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1 Fakultät für Mathematik und Informatik Preprint Susanne Franke, Patrick Mehlitz, Maria Pilecka Optimality conditions for the simple convex bilevel programming problem in Banach spaces ISSN

2 Susanne Franke, Patrick Mehlitz, Maria Pilecka Optimality conditions for the simple convex bilevel programming problem in Banach spaces TU Bergakademie Freiberg Fakultät für Mathematik und Informatik Prüferstraße FREIBERG

3 ISSN Herausgeber: Herstellung: Dekan der Fakultät für Mathematik und Informatik Medienzentrum der TU Bergakademie Freiberg

4 TU Bergakademie Freiberg Preprint Optimality conditions for the simple convex bilevel programming problem in Banach spaces Susanne Franke, Patrick Mehlitz, Maria Pilecka Abstract The simple convex bilevel programming problem is a convex minimization problem whose feasible set is the solution set of another convex optimization problem. Such problems appear frequently when searching for the projection of a certain point onto the solution set of another program. Due to the nature of the problem, Slater s constraint qualification generally fails to hold at any feasible point. Hence, one has to formulate weaker constraint qualifications or stationarity notions in order to state optimality conditions. In this paper, we use two different single-level reformulations of the problem, the optimal value and the Karush-Kuhn-Tucker approach, to derive optimality conditions for the original program. Since all these considerations are carried out in Banach spaces, the results are not limited to finitedimensional optimization. On the route, we introduce and discuss a certain concept of M-stationarity for mathematical programs with complementarity constraints in Banach spaces. Keywords Bilevel Programming Constraint Qualifications Mathematical Program with Complementarity Constraints Programming in Banach Spaces Mathematics Subject Classification (2000) 46N10 49K27 90C25 90C33 Dedicated to Professor Stephan Dempe on the occasion of his 60th birthday. 1 Introduction Consider the simple convex bilevel programming problem (SCBPP for short) which is given as stated below: F (x) min x Ψ := Argmin{f(x) x Θ}. (SCBPP) Therein, F, f : X R are convex, continuous, and sufficiently smooth functionals of a Banach space X, while Θ X is a nonempty, closed, and convex set. Note that these assumptions guarantee that the set Ψ is convex, i.e. (SCBPP) is a convex optimization problem. However, the derivation of optimality conditions is a challenging problem since most of the standard constraint qualifications generally fail to hold at any feasible point of (SCBPP). In the finite-dimensional situation, (SCBPP) was already discussed in [6] where its relationship to standard bilevel programming problems was mentioned. For a detailed introduction to the topic of bilevel programming, we refer the interested reader to [5]. In order to stay as close as possible to the terminology of bilevel programming, we call f(x) min x Θ the lower level problem of (SCBPP). Problems of type (SCBPP) arise frequently when a best point (in a certain sense) among the optimal solutions of a convex optimization problem has to be found. This S. Franke, P. Mehlitz (corresponding author), and M. Pilecka Technische Universität Bergakademie Freiberg, Faculty of Mathematics and Computer Science, Freiberg, Germany {susanne.franke,mehlitz,pilecka}@math.tu-freiberg.de, the work of the first author has been supported by the Deutsche Forschungsgemeinschaft, grant DE-650/7-1

5 2 Susanne Franke, Patrick Mehlitz, Maria Pilecka situation is modelled insufficiently by a vector optimization approach with two objective functionals and, hence, is discussed in the context of bilevel programming. If the lower level feasible set Θ is described by conic constraints, it is not difficult to show, see Theorem 5.5, that (SCBPP) is equivalent to a certain mathematical program with complementarity constraints (MPCC for short) in Banach spaces, recently introduced and studied in [18, 22, 23]. However, it turns out that the strong stationarity conditions of the surrogate problem are, in general, too strong to hold at the local minimizers of (SCBPP), see Section 5. Hence, weaker stationarity notions have to be considered. Therefore, in this paper, we want to consider a generalized concept of Mordukhovichstationarity (M-stationarity for short) which is known to be weaker than strong stationarity for common finite-dimensional or semidefinite MPCCs. We organized this article as follows: In Section 2, we subsume notations and preliminary results we use throughout the paper. Section 3 is dedicated to the study of constraint qualifications for optimization problems in Banach spaces. Afterwards, MPCCs in Banach spaces are discussed in Section 4. Therein, we recall the notions of weak and strong stationarity already known from [18,22]. Furthermore, we introduce a concept of M-stationarity which is shown to be reasonable in the case where the cone which generates the complementarity constraint is polyhedral. Finally, we consider (SCBPP) in Section 5. The lower level optimal objective value and the lower level Karush-Kuhn-Tucker (KKT for short) conditions of (SCBPP) are used to state two single-level surrogate problems which are exploited to derive optimality conditions for (SCBPP). The aforementioned KKT approach leads to necessary optimality conditions of M-stationarity-type. 2 Notation and preliminary results 2.1 Notation Let us start this section by introducing the notation we exploit in this paper. We use N, R, R, R n, R n,+ 0, and R n m to denote the natural numbers (without zero), the real numbers, the extended real line, the set of all real vectors with n components, the cone of all vectors from R n possessing nonnegative components, and the set of all real matrices with n rows and m columns, respectively. Furthermore, we stipulate R + 0 := R1,+ 0. For an arbitrary matrix Q R n m and an index set I {1,..., n}, Q I R I m denotes the matrix composed of the rows of Q whose indices come from I. Furthermore, O R n m and I R n n represent the zero matrix and the identity matrix of appropriate dimension, respectively. Let X be a real Banach space with norm X and zero vector 0, and let A X be nonempty. We denote by lin(a), cone(a), conv(a), int(a), and cl(a) the smallest subspace of X containing A, the smallest convex cone containing A, the convex hull of A, the interior of A, and the closure of A, respectively. The indicator function δ A : X R of the set A is defined by { 0 if x A, x X : δ A (x) := + if x / A. Now fix some x A. Then we define the radial cone, the tangent (or Bouligand) cone, and the weak tangent cone to A at x as stated below: R A (x) := {d X α > 0 t (0, α): x + td A}, T A (x) := {d X {d k } X {t k } R: d k d, t k 0, x + t k d k A k N}, T w A (x) := {d X {d k } X {t k } R: d k d, t k 0, x + t k d k A k N}. Therein, and denote the norm and the weak convergence in X, respectively. Observe that the cone T A (x) is always a closed set which satisfies R A (x) T A (x) T w A (x). Let B X be the closed unit ball of X, while for any ε > 0 and x X we use U ε X (x) and Bε X (x) to denote the open and closed ε-ball around x, respectively. The (topological) dual space of X is denoted by X, and, : X X R is the corresponding dual pairing. Let B X be nonempty. Then cl (B) denotes the weak- -closure of B, i.e. the closure with respect to (w.r.t.) the weak- -convergence. Now

6 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 3 suppose that X is reflexive, i.e. X = X holds true. Then we define the polar cones and annihilators of the sets A and B by A := {x X x A: x, x 0}, B := {x X x B : x, x 0}, A := {x X x A: x, x = 0}, B := {x X x B : x, x = 0}, respectively. Note that due to the reflexivity of X, the above notation is consistent. For the purpose of simplicity, we omit curly brackets when considering singletons, i.e. we set lin(x) := lin({x}) and x := {x} for any x X, and similar definitions shall hold for all x X as well as, if consistent, in the case of a nonreflexive Banach space X. Now X and Y may be arbitrary Banach spaces again. Then the product space X Y is a Banach space, too, when, e.g., equipped with the sum norm induced by X and Y. In the case where Y = X holds, we use X 2 := X X and the components of any x X 2 are addressed by x 1, x 2 X. We exploit a similar notation for all n N satisfying n 3 as well as arbitrary subsets of A X, i.e., A n denotes the Cartesian product of order n of A. For a closed, convex set C X and a fixed point x C, we obtain R C (x) = cone(c {x}) and T C (x) = cl(r C (x)), respectively. Moreover, if x T C (x) is chosen, then the critical cone to C w.r.t. x and x is defined by K C (x, x ) := T C (x) (x ). In the case that C is additionally a cone, we easily obtain R C (x) = C + lin(x), T C (x) = cl(c + lin(x)), T C (x) = C x. The (not necessarily conic) set C is called polyhedric w.r.t. (x, x ), where x C and x T C (x) hold, if cl(r C (x) (x ) ) = K C (x, x ) is satisfied. Moreover, C is called polyhedric if it is polyhedric w.r.t all points (x, x ) satisfying x C and x T C (x). Recall that C is called polyhedral if there exist functionals x 1,..., x n X and scalars β 1,..., β n R such that C possesses the representation C = {x X x, x i β i i {1,..., n}}. For a reflexive Banach space X and a closed (but not necessarily convex) set D X satisfying x D we define the Fréchet normal cone to D at x as stated below: { N D (x) := x X y x, x } lim sup 0. y x, y D y x X It follows from [19, Theorem 1.10] that N D (x) = T w D (x) is satisfied. Next, we introduce the basic (or limiting, Mordukhovich) normal cone to D at x by N D (x) := lim sup y x, y D N D (y) where the limit superior has to be understood in the sense of Painlevé-Kuratowski and is taken w.r.t. the weak- -convergence in X. Note that due to the reflexivity of X, weak- - and weak convergence are equivalent. In the case of the set D being convex, we have N D (x) = N D (x) = T D (x). We call D sequentially normally compact (SNC for short) at x if for any sequences {x k } D and {x k } X such that x k N D (x k ) is valid for all k N and x k x as well as x k 0 hold, x k 0 is satisfied. Obviously, any subset of a finite-dimensional Banach space is SNC at any of its points. On the other hand, a singleton in X is SNC iff X is finite-dimensional, see [19, Theorem 1.21]. Let U, V, and W be arbitrary Banach spaces. By L[U, V] we denote the space of bounded linear operators mapping from U to V. For any operator F L[U, V], the operator F L[V, U ] denotes the

7 4 Susanne Franke, Patrick Mehlitz, Maria Pilecka adjoint of F. The identical mapping of U, denoted by I U, is always a continuous operator as long as U is equipped with the same norm in its definition and image space. For another operator G L[U, W], the linear operator (F, G) L[U, V W] is defined as stated below: u U : (F, G)[u] := (F[u], G[u]). Let Γ : U 2 V be an arbitrary set-valued mapping. Then gph Γ := {(u, v) U V v Γ (u)} denotes its graph. Recall that Γ is called Lipschitz-like at some point (ū, v) gph Γ if there are constants ε > 0, ρ > 0, and L > 0 such that u, u U ε U(ū): Γ (u) U ρ V ( v) Γ (u ) + L u u U B V is satisfied. If the above condition holds whenever u := ū is fixed, then Γ is called calm at (ū, v). Obviously, calmness is implied by the Lipschitz-like property. Assume that U and V are reflexive Banach spaces. The normal coderivative DN Γ (ū, v): V 2 U is defined by v V : D NΓ (ū, v)(v ) = {u U (u, v ) N gphγ (ū, v)}. We call Γ partially sequentially normally compact (PSNC for short) at (ū, v) if for any sequences {(u k, v k )} gph Γ and {(u k, v k )} U V which satisfy (u k, v k ) (ū, v), u k 0, v k 0, and (u k, v k ) N gph Γ (u k, v k ) for any k N, we obtain u k 0. Note that if Γ is PSNC at (ū, v), satisfies DN Γ (ū, v)(0) = {0}, and possesses a closed graph in a neighborhood of (ū, v), then Γ is Lipschitz-like at the latter point, see [19, Theorem 4.10]. We drop the reflexivity assumption on U and V. Let φ: U R be a convex functional and ū U be a point where φ(ū) < is satisfied. Then φ(ū) := {u U φ(u) φ(ū) + u ū, u u U} denotes the subdifferential of φ at ū. For a nonempty, closed, and convex set Ω U, the functional δ Ω is convex, and it is easy to see that for any point ū Ω, we have δ Ω (ū) = T Ω (ū). For some convex cone K V, we say that a mapping ψ : U V is K-convex if the following condition holds: u, u U σ (0, 1): σψ(u) (1 σ)ψ(u ) + ψ(σu + (1 σ)u ) K. Note that for a K-convex mapping, the set {u U ψ(u) K} is convex. Suppose that θ : U V is a twice Fréchet differentiable mapping and ū U is fixed. Then the linear operators θ (ū) L[U, V] and θ (2) (ū) L[U, L[U, V]] denote its first and second order Fréchet derivative at ū, respectively. For any ξ V, we use θ (2) (ū), ξ to represent the linear operator in L[U, U ] defined below: d u U : θ (2) (ū), ξ [d u ] := θ (2) (ū)[d u ], ξ = ( θ (2) (ū)[d u ] ) [ξ] = ξ θ (2) (ū)[d u ]. Therein, denotes the composition of mappings. 2.2 Preliminary results The first part of the following result is called Generalized Farkas Lemma and is stated in a more abstract setting, e.g., in [10, Theorem 1, Lemma 3]. Lemma 2.1 Let X and Y be Banach spaces, φ: X R be a positively homogeneous and convex functional, A L[X, Y] be a bounded linear operator, and K Y be a nonempty, closed, convex cone. 1. For any ξ X, the following assertions are equivalent: (a) ξ cl ( φ(0) + A [K ] ), (b) x X : A[x] K = x, ξ φ(x). Particularly, setting φ( ) := δ TΩ ( x)( ) (indicator function of the tangent cone to Ω at x) for some closed, convex set Ω X and x Ω, we have cl ( T Ω ( x) + A [K ] ) = {x T Ω ( x) A[x] K}.

8 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 5 2. Suppose that A[X ] K = Y is satisfied. Then we have A [K ] = {x X A[x] K}. Note that if X is reflexive, then we can replace the weak- -closure in Lemma 2.1 by the common closure since weak and weak- -convergence in X are equivalent, and convex sets are closed iff they are weakly closed by Mazur s theorem. Indeed, the first statement of Lemma 2.1 is a generalization of the famous Farkas Lemma. Remark 2.1 Choosing X = R n, Y = R m, A R m n, b R n, K := R m,+ 0, as well as ξ = 0 and defining φ: R n R by φ(x) := b x for any x R n, Lemma 2.1 yields that the assertions 1. y R m : A y = b, y 0, 2. x R n : Ax 0 = b x 0 are equivalent. Hence, precisely one of the following systems possesses a solution: A y = b Ax 0 y 0 b x < 0. Moreover, we can characterize the adjoint of a dense range operator simply by applying Lemma 2.1. Remark 2.2 Let X and Y be reflexive Banach spaces, while A L[X, Y] is an arbitrary operator. Then A is injective iff A possesses a dense range. Lemma 2.2 Let A L[X, Y] be an injective bounded linear operator with a closed range between Banach spaces X and Y. Then there is a constant c > 0 satisfying c x X A[x] Y for all x X. Proof Clearly, due to its closedness, A[X ] is a Banach space as well. Hence, we can define a bijection à L[X, A[X ]] by means of Ã[x] := A[x] for any x X. Then à 1 is a bounded linear operator as well, i.e., there is a constant c > 0 which satisfies à 1 [y] X c y Y for all y A[X ]. For arbitrary x X, we obtain x X = à 1 [A[x]] X c A[x] Y. Hence, the statement of the lemma follows by choosing c := 1 c. Later we will introduce a generalized notion of M-stationarity for MPCCs where the complementarity cone is polyhedral. Therefore, we need the following lemma which summarizes the results obtained in [13]. Lemma 2.3 Let Z be a reflexive Banach space and {z 1,..., z k } Z be a set of linear independent functionals. We consider the closed, convex cone K := {z Z z, z i 0 i {1,..., k}}. (1) Then we have K = We analyze the complementarity set { k } i=1 α izi Z αi 0 i {1,..., k}. F := {(u, v) K K u, v = 0}. Choosing (ū, v) F, we define index sets I(ū, v) and J(ū, v) by I(ū, v) := {i {1,..., k} ū, z i = 0}, J(ū, v) := {i I(ū, v) ᾱ i > 0}. (2) Therein, we exploited the unique representation v = k i=1 ᾱiz i for some ᾱ 1,..., ᾱ k 0. For any index sets J := J(ū, v) P Q I(ū, v) =: I, we define Then the following relations hold: C Q,P := cone({z i i Q \ P }) + lin({z i i P }), D Q,P := {z Z z, z i 0 i Q \ P } {z i i P }. N F (ū, v) = C I,J D I,J, N F (ū, v) = J P Q I C Q,P D Q,P.

9 6 Susanne Franke, Patrick Mehlitz, Maria Pilecka Although it is well-known, we want to recall the following result for the sake of completeness. Lemma 2.4 Let the polyhedral cone K be given as stated in (1). Then K is polyhedric. Proof The assertion clearly follows if we can show R K (ū) = T K (ū) for arbitrary ū K. Therefore, choose ū K arbitrarily, define I(ū) := {i {1,..., k} ū, zi = 0}, and observe R K (ū) = K + lin(ū) = {z + κū Z κ R, z, z i 0 i {1,..., k}} = {d Z κ R: d κū, z i 0 i {1,..., k}} = {d Z κ R: d, z i 0 i I(ū), d, z i κ ū, z i i {1,..., k} \ I(ū)} = {d Z d, z i 0 i I(ū)}. Obviously, R K (ū) is closed and, hence, it coincides with T K (ū). This completes the proof. 3 Constraint qualifications in Banach spaces Let X and Y be arbitrary Banach spaces, H : X Y be continuously Fréchet differentiable, and let C Y and Ω X be nonempty and closed. Feasible regions of mathematical programs are often described by sets of the type M := {x Ω H(x) C}. In order to guarantee that necessary optimality conditions of KKT-type hold at local minimizers of h(x) min x M, (3) where h: X R is continuously Fréchet differentiable, one generally has to postulate certain constraint qualifications. The aforementioned KKT conditions of (3) at a certain feasible point x M are stated below: λ N C (H( x)) ξ N Ω ( x): 0 = h ( x) + H ( x) [λ] + ξ. (4) We introduce the set Λ( x) of regular Lagrange multipliers of (3) at x by Λ( x) := {(λ, ξ) N C (H( x)) N Ω ( x) 0 = h ( x) + H ( x) [λ] + ξ}. Consequently, the KKT conditions are satisfied at x M iff the set Λ( x) is nonempty. In this section, we give a short overview of regularity conditions and discuss the relationships between them. 3.1 Weak constraint qualifications Without mentioning it again, we assume throughout this section that the sets C and Ω are convex. This situation is well-studied in literature. For the upcoming considerations, we choose an arbitrary point x M. The most common constraint qualification in our setting is KRZCQ (cf. [3, 17, 20]), the so-called Kurcyusz Robinson Zowe Constraint Qualification, which postulates Polarizing this equation leads to H ( x)[r Ω ( x)] R C (H( x)) = Y. {(λ, ξ) T C (H( x)) T Ω ( x) H ( x) [λ] + ξ = 0} = {(0, 0)}. (5) Note that due to the convexity of C and Ω, we have N C (H( x)) = T C (H( x)) and N Ω ( x) = T Ω ( x). Hence, in terms of (4), the set on the left in (5) is called the set of singular Lagrange multipliers. Observe that this set is always a nonempty, closed, and convex cone. We polarize (5) once more to obtain the condition cl ( H ( x)[t Ω ( x)] T C (H( x)) ) = Y

10 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 7 which is equivalent to (5) by the bipolar theorem, cf. [3, Proposition 2.40]. Now it is easy to see that the latter condition is equivalent to cl(h ( x)[r Ω ( x)] R C (H( x)) ) = Y. (6) It follows from [3, Corollary 2.98] that the constraint qualifications KRZCQ, (5), and (6) are equivalent in the case where Ω = X holds and C possesses a nonempty interior. Let us consider the situation X = R n, Y = R m, C = R m,+ 0, Ω = R n which describes a standard nonlinear program with inequality constraints. It is common knowledge that the dual form of the well-known MFCQ, the Mangasarian Fromovitz Constraint Qualification, takes precisely the form (5), and, consequently, KRZCQ equals MFCQ in this case. A condition which obviously implies KRZCQ is given by H ( x)[r Ω ( x)] = Y, which reduces to the surjectivity of H ( x) if either Ω = X or x int(ω) is satisfied. We call this condition FRCQ, the Full Range Constraint Qualification. Observe that for standard nonlinear programs, FRCQ is strictly stronger than LICQ, the Linear Independence Constraint Qualification, as long as not all constraints are active at x. In order to state weaker constraint qualifications, we introduce two convex cones. First, the linearized tangent cone to M at x, denoted by L M ( x), is given below: L M ( x) := {d T Ω ( x) H ( x)[d] T C (H( x))}. Obviously, this cone is closed. Furthermore, we will deal with the linearized normal cone S M ( x) to M at x defined by S M ( x) := {H ( x) [λ] + ξ X λ T C (H( x)), ξ T Ω ( x) }. Note that this cone is not necessarily closed, e.g., in the case where C = {0} and Ω = X hold, while the image of H ( x) is dense in X. In the following proposition, we state some results on the cones L M ( x) and S M ( x). Proposition 3.1 Let x M be arbitrarily chosen. Then the following assertions are satisfied: 1. T M ( x) L M ( x), 2. L M ( x) = cl ( S M ( x) ), 3. if KRZCQ holds at x, then L M ( x) T M ( x) is satisfied and L M ( x) = S M ( x) holds, i.e. S M ( x) is closed w.r.t. the weak- -topology. Proof The proof of the first assertion is straightforward and, hence, omitted. For the proof of the second statement, we refer the reader to the first part of Lemma 2.1. The first statement of the third assertion follows from [3, Corollary 2.91]. Hence, we only need to show that L M ( x) = S M ( x) is valid under KRZCQ. Therefore, observe that the latter regularity condition is equivalent to ( H ( x), I X ) [X ] RC (H( x)) R Ω ( x) = Y X which implies ( H ( x), I X ) [X ] TC (H( x)) T Ω ( x) = Y X. Now we apply the second statement of Lemma 2.1 with A = ( ) H ( x), I X and K = TC (H( x)) T Ω ( x) to obtain L M ( x) = S M ( x). In the following definition, we present two constraint qualifications which are weaker than KRZCQ. These conditions can be found in a slightly different setting in [1, 11]. Definition 3.1 Let x M be arbitrarily chosen. We say that ACQ, the Abadie Constraint Qualification, holds true at x if T M ( x) = L M ( x) is valid, while S M ( x) is closed w.r.t. the weak- -topology. Furthermore, GCQ, the Guignard Constraint Qualification, is said to be satisfied at x if the relation T M ( x) = S M ( x) holds.

11 8 Susanne Franke, Patrick Mehlitz, Maria Pilecka From Proposition 3.1, we derive the following result. Lemma 3.1 Let x M be arbitrarily chosen. Then the following implications hold for the constraint qualifications from above which might be satisfied at x: FRCQ = KRZCQ = ACQ = GCQ. Furthermore, if x is a local minimizer of (3) where at least GCQ holds, then the KKT conditions are satisfied at this point. Hence, any of the above constraint qualifications is sufficient for the KKT conditions to be necessary optimality conditions for (3). Proof We start to verify the implications for the presented constraint qualifications. Obviously, FRCQ implies KRZCQ. If KRZCQ holds at x, then by Proposition 3.1, T M ( x) = L M ( x) is satisfied, while S M ( x) is closed w.r.t. the weak- -topology, i.e. ACQ holds at x. Now suppose that ACQ is valid at x. Then we can easily deduce T M ( x) = L M ( x) = cl ( S M ( x) ) = S M ( x). Consequently, GCQ holds at x as well. Now assume that x is a local minimizer of (3) where GCQ is satisfied. Then the local optimality of x for (3) implies that the first order optimality condition d T M ( x): h ( x)[d] 0 holds, see [15, Theorem 4.14]. This implies h ( x) T M ( x) = S M ( x) by GCQ. Hence, the KKT conditions are valid at x. A special setting where ACQ always holds is described in the following example. Example 3.1 Let X and Y be Banach spaces, A L[X, Y] be a bounded linear operator with closed range, b Y as well as a 1,..., a k X and β 1,... β k R be fixed, and consider the set M := {x X A[x] = b, x, a i β i i {1,..., k}}. For some x M, we define I( x) := {i {1,..., k} x, a i = β i} and obtain L M ( x) = {d X A[d] = 0, d, a i 0 i I( x)}, S M ( x) = A [Y ] + { i I( x) α ia i X αi 0 i I( x) }. It is not difficult to see that T M ( x) = L M ( x) holds true, and from [3, Proposition 2.201] it follows L M ( x) = S M ( x), i.e. S M ( x) is closed w.r.t. the weak- -topology. Hence, ACQ is valid at x. 3.2 Constraint qualifications for nonconvex abstract constraints Now we want to take a closer look at the situation where C and Ω are not necessarily convex. Then the constraint qualifications FRCQ, KRZCQ, ACQ, and GCQ are not applicable. In [23, Section 3], the author applies a generalized concept of tangent approximations to derive a KRZCQ-type regularity condition for (3). We want to use another approach here which dates back to the variational calculus introduced by Mordukhovich, see [19]. Definition 3.2 Let x M be arbitrarily chosen. Then BCQ, the Basic Constraint Qualification, is satisfied at x if the following two conditions are valid: H H ( x) } [λ] + ξ = 0, ( x) is surjective, = ξ = 0. λ N C (H( x)), ξ N Ω ( x) Furthermore, we say that PCBCQ, the Partially Convex Basic Constraint Qualification, is satisfied at x if the sets C and Υ := {x X H(x) C} are convex and the following two conditions are valid: H H ( x) } [λ] + ξ = 0, ( x)[x ] R C (H( x)) = Y, λ T C (H( x)) = ξ = 0., ξ N Ω ( x) We start our considerations with a short statement on these conditions.

12 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 9 Corollary 3.1 Choose x M arbitrarily. Then PCBCQ implies the following condition: H ( x) } [λ] + ξ = 0, λ T C (H( x)) = λ = 0, ξ = 0. (7), ξ N Ω ( x) On the other hand, if C possesses a nonempty interior and the sets C as well as Υ are convex, then (7) implies PCBCQ. Furthermore, BCQ implies the following condition H ( x) [λ] + ξ = 0, λ N C (H( x)), ξ N Ω ( x) } = λ = 0, ξ = 0. (8) Proof Let PCBCQ hold at x M and suppose that H ( x) [λ]+ξ = 0 is valid for some λ T C (H( x)) and ξ N Ω ( x). Then we automatically have ξ = 0 from the second condition of PCBCQ, i.e. H ( x) [λ] = 0 is satisfied. Polarizing the first condition of PCBCQ yields {λ T C (H( x)) H ( x) [λ] = 0} = {0}. This leads to λ = 0. Consequently, condition (7) is valid. Now assume that condition (7) holds. Then the second condition of PCBCQ is trivially satisfied. Furthermore, choosing 0 N Ω ( x) in (7) leads to {λ T C (H( x)) H ( x) [λ] = 0} = {0}. Since C possesses a nonempty interior and due to the postulated convexity assumptions, the statements of Section 3.1 are applicable, and, consequently, this condition equals the first condition of PCBCQ. The fact that BCQ implies (8) is obvious since the surjectivity of H ( x) leads to the injectivity of H ( x) by the Closed-Range-Theorem [24, Theorem IV.5.1]. Now we want to show that BCQ and PCBCQ are sufficient for the KKT conditions to be necessary optimality conditions for (3) under an additional assumption. Proposition 3.2 Let X be reflexive and x M be a local minimizer of (3) where Ω is SNC and BCQ or PCBCQ holds. Then the KKT conditions are satisfied at x. Proof From [19, Proposition 5.1], we have h ( x) N M ( x). Note that M = Υ Ω holds. Using the first condition of BCQ and [19, Theorem 1.17], we obtain If PCBCQ holds, by Lemma 3.1 N Υ ( x) = H ( x) [N C (H( x))]. N Υ ( x) = T Υ ( x) = L Υ ( x) = S Υ ( x) = H ( x) [T C (H( x)) ] holds as well. Hence, the second condition of BCQ and PCBCQ is equivalent to N Υ ( x) ( N Ω ( x) ) = {0}. The latter condition and the property of Ω to be SNC at x imply N M ( x) N Υ ( x) + N Ω ( x) by [19, Corollary 3.5]. Consequently, we obtain which yields the KKT conditions at x. h ( x) H ( x) [N C (H( x))] + N Ω ( x) In the case where the Banach spaces X and Y are finite-dimensional, we can drop the surjectivity assumption on the constraint mapping H and still obtain an appropriate constraint qualification. This result follows, e.g., from [8, Theorem 2.1, Proposition 2.2, Proposition 2.3]. Remark 3.1 Let X as well as Y be finite-dimensional Banach spaces and assume that x M is a local minimizer of (3) where the condition (8) is satisfied. Then the KKT conditions hold at x. That is why in the finite-dimensional setting, the condition (8) is called basic constraint qualification as well. Let us consider the set-valued mapping M: Y 2 X as stated below: y Y : M(y) := {x Ω H(x) + y C}. (9) Obviously, M(0) = M is satisfied. It is well-known that certain stability properties of the perturbation mapping M may serve as a constraint qualification for (3) as well. We obtain the following result which is similar to [8, Theorem 2.1] or [19, Lemma 5.47] where slightly different settings are analyzed.

13 10 Susanne Franke, Patrick Mehlitz, Maria Pilecka Proposition 3.3 Let x M be a local optimal solution of (3) where X and Y are reflexive and assume that M is calm at (0, x). Then the KKT conditions are valid. Proof Due to the calmness of M at (0, x), we find ε, ρ > 0 and L M > 0 such that y U ε Y(0): M(y) U ρ X ( x) M(0) + L M y Y B X is satisfied. Furthermore, we find a constant ϱ > 0 such that h and H are Lipschitz continuous on U ϱ X ( x) with modulus L h and L H, respectively, and h(x) h( x) holds true for all x M(0) U ϱ X ( x). Now set ɛ := min { εl M ; ρ; ϱ } { } { } 2, α := min ɛ; ɛ 2L H L M, as well as β := min ɛ; ɛ 2L M and choose points x Ω U α X ( x) and z C Uβ Y (H( x)) arbitrarily. Then it follows z H(x) Y z H( x) Y + L H x x X < ɛ L M ε. Observe that x M(z H(x)) U ρ X ( x) is satisfied, which is why we obtain the existence of x M(0) satisfying x x X L M z H(x) Y < ɛ from the calmness condition. Consequently, we derive x x X x x X + x x X 2ɛ. Hence, x M(0) U ϱ X ( x) holds. Now we easily obtain h( x) h(x ) = h(x) + (h(x ) h(x)) h(x) + L h x x X h(x) + L h L M z H(x) Y. That is why ( x, H( x)) is a local optimal solution of h(x) + L h L M z H(x) Y min x,z (x, z) Ω C. Applying [19, Proposition 1.2, Proposition 5.3], which is possible due to the reflexivity of X and Y, and some calculus rules for subdifferentials, see [19, Section 1.3.4], yields the existence of y B Y satisfying 0 {h ( x) L h L M H ( x) [y ]} + N Ω ( x), 0 {L h L M y } + N C (H( x)). Finally, λ := L h L M y is defined in order to show that the KKT conditions hold. In the following lemma, we show a relationship between BCQ and the calmness of M. Lemma 3.2 Let x M be a point where BCQ holds, assume that X as well as Y are reflexive, and suppose that Ω is SNC at x. Then M is calm at x. Proof Due to the closedness of Ω and C and the continuity of H, gph M is closed. Now we show DN M(0, x)(0) = {0} and that M is PSNC at (0, x). By means of [19, Theorem 4.10], this is sufficient for M to be Lipschitz-like and, hence, calm at this point. Since Ω is SNC at x, Y Ω is SNC at (0, x). Due to the special structure of the perturbation mapping M, we deduce the relationship DN M(0, x)(0) = {0} from [19, Corollary 1.69(ii), Theorem 4.32(b)] using the second condition postulated in the definition of BCQ. Now we need to show that M is PSNC at (0, x). Exploiting the fact that Y Ω is SNC at this point, by means of [19, Corollary 3.80] we only need to show that the set-valued map M : Y 2 X such that M (y) = {x X H(x) + y C} is satisfied for any y Y is PSNC at (0, x). Hence, we choose sequences {(y k, x k )} gph M and {(yk, x k )} Y X such that (y k, x k ) (0, x), yk 0, x k 0, and (yk, x k ) N gph M (y k, x k ) for all k N hold true. Applying [19, Corollary 1.15], we obtain N gph M (y k, x k ) = {(v k, H (x k ) [v k]) Y X v k N C (H(x k ) + y k )}. Consequently, there is a sequence {vk } Y satisfying vk N C (H(x k ) + y k ), yk = v k, as well as x k = H (x k ) [vk ] for any k N. The observation H ( x) [vk] + ( H (x k ) H ( x) ) [vk] = x k 0 }{{} 0

14 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 11 leads to H ( x) [vk ] 0 since {v k } converges weakly and, hence, is bounded. From BCQ, we know that H ( x) is surjective which is why H ( x) is injective, see Remark 2.2. Moreover, H ( x) [Y ] is closed by [24, Theorem IV.5.1]. Applying Lemma 2.2, we find a constant c > 0 such that 0 c v k Y H ( x) [v k] X 0 holds, i.e. vk 0 is satisfied which is equivalent to y k 0. Hence, M is PSNC at (0, x). Due to the above argumentation, M is PSNC at (0, x) as well. This completes the proof. Once more we want to take a look at the finite-dimensional situation. The upcoming result and its proof can be found in [8, Proposition 2.3]. Remark 3.2 Let X as well as Y be finite-dimensional Banach spaces and assume that x M is a point where the condition (8) is satisfied. Then the corresponding perturbation mapping M is calm at (0, x). Remark 3.3 Be aware that the conditions in Lemma 3.2 and Remark 3.2 already imply that the perturbation mapping M is Lipschitz-like at the reference point, which is a stronger property than calmness. That is why the calmness assumption is a much weaker constraint qualification than BCQ in general, see Proposition MPCCs in Banach spaces In this section, we take a closer look at MPCCs in Banach spaces which can be stated as given below: h(x) min a(x) C A(x) K B(x) K A(x), B(x) = 0. (MPCC) Therein, h: X R, a: X Y, A: X Z, and B : X Z are continuously Fréchet differentiable mappings between real Banach spaces X, Y, and Z such that Z is reflexive. Furthermore, C Y is a nonempty, closed, convex set, while K Z is a nonempty, closed, convex cone. Note that (MPCC) is a quite irregular problem. The constraint qualification KRZCQ fails to hold at any feasible point of it, cf. [18, Lemma 3.1]. Let x be a feasible point of (MPCC). Then the tangent cone to the feasible set of (MPCC) at x turns out to be generally nonconvex, while the corresponding linearized tangent cone is convex. Hence, we cannot expect ACQ to be satisfied at x in general. As it was shown in [9] by means of a simple example, GCQ may be satisfied at a feasible point of (MPCC) under appropriate assumptions on the given problem. Hence, the KKT conditions of (MPCC) at x, i.e., the existence of λ Y, µ Z, and ν Z which satisfy 0 = h ( x) + a ( x) [λ] + A ( x) [µ] + B ( x) [ν], (10a) λ T C (a( x)), (10b) µ R K (B( x)) A( x), (10c) ν R K (A( x)) B( x), (10d) see [22, Lemma 5.1], may hold. Nevertheless, these conditions turn out to be too strong to yield applicable necessary optimality conditions for (MPCC) in general. Here we recall the concepts of weak and strong stationarity for (MPCC), recently introduced in [18] and [22], respectively. Afterwards, we define and study a generalized notion of M-stationarity which turns out to be reasonable w.r.t. common finite-dimensional, semidefinite, and polyhedral MPCCs. A feasible point x of (MPCC) is called strongly stationary if there are multipliers λ Y, µ Z, and ν Z which satisfy (10a), (10b), as well as µ K K (B( x), A( x)), (11a) ν K K (A( x), B( x)). (11b)

15 12 Susanne Franke, Patrick Mehlitz, Maria Pilecka Observe that the conditions (11a) and (11b) are weaker than the corresponding conditions (10c) and (10d), respectively. Hence, any KKT point of (MPCC) is strongly stationary, but the converse implication, which holds for common MPCCs in finite dimensions, is generally not satisfied. It was shown in [22] that the strong stationarity conditions provide a necessary optimality condition of reasonable strength whenever the cone K is polyhedric. In the absence of polyhedricity, one could use linearization approaches in order to obtain better optimality conditions than strong stationarity, see [22, Section 6] and [23]. A feasible point x of (MPCC) is called weakly stationary whenever there exist multipliers λ Y, µ Z, and ν Z such that (10a) and (10b) as well as µ cl ( K K A( x) ) A( x), (12a) ν cl ( K K B( x) ) B( x) (12b) are satisfied. Note that any strongly stationary point is also weakly stationary, see [18, Lemma 3.4]. It was shown in [22, Proposition 5.2] that a local minimizer x of (MPCC) is strongly stationary whenever the linear operator (a ( x), A ( x), B ( x)) is surjective. Observe that this condition is a stronger version of MPCC-LICQ (cf. [9, Definition 4.4]) for common finite-dimensional MPCCs. We state the following result which is a direct consequence of [22, Lemma 5.1, Proposition 5.2] and the definition of the critical cones appearing in the strong stationarity conditions. Corollary 4.1 Let x X be a local optimal solution of (MPCC) where (a ( x), A ( x), B ( x)) is a surjective operator, whereas the cones R K (A( x)) and R K (B( x)) are closed. Then x is a KKT point of (MPCC). Remark 4.1 Let x X be a feasible point of (MPCC). Observe that in the case where Z is infinitedimensional, the cones R K (A( x)) and R K (B( x)) are barely closed. A similar situation appears for semidefinite MPCCs, see [3, Section 5.3.1]. On the other hand, the closedness assumption is always satisfied if K is a polyhedral cone, see proof of Lemma 2.4. Note that this closedness assumption implies that the strong stationarity and KKT conditions are equivalent which, as we already mentioned above, is not the case for general MPCCs. Recall that a feasible point x of a common finite-dimensional MPCC (i.e. X = R n, Y = R m, Z = R k, C = R m,+ 0, K = R k,+ 0 ) is called M-stationary if there are λ R m, µ, ν R k which solve the system 0 = h ( x) + m j=1 λ ja j( x) + k i=1 µ ia i( x) + k i=1 ν ib i( x), λ 0, λ a( x) = 0, i I +0 ( x): µ i = 0, i I 0 ( x): ν i = 0, i I 00 ( x): µ i ν i = 0 (µ i < 0 ν i > 0). Therein, the index sets I +0 ( x), I 0 ( x), and I 00 ( x) are defined as stated below: I +0 ( x) := {i {1,..., k} A i ( x) > 0 B i ( x) = 0}, I 0 ( x) := {i {1,..., k} A i ( x) = 0 B i ( x) < 0}, I 00 ( x) := {i {1,..., k} A i ( x) = 0 B i ( x) = 0}. Observe that the last three conditions in (13) are equivalent to (µ, ν) N gph TK ( ) (A( x), B( x)), where T K ( ) denotes the normal cone mapping generated by the convex cone K, i.e., the mapping x T K (x). This observation gives rise to the following equivalent reformulation of (MPCC): h(x) min a(x) C (A(x), B(x)) gph T K ( ). Indeed, this problem is equivalent to (MPCC) since we have (u, v) gph T K ( ) u K v K v u. (13)

16 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 13 Now we introduce dummy variables in order to state (MPCC) as a problem of type (3) with a nonconvex set Ω: This justifies the following definition. h(x) min x,u,v a(x) C A(x) u = 0 B(x) v = 0 (u, v) gph T K ( ). Definition 4.1 A feasible point x X of (MPCC) is called M-stationary if there exist multipliers λ Y, µ Z, and ν Z which satisfy (10a), (10b), and (µ, ν) N gph TK ( ) (A( x), B( x)). Observe that without a specific setting this definition is barely applicable since the appearing basic normal cone is usually difficult to compute. However, as presented above, if a common finite-dimensional MPCC is considered, then this generalization of M-stationarity equals the already existing notion. Furthermore, in the setting of semidefinite complementarity programming, the notion of M-stationarity is derived in a similar way, see [8]. A formula for the corresponding basic normal cone to the graph of the normal cone mapping for the cone of all positively semidefinite symmetric matrices is presented in [8, Theorem 3.1]. Here, we want to comment on the relationship between weak, M-, and strong stationarity. From common finite-dimensional or semidefinite MPCCs, we expect strong stationarity = M-stationarity = weak stationarity (15) for any feasible point of (MPCC). However, as mentioned earlier, without a deeper knowledge of K it is difficult to verify these relations in general. In the upcoming lemma, we show that strong stationarity implies M-stationarity whenever the corresponding complementarity cone K is polyhedric. Lemma 4.1 Let x X be a feasible point of (MPCC) where the complementarity cone K is polyhedric w.r.t. (A( x), B( x)). Then N gph TK ( ) (A( x), B( x)) = K K (B( x), A( x)) K K(A( x), B( x)) is satisfied. Particularly, if x is a strongly stationary point of (MPCC), it is M-stationary as well. Proof We proceed in two steps. First, we show w T gph TK ( ) (A( x), B( x)) Tgph T K ( ) (A( x), B( x)) conv ( T gph TK ( ) (A( x), B( x))). (16) The first of these inclusions is clear because of the definition of the appearing tangent cones. Hence, we verify the second one. Therefore, choose (u, v) Tgph w T K ( ) (A( x), B( x)) arbitrarily. Then there are sequences {(u k, v k )} gph T K ( ) and {t k } R satisfying u k A( x), v k B( x), t k 0, and (14) u k A( x) t k u, v k B( x) t k v. Obviously, we have u k A( x) t k R K (A( x)) and v k B( x) t k R K (B( x)) for all k N. Hence, by Mazur s theorem and the convexity of K and K, we obtain u T K (A( x)) and v T K (B( x)). Now choose some w K B( x). Then we have w u k, v k B( x) t k and passing to the limit k yields w u k, B( x) t k = A( x) u k, B( x) t k = w A( x), v u, B( x) 0. uk A( x) t k, B( x) 0,

17 14 Susanne Franke, Patrick Mehlitz, Maria Pilecka Using w := A( x), we obtain u B( x) and, hence, u K K (A( x), B( x)) holds. Moreover, we derive v A( x) by fixing w := 0 and w := 2A( x). Consequently, v K K (B( x), A( x)) is satisfied. From [23, Lemma 3.11], we have K K (A( x), B( x)) K K (B( x), A( x)) = conv ( T gph TK ( ) (A( x), B( x))) which finally yields the second inclusion. Now we simply polarize (16). From M = conv(m) for arbitrary sets M, we obtain w N gph TK ( ) (A( x), B( x)) = Tgph T K ( ) (A( x), B( x)) = K K (A( x), B( x)) K K (B( x), A( x)) = K K (B( x), A( x)) K K (A( x), B( x)) where the last equality holds due to the polyhedricity of K w.r.t. (A( x), B( x)), see [22, Lemma 5.2]. This completes the proof. Note that the polyhedricity assumption is not necessary for strong stationarity to imply M-stationarity since this relation is satisfied for semidefinite MPCCs as well and the positive semidefinite cone is not polyhedric. Now we present a general approximation of the basic normal cone to the complementarity set gph T K ( ) from above. However, this approximation turns out to be even larger than the multiplier set used in the definition of weak stationarity. Lemma 4.2 Let x X be a feasible point of (MPCC). Then the following formula holds true: N gph TK ( ) (A( x), B( x)) A( x) B( x). Proof Choose (µ, ν) N gph TK ( ) (A( x), B( x)) arbitrarily. We find sequences {(u k, v k )} gph T K ( ) and {(µ k, ν k )} N gph TK ( ) (u k, v k ) such that (u k, v k ) (A( x), B( x)) and (µ k, ν k ) (µ, ν) hold. Observe that for any k N we obtain u u k, µ k + ν k, v v k lim sup 0. (u,v) (u k,v k ), u u k Z + v v k Z (u,v) gph T K ( ) Choosing v l := v k, defining u l := (1 + 1 l )u k, and considering l, we arrive at u k, µ k 0. On the other hand, the choices v l := v k and u l := (1 1 l )u k yield u k, µ k 0 when l. Hence, we derive u k, µ k = 0, and, similarly, we obtain ν k, v k = 0. Since {µ k } is weakly convergent, it is bounded which is why we have and, hence, lim u k A( x), µ k lim u k A( x) Z µ k Z = 0, k k A( x), µ = lim A( x), µ ( k = lim A( x) uk, µ k + u k, µ k ) = lim u k, µ k = 0 k k k is satisfied. A similar argumentation yields ν, B( x) = 0. In the next lemma, we show that for polyhedral complementarity cones, the relations in (15) are valid. Lemma 4.3 Let x X be a feasible point of (MPCC) where the complementarity cone K is given as stated in (1). Then the implications from (15) hold for x.

18 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 15 Proof Note that due to Lemma 2.4, the cone K is polyhedric. Hence, by Lemma 4.1, strong stationarity implies M-stationarity. Consequently, we only need to show N gph TK ( ) (A( x), B( x)) cl( K K A( x) ) A( x) cl ( K K B( x) ) B( x). (17) First, we define the index sets I := I(A( x), B( x)) and J := J(A( x), B( x)) as stated in (2). By means of Lemma 2.3, it is sufficient to show cl ( K K A( x) ) A( x) = lin({z i i I}), cl ( K K B( x) ) B( x) = {z i i J}. We start with the verification of the first equality. Therefore, observe K A( x) = cone({z i i I}). This leads to K K A( x) = cone({z i i {1,..., k} \ I}) + lin({z i i I}) which is a closed set by means of [3, Proposition 2.201]. Hence, we obtain cl ( K K A( x) ) A( x) = (K K A( x) ) A( x) = lin({z i i I}), i.e., the first assertion is correct. In order to prove the second one, we show both inclusions separately. Therefore, we first observe This yields K B( x) = {z Z z, z i 0 i {1,..., k} \ J} {z i i J}. Since the set on the right is closed, we have K K B( x) {z Z z, z i 0 i J}. cl ( K K B( x) ) B( x) {z Z z, z i 0 i J} B( x) = {z i i J}. On the other hand, the set {z i i J} is obviously contained in B( x). Moreover, we obtain {z i i J} K B( x) K B( x) K K B( x) cl ( K K B( x) ) which shows the other inclusion, i.e. the second equation is satisfied as well. From Lemma 2.3 and the proof of the above lemma, we obtain the following corollary which states explicit representations of strong, M-, and weak stationarity for MPCCs whose complementarity cone is polyhedral. Corollary 4.2 Let x X be a feasible point of (MPCC) where the complementarity cone K is given as stated in (1). Let I := I(A( x), B( x)) and J := J(A( x), B( x)) be the index sets defined in (2). Then x is 1. weakly stationary iff there exist λ Y, µ 1,..., µ k R, and ν Z which solve the system 0 = h ( x) + a ( x) [λ] + k i=1 µ ia ( x) [zi ] + B ( x) [ν], (18a) λ T C (a( x)), (18b) i {1,..., k} \ I : µ i = 0, i J : ν, z i = 0; (18c) (18d) 2. M-stationary iff there exist multipliers λ Y, µ 1,..., µ k R, and ν Z as well as a partition {I 1, I 2, I 3 } of I \ J which satisfy the conditions in (18) and i I 1 : µ i = 0, i I 2 : ν, z i = 0, i I 3 : µ i > 0 ν, z i < 0; (19a) (19b) (19c) 3. strongly stationary iff there exist multipliers λ Y, µ 1,..., µ k R, and ν Z which satisfy the conditions in (18) and i I \ J : µ i 0 ν, z i 0. (20)

19 16 Susanne Franke, Patrick Mehlitz, Maria Pilecka Observe that the conditions (19) can be equivalently represented by i I \ J : ν, µ i z i = 0 (µ i > 0 ν, z i < 0). Recalling the standard finite-dimensional MPCC, where Z = R k and K = R k,+ 0 hold, we have K = {z R k z, e i 0 i {1,..., k}}. Therein, e i R k denotes the i-th unit vector in R k. Hence, it is easily seen that the aformentioned notion of M-stationarity for standard MPCCs coincides with the M-stationarity notion w.r.t. an MPCC whose complementarity cone is polyhedal. In [2], the authors discuss a linear problem with conic constraints comprising the nonpolyhedral (and nonpolyhedric) second-order cone K k R k+1 given by K k = {(x, t) R k R t x 2 } where 2 denotes the Euclidean norm in R k. They use a polyhedral approximation K of the cone K k in order to simplify the given conic problem. It is shown that this approximation is reasonably good under mild assumptions. Transferring this idea to (MPCC), one could think of approximating the possibly nonpolyhedral complementarity cone by a polyhedral one in certain situations to obtain a surrogate MPCC of a type studied in Lemma 4.3 and Corollary 4.2. How this can be done and how the two problems are related is, however, beyond the scope of this paper and a question of future research. Recalling semidefinite MPCCs, we strongly believe that the relations in (15) hold in much more general situations than the ones described in Lemmata 4.1 and 4.3. However, a detailed study of M- stationarity is beyond the scope of this paper and a topic of future research as well. We want to close this section with the presentation of general situations where M-stationarity is a necessary optimality condition for (MPCC). Applying Proposition 3.2 to (14), we obtain the following result. Proposition 4.1 Let x X be a a local optimal solution of (MPCC) such that gph T K ( ) is SNC at (A( x), B( x)), and let X be reflexive. Furthermore, let a ( x) be surjective, while the following constraint qualification holds: 0 = a ( x) [λ] + A ( x) [µ] + B ( x) } [ν], λ T C (a( x)), (µ, ν) N gph TK ( ) (A( x), B( x)) = µ = 0, ν = 0. (21) Then x is an M-stationary point of (MPCC). Let X and Y be reflexive. Be aware that the constraint qualification (21) is satisfied for some feasible point x X of (MPCC) whenever the linear operator (a ( x), A ( x), B ( x)) has a dense range since in this case, its adjoint is injective, see Remark 2.2. This condition may be strictly weaker than the surjectivity of (a ( x), A ( x), B ( x)) if at least one of the spaces Y or Z is infinite-dimensional. Clearly, the SNC property postulated above holds whenever the space Z is finite-dimensional. On the other hand, we have to mention that it is pretty restrictive in the infinite-dimensional setting as presented in [16] for the Sobolev space H 1 (D) where D R n is some bounded domain. 5 Optimality conditions for the SCBPP Now we want to deduce optimality conditions for (SCBPP) using some reformulations of the problem. Therefore, we first use an optimal value approach where we replace the lower level problem by two new upper level constraints ensuring lower level feasibility and lower level optimality by bounding the lower level objective function value appropriately from above. On the other hand, we can replace the lower level problem by its KKT conditions provided the set Θ is stated in a certain form. Due to the results in [7], it seems necessary to study the relationship between the original and the surrogate problem. In order to characterize the solutions of (SCBPP), one first needs to ensure that this convex optimization problem possesses a solution. Therefore, we included the subsequent theorem. Recall that the general assumptions on (SCBPP) only comprise that F and f are convex and continuous, while Θ is nonempty, closed, and convex.

20 Optimality conditions for the simple convex bilevel programming problem in Banach spaces 17 Theorem 5.1 Problem (SCBPP) possesses an optimal solution if X is reflexive, while Θ is bounded or f is coercive. Proof Suppose that Θ is bounded, while X is reflexive. Then Θ is weakly sequentially compact, see [15, Appendix B]. Since f is weakly lower semicontinuous, see [15, Lemma 2.11], Ψ is nonempty, see [15, Theorem 2.3]. Due to the convexity and continuity of f, Ψ is convex and closed. Moreover, Ψ is bounded since Θ is bounded. Consequently, Ψ is a nonempty, weakly sequentially compact set. Finally, F is weakly lower semicontinuous, i.e., (SCBPP) possesses a solution. Now we assume that X is reflexive, while f is coercive. Observe that due to the nonemptiness of Θ, there exists some x Θ and the set Θ := {x Θ f(x) f( x)} is nonempty. Furthermore, since f is convex and continuous, Θ is convex and closed. Finally, the coercivity of f yields the boundedness of Θ. Observe that we have Ψ = Argmin{f(x) x Θ}. Now the first situation described in this proof applies. 5.1 Optimal value approach In this section, we assume that F and f are continuously Fréchet differentiable. Suppose that the set Ψ is nonempty. Then we denote the optimal value of the lower level problem by α R, i.e., α := f(x ) for arbitrary x Ψ. Consequently, we have Ψ = {x Θ f(x) α 0}, and (SCBPP) is equivalent to F (x) min f(x) α 0 x Θ (22) which is a standard nonlinear but convex optimization problem in Banach spaces. Slater s constraint qualification generally fails to hold everywhere, since there is no feasible point which satisfies the first constraint in (22) as a strict inequality. Checking the standard constraint qualifications, we easily obtain the following result. Lemma 5.1 KRZCQ fails to hold at any feasible point of (22). Proof Let x X be a feasible point of (22). Hence, it follows from x Ψ that f ( x)[d] 0 holds for all d T Θ ( x), see [15, Theorem 4.14]. Furthermore, we have f( x) = α. This leads to i.e., KRZCQ fails to hold at x. f ( x)[r Θ ( x)] R R + (0) R R+ 0 = R+ 0, 0 On the other hand, the constraint qualification ACQ may be satisfied at a feasible point of (SCBPP), as the following example illustrates. Example 5.1 Consider X := R 2, Θ := [0, 1] 2, and F, f : R 2 R given as stated below: x = (x 1, x 2 ) R 2 : F (x) := x x 2 2, f(x) := (x 1 2) 2. Then we have Ψ = {1} [0, 1], and the global optimal solution of the corresponding (SCBPP) is x = (1, 0). Observe that we have T Ψ ( x) = cone({(0, 1)}) and L Ψ ( x) = {(d 1, d 2 ) cone({( 1, 0), (0, 1)}) 2d 1 0} = cone({(0, 1)}), S Ψ ( x) = {( 2λ + ξ 1, ξ 2 ) R 2 λ 0, (ξ 1, ξ 2 ) cone({(1, 0), (0, 1)})} = R ( R + ) 0. Consequently, ACQ holds at x. The latter observation yields that there are situations where the KKT conditions of (22) may be necessary and sufficient optimality conditions since ACQ (and, hence, GCQ) may hold for (22). Theorem 5.2 Let x X be a feasible point of (SCBPP) where GCQ holds. Then x is an optimal solution of (SCBPP) iff the following condition is satisfied: λ 0 ξ T Θ ( x) : 0 = F ( x) + λ f ( x) + ξ. (23)