof Remark . We know from the proof of Theorem . that the calmness of M(·) at ¯ is ensured by the CN C condition, which means the condition in Theorem . (, x¯ , y¯ , λ) is weaker than the conditions in [], Theorem . In the following, we apply the results obtained to derive a necessary and sufficient optimality condition for the bilevel programming () Theorem . Suppose the function f in BP () is convex, ∇y ψ(x, y) is a linear function and the conditions in Theorem . hold at (¯x, y¯ ) with the involved function F(x, y) := ∇y ψ(x, y) Then (¯x, y¯ ) is a locally optimal solution of BP () if and only if there exists (w, u) ∈ p × m satisfying w ∈ D∗ NKp (A(¯x)¯y + b, λ¯ )(A(¯x)u) for some λ¯ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b) such that ¯ u + J G(¯x, y¯ ) w, = ∇f (¯x, y¯ ) + Jx,y L(¯x, y¯ , λ) T T () where G(x, y) := A(x)y + b, L(x, y, λ) = ∇y ψ(x, y) + A(x)T λ and (x, y) = {λ : L(x, y, λ) = } Proof Since S(x) is the optimal solution set of the parametric problem () and for any x ∈ n , () is a convex optimization problem, S(x) can be written as S(x) = y : ∈ ∇y ψ(x, y) + Q(x, y) , where (x) = {y : A(x, y) + b ∈ Kp } and Q(x, y) = N can be reformulated as f (x, y) s.t (x, y) ∈ gph S, () (x) (y) As a result, the bilevel problem Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page of 12 where ⎧ ⎪ ⎨ gph S = ⎪ ⎩ ⎫ ⎤ ⎪ x ⎬ ⎢ ⎥ y ⎣ ⎦ ∈ gph Q ⎪ ⎭ –∇y ψ(x, y) ⎡ (x, y) ∈ n × m We next show that gph Q is normally regular at (¯x, y¯ , –∇ψ(¯x, y¯ )) We know from the proof of [], Theorem . that, under conditions (a) and (b) in Theorem ., D∗ Q(¯x, y¯ , v¯ )(u) = Jx,y A(¯x)T λ holds for any λ ∈ T u + D∗ (NKp ◦ G)(¯x, y¯ , λ) A(¯x)u (¯x, y¯ ), which, by Definition ., means that w ∈ Ngph Q (¯x, y¯ , v¯ ) –u ⇐⇒ w ∈ D∗ Q(¯x, y¯ , v¯ )(u) ⇐⇒ w ∈ Jx,y A(¯x)T λ T u + D∗ (NKp ◦ G)(¯x, y¯ , λ) A(¯x)u ⇐⇒ w – (Jx,y (A(¯x)T λ))T u ∈ Ngph NKp ◦G (¯x, y¯ , λ) –A(¯x)u ⇐⇒ In+m (Jx,y (A(¯x)T λ))T A(¯x) w ∈ Ngph NKp ◦G (¯x, y¯ , λ) –u () holds for any λ ∈ (¯x, y¯ ) Under conditions (a) and (b) in Theorem ., we know from the proof of [], Lemma . that D∗ Q(¯x, y¯ , v¯ )(u) = Jx,y A(¯x)T λ T u + D∗ (NKp ◦ G)(¯x, y¯ , λ) A(¯x)u () Under the SC condition, by Proposition ., we have Ngph NKp ◦G (¯x, y¯ , λ) = Ngph NKp ◦G (¯x, y¯ , λ) () Consequently, combining with (), (), and (), we see that gph Q is normally regular at (¯x, y¯ , –∇ψ(¯x, y¯ )) We know from the proof of [], Theorem . that if the set-valued ¯ with λ¯ ∈ (¯x, y¯ ) ∩ NKp (A(¯x)¯y + b), then mapping P () is calm at the points (, , x¯ , y¯ , λ) n m m the set-valued mapping : × × ⇒ n × m defined by ⎧ ⎪ ⎨ (ζ ) := (x, y) ∈ ⎪ ⎩ ⎫ ⎤ ⎪ x ⎬ ⎥ ⎢ + ζ ∈ gph Q y ⎦ ⎣ ⎪ ⎭ –∇y ψ(x, y) ⎡ n × m is calm at (, x¯ , y¯ ), which, by Proposition ., implies that Ngph S (¯x, y¯ ) ⊆ In Im –Jx (∇y ψ(¯x, y¯ ))T ◦ Ngph Q x¯ , y¯ , –∇y ψ(¯x, y¯ ) –Jy (∇y ψ(¯x, y¯ ))T () Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page 10 of 12 On the other hand, we know from [], Theorem . that Ngph S (¯x, y¯ ) ⊇ Im In –Jx (∇y ψ(¯x, y¯ ))T ◦ Ngph Q x¯ , y¯ , –∇y ψ(¯x, y¯ ) –Jy (∇y ψ(¯x, y¯ ))T () Notice that Ngph S (¯x, y¯ ) ⊆ Ngph S (¯x, y¯ ) Then combining () and (), the normal regularity of gph S at (¯x, y¯ ) is directly from the normal regularity of gph Q at (¯x, y¯ , –∇ψ(¯x, y¯ )) Therefore, (¯x, y¯ ) is a locally optimal solution if and only if (¯x, y¯ ) satisfying ∈ ∇f (x, y) + Ngph S (¯x, y¯ ), i.e., ∈ ∇x f (¯x, y¯ ) + D∗ S(¯x, y¯ ) ∇y f (¯x, y¯ ) () Under the conditions in Theorem ., we have D∗ S(¯x, y¯ ) y∗ = Jx L(¯x, y¯ , λ¯ ) u + Jx A(¯x)¯y T T w| ∈ y∗ + Jy L(¯x, y¯ , λ¯ ) u + A(¯x)T w, T w ∈ D∗ NKp A(¯x)¯y + b, λ¯ A(¯x)u () Consequently, the conclusion is directly from () and () In [], Theorem ., a necessary and sufficient global optimality condition for the bilevel programming () has been derived under some strong condition such as G(x, y) + λ ∈ (int Kp ) ∪ (int(Kp )– ) In the case when one of the conditions in [], Theorem . is not satisfied at a point, we not know whether the point is a global optimal solution However, by Theorem ., we may verify that the point is a local optimal solution for the bilevel programming () We next give an example to show this Example . Consider f (x , x , y , y ) := ex + x + y – y + y – y s.t y ∈ S(x), () where S(x) is the optimal solution set of the following problem: ψ(x , x , y , y ) := y – y + ex + x s.t G(x, y) := x + x + y ∈ K , y where x , x , y , y ∈ Consider a point (¯x, y¯ ) = (, , , )T ∈ By simple computing, we have the multiplier set (¯x, y¯ ) = {λ¯ } = {(–, )T } Then we have G(¯x, y¯ ) + λ¯ = (–, )T ∈/ (int K ) ∪ (int(K )– ), which means that one of the conditions in [], Theorem . is not satisfied at (¯x, y¯ ) and hence we not know whether it is a global solution to problem Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page 11 of 12 () Next by Theorem ., we verify that (¯x, y¯ ) = (, , , )T is a locally optimal solution of problem () (i) Since there exists yˆ = (, )T such that A(¯x)ˆy = yˆ ∈ int K , the SCQ () holds for x¯ (ii) Since A(¯x) = I , the CN C () holds at (¯x, y¯ ) (iii) The multifunction P(·) ¯ In fact, by a simple computation, we defined by () is calm at the points (, , x¯ , y¯ , λ) obtain Jx,y L(¯x, y¯ , λ¯ ) u + Jx,y A(¯x)¯y T – = T u+ T w w, which means that ∈ (Jx,y L(¯x, y¯ , λ))T u + (Jx,y (A(¯x)¯y))T w, w ∈ D∗ NKp (G(¯x, y¯ ), λ)(A(¯x)u) ⇒ w = , u = This, by the Mordukhovich criterion (), is a condition ensuring the Lipschitz-like property of P at (, , x¯ , y¯ , λ), which ensures the calmness of P at (, , x¯ , y¯ , λ) (iv) By simple computing, the SC condition holds (v) Next we show there exists (w, u) ∈ × sat¯ isfying w ∈ D∗ NKp (A(¯x)¯y + b, λ)(A(¯ x)u) such that () holds If the equality () holds for ¯ (¯x, y¯ , λ) and (w, u), then we have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ – ⎜ ⎟ ⎜ ⎟ u ⎜ ⎟ w ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ =⎜ ⎟+⎜ +⎜ ⎟ ⎟ ⎝–⎠ ⎝ ⎠ u ⎝ ⎠ w () We know from Proposition . that w ∈ D∗ NKp (A(¯x)¯y + b, λ¯ )(A(¯x)u) means that w ∈ |u| w ⊥ + –u u () Then, combining () and (), we have w = (, –)T and u = (/, )T satisfying w ∈ ¯ D∗ NKp (A(¯x)¯y + b, λ)(A(¯ x)u) such that () holds Therefore, by Theorem ., (¯x, y¯ ) is a locally optimal solution of problem () Remark . By a similar computation, we can infer the point (¯x, y¯ ) = (, , , )T is a locally optimal solution of problem () with f (x, y) := x + x + (y – ) + (y – ) and it is not a locally optimal solution of problem () with f (x, y) := ex + x – x + y + /y – y Conclusion In this paper, an equality type representation of the coderivative of the solution mapping S () is obtained, which is an improvement of [], Theorem . The result obtained is then used to develop a necessary and sufficient local optimality condition for a bilevel programming with SOC as its lower level problem Competing interests The authors declare that they have no competing interests Zhang et al Journal of Inequalities and Applications (2015) 2015:361 Page 12 of 12 Authors’ contributions All authors contributed equally to the writing of this paper All authors read and approved the final manuscript Acknowledgements This work was supported by the National Natural Science Foundation of China under Project No 11201210, CPSF grant 2014M560200 and Program for Liaoning Excellent Talents in University No LJQ2015059 Received: 29 June 2015 Accepted: November 2015 References Dempe, S: Foundations of Bilevel Programming Kluwer Academic, Dordrecht (2002) Ye, J: Constraint qualifications and KKT conditions for bilevel programming problems Math Oper Res 31, 811-824 (2006) Outrata, JV, Sun, DF: On the coderivative of the projection operator onto the second-order cone Set-Valued Anal 16, 999-1014 (2008) Mordukhovich, BS: Variational Analysis and Generalized Differentiation I: Basic Theory, II: Applications Springer, Berlin (2006) Mordukhovich, BS, Outrata, JV: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization SIAM J Optim 18, 389-412 (2007) Shen, J, Pang, LP: A bundle-type auxiliary problem method for generalized variational-like inequality Comput Math Appl 55, 2993-2998 (2008) Shen, J, Pang, LP: An approximate bundle-type auxiliary problem method for generalized variational inequality Math Comput Model 48, 769-775 (2008) Shen, J, Pang, LP: A proximal analytic center cutting plane algorithm for solving variational inequality problems J Appl Math 2012, Article ID 503242 (2012) Mordukhovich, BS: Coderivatives of set-valued mappings: calculus and applications Nonlinear Anal 30, 3059-3070 (1997) 10 Zhang, J, Li, Y, Zhang, L: On the coderivative of the solution mapping to a second-order cone constrained parametric variational inequality J Glob Optim 61, 379-396 (2015) 11 Rockafellar, RT, Wets, RJB: Variational Analysis Springer, Berlin (1998) 12 Mordukhovich, BS: Metric approximations and necessary optimality conditions for general classes of extremal problems Sov Math Dokl 22, 526-530 (1980) 13 Mordukhovich, BS: Generalized differential calculus for nonsmooth and set-valued mappings J Math Anal Appl 183, 250-288 (1994) 14 Henrion, R, Jourani, A, Outrata, JV: On the calmness of a class of multifunctions SIAM J Optim 13, 603-618 (2002) 15 Bonnans, JF, Shapiro, A: Perturbation Analysis of Optimization Problems Springer, New York (2000) 16 Zhang, J, Zhang, L, Pang, L: On the convergence of coderivative of SAA solution mapping for a parametric stochastic variational inequality Set-Valued Var Anal 20, 75-109 (2012)