Preprint Patrick Mehlitz, Gerd Wachsmuth Weak and strong stationarity in generalized bilevel programming and bilevel optimal control

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1 Fakultät für Mathematik und Informatik Preprint Patrick Mehlitz, Gerd Wachsmuth Weak and strong stationarity in generalized bilevel programming and bilevel optimal control ISSN

2 Patrick Mehlitz, Gerd Wachsmuth Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 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 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control Patrick Mehlitz Gerd Wachsmuth Abstract In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints MPCC. We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented. Keywords Bilevel Programming Programming in Banach Spaces Mathematical Program with Complementarity Constraints Stationarity Bilevel Optimal Control Mathematics Subject Classification 2 46N1 49K15 9C33 9C48 1 Introduction It is well-known in optimization theory that mathematical programs with complementarity constraints MPCCs are difficult to handle since all constraint qualifications of reasonable strength fail to hold at any feasible point. This applies to both, the finite-dimensional and the infinite-dimensional situation. On the other hand, such problems arise frequently in many different applications. Hence, in the past a huge effort was put in constructing suitable regularity conditions, stationarity concepts, and relaxation techniques, which are applicable to MPCCs in theoretical and numerical procedures, cf. 16, 22, 24, 25, 28 and the references therein. Mainly, this analysis was done in finite-dimensional spaces. To the best of our knowledge, 26 is the first contribution which introduced a concept of strong stationarity applicable to a generalized MPCC in reflexive Banach spaces. The consideration of this abstract setting is rather important from the practical perspective since, e.g., there exist many dynamic optimization problems resulting in optimal control problems with complementarity constraints. Especially, when considering bilevel optimal control problems, it is possible to replace the original hierarchical optimization problem by a single-level optimal control problem which possesses a complementarity constraint provided the lower level problem is regular, convex, and equipped with pure state, mixed, or pure control inequality constraints. After mentioning some notations and preliminary results in Section 2 we are going to recall the generalized concept of strong stationarity and introduce weak stationarity for MPCCs in Section 3. Afterwards, P. Mehlitz Technische Universität Bergakademie Freiberg, Faculty of Mathematics and Computer Science, 9596 Freiberg, Germany mehlitz@math.tu-freiberg.de G. Wachsmuth Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Methods Partial Differential Equations, 917 Chemnitz, Germany gerd.wachsmuth@mathematik.tu-chemnitz.de

5 2 Patrick Mehlitz, Gerd Wachsmuth we apply these concepts to a bilevel programming problem in Banach spaces whose lower level problem is regular and convex in Section 4. We present regularity conditions, which imply that these stationarity conditions hold at local minimizers of the original bilevel programming problem. Furthermore, we specify the results for finite-dimensional problems in order to show, that our concepts are an appropriate generalization. Section 5 is dedicated to the study of a certain bilevel optimal control problem with pure control inequality constraints in the lower level. After a brief introduction to bilevel optimal control we apply the aforementioned theory in order to derive necessary optimality conditions of Pontryagin-type for the latter problem. It will be shown that the controllability of a linear autonomous system is sufficient for any local minimizer of the given bilevel optimal control problem to be strongly stationary. 2 Notation and preliminary results We use N, N +, R, R n, R n,+, and R n m to denote the natural numbers with zero, the positive natural numbers, the real numbers, 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. For an arbitrary matrix Q R n m, Q denotes its transpose. Furthermore, we use R n, O R n m, and I R n n to represent the zero vector, the zero matrix, and the identity matrix of appropriate dimensions, respectively. Let X be a real reflexive Banach space with norm X and zero vector o X, and let A X be a nonempty subset of X. We denote by lina, conea, conva, and cla the smallest subspace of X containing A, the conic hull of A, the convex hull of A, and the closure of A, 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 as well. Then we define the polar cones and annihilators of the sets A and B by A := {x X x A: x, x } A := {x X x A: x, x = } B := {x X x B : x, x } B := {x X x B : x, x = }, respectively. Note that due to the reflexivity of X, the above notation is consistent. For the purpose of simplicity, we leave curly brackets away when considering singletons, i.e. we set linx := lin{x}, conex := cone{x}, x := {x}, and x := {x} for any x X and similar definitions shall hold for all x X. For a closed, convex set C X and a fixed vector x C, we define the cone of feasible directions and the Bouligand tangent cone to C at x by R C x := conec {x} and T C x := clr C x, respectively. Moreover, if x T C x is chosen, then the critical cone to C with respect to w.r.t. x and x is defined by In case that C is additionally a cone, we have K C x, x := T C x x. R C x = C + linx T C x = clc + linx T C x = C x. 1 The next lemma summarizes some properties of the above operators. Lemma 2.1 Let K 1, K 2 X be nonempty, closed, convex cones. Then we have K 1 = K 1 2a K 1 + K 2 = K 1 K 2 2b K 1 K 2 = clk 1 + K 2 2c K 1 = K 1 K 1 2d K 1 = K 1 K 1. 2e

6 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 3 Proof For the statements 2a, 2b, and 2c, we refer to 5, Proposition 2.4, 2.31, and 2.32, respectively. Using 2a, we obtain 2d by K = {x X x K 1 : x, x = } = {x X x K 1 : x, x } {x X x K 1 : x, x } = K 1 K 1 = K 1 K 1. A similar argument yields 2e. Additionally, observe that linx = x = x holds for an arbitrary point x X. We stipulate that throughout the whole paper the intersection is a stronger operation than the Minkowski sum, i.e. S 1 + S 2 S 3 is to be read as S 1 + S 2 S 3, for arbitrary subsets S 1, S 2, and S 3 of a linear space. By a simple calculation, we obtain the following lemma. Its proof is straightforward and, hence, omitted. Lemma 2.2 Let A X be a nonempty set, U X be a linear subspace of X, and ξ U. Then we have A + linξ U = A U + linξ. Let U, V, and W be arbitrary Banach spaces. By LU, V we denote the space of bounded linear operators mapping from U to V. For any operator F LU, V, the operator F LV, U denotes the adjoint of F. Suppose that θ : U V W is a twice Fréchet differentiable mapping and û, ˆv U V is fixed. Then the linear operators θ û, ˆv LU V, W, θ uû, ˆv LU, W, θ 2 û, ˆv LU V, LU V, W, θ uu 2 û, ˆv LU, LU, W, and θ uv 2 û, ˆv LV, LU, W denote its first order Fréchet derivative, partial first order Fréchet derivative w.r.t. u, second order Fréchet derivative, partial second order Fréchet derivative w.r.t. to u, and mixed partial second order Fréchet derivative w.r.t. u and v at û, ˆv, respectively. As it is common in literature, we will identify any second order Fréchet derivative with a bounded bilinear form. Suppose U = R n, V = R m as well as W = R k, choose an arbitrary point û, ˆv R n R m, and let ϑ: R n R m R k be differentiable. Then the real matrices ϑû, ˆv R k n+m and u ϑû, ˆv R k n denote the Jacobian and the partial Jacobian w.r.t. u of ϑ at û, ˆv. For k = 1 these matrices are rows and represent the transposed gradient and the partial transposed gradient w.r.t. u of ϑ at û, ˆv. Assume that k = 1 holds true, while ϑ is twice differentiable. Then the matrices 2 ϑû, ˆv R n+m n+m, 2 uuϑû, ˆv R n n, and 2 uvϑû, ˆv R n m denote the Hessian, the Hessian w.r.t. u and the mixed Hessian w.r.t. u and v of ϑ at û, ˆv, respectively. One may identify these matrices with the Fréchet derivatives introduced earlier. Recall that a point û U is called a Karush-Kuhn-Tucker-point, KKT-point for short, of the optimization problem min{hu Hu C}, 3 where h: U R, H : U V are continuously Fréchet differentiable and C V is a nonempty, closed, and convex set, provided it satisfies Hû C and there exists a multiplier ω T C Hû such that h û + H û ω = o U holds. The latter conditions are called KKT-conditions of 3. Furthermore, the Kurcyusz-Robinson-Zowe-Constraint-Qualification, KRZCQ for short, is said to be valid at a feasible point u U of 3 whenever H uu R C Hu = V is satisfied. The following results and their proofs are well-known in the theory of optimization in Banach spaces, see 31, Theorems 3.1 and 4.1 or 5, Theorem 3.9 and Proposition Lemma 2.3 Let û U be a feasible point of 3 where KRZCQ is satisfied. 1. If û is a local optimal solution of 3, then û is a KKT-point of The nonempty, closed, convex cone Λ û := { ω T C Hû H û ω = o U }, which is called the set of singular Lagrange multipliers of 3 at û, contains only the zero vector o V.

7 4 Patrick Mehlitz, Gerd Wachsmuth 3 MPCCs in Banach spaces 3.1 Formulation of the problem and failure of KRZCQ In this section, we are going to study the complementarity constrained optimization problem ax min bx cx dx K C C cx, dx =. Therein, a: X R, b: X W, c: X Z, and d: X Z are continuously Fréchet differentiable mappings between reflexive Banach spaces X, W, and Z. The set K W is nonempty, closed, and convex, while C Z is a nonempty, closed, convex cone. For an arbitrary point ˆx X which satisfies cˆx C and dˆx C, we already have cˆx, dˆx. Hence, the last three constraints in 4 are called complementarity constraint. That is why 4 is often referred to as mathematical program with complementarity constraint, abbreviated by MPCC. Let us define the set M X which contains all feasible points of the MPCC 4 by means of M := {x X bx K, cx C, dx C, cx, dx = }. The next lemma yields that KRZCQ fails to hold at any point of M. Lemma 3.1 The constraint qualification KRZCQ is violated at any feasible point ˆx M of 4. Proof We proceed by showing that there exist nonzero singular Lagrange multipliers at ˆx. By means of Lemma 2.3, this shows the failure of KRZCQ. The quadruple ϱ, ν, µ, ξ T K bˆx C C R is a singular Lagrange multiplier of ˆx M provided o X = b ˆx ϱ + c ˆx ν + d ˆx µ + ξ c ˆx dˆx + d ˆx cˆx = cˆx, ν = µ, dˆx is satisfied. The above relation can be transformed into o X = b ˆx ϱ + c ˆx ν + ξ dˆx + d ˆx µ + ξ cˆx = cˆx, ν = µ, dˆx =. Observe that ϱ := o W, ν := dˆx, µ := cˆx, and ξ := 1 satisfy all these conditions due to the complementarity condition cˆx, dˆx =. Hence, ϱ, ν, µ, ξ is a nonvanishing singular Lagrange multiplier of 4, i.e. KRZCQ fails to hold at ˆx M according to Lemma Auxiliary problems and stationarity concepts In this section, we present the definitions of weak and strong stationarity for the MPCC 4 as defined in 26, Section 5.1. Moreover, we give some constraint qualifications CQs which imply that these stationarity concepts are necessary optimality conditions. To this end, we follow 26 and introduce auxiliary problems without a complementarity constraint, which can be used in order to define the stationarity concepts for problem 4. The definition of these auxiliary problems parallels the corresponding definition in the finite-dimensional situation, see 26, Section 5.1. Let ˆx M be an arbitrary feasible point of 4. We define the relaxed nonlinear problem, abbreviated by RNLP, of 4 at ˆx by ax min bx cx K C dˆx dx C cˆx. 5

8 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 5 Obviously, ˆx is a feasible point of 5 but an arbitrary feasible point x X of the latter problem may not satisfy the complementarity condition cx, dx =. On the other hand, a feasible point of 4 is, in general, not feasible for 5. In order to guarantee that the complementarity constraint cx, dx = holds, we have to modify 5 by shrinking its feasible set. Therefore, we introduce the tightened nonlinear problem, TNLP for short, of 4 at ˆx by ax min bx K cx C dˆx C cˆx dx C cˆx C dˆx. 6 The feasible set of 6 is a subset of the feasible set M of 4 and contains ˆx. Hence, if ˆx M is a local minimizer of 4, it is a local minimizer for 6. Consequently, under suitable regularity conditions it is a KKT-point of 6. In order to characterize KKT-points of 6, we need to calculate the polar cones of C dˆx C cˆx and C cˆx C dˆx. We summarize the corresponding calculations in the next lemma. Lemma 3.2 Let ˆx M be a feasible point of the MPCC 4. Then lin dˆx lin C cˆx 7a lin cˆx lin C dˆx 7b C dˆx C cˆx = cl C C cˆx 7c C cˆx C dˆx = cl C C dˆx. 7d Proof From the feasibility of ˆx X we deduce dˆx C and cˆx, dˆx =, i.e. dˆx cˆx. This leads to dˆx C cˆx. 7a follows from the monotonicity of the operator lin. Analogously, we obtain 7b. Let us show 7c. Observe that due to the inclusion lindˆx linc cˆx, the equality lindˆx + lin C cˆx = lin C cˆx holds. Due to the property of C to be a convex cone, we have C + C cˆx = C. Finally, recall that for any convex cone C the relation linc = C C is satisfied. Putting these three observations together, we obtain C dˆx C cˆx = cl C + lindˆx + linc cˆx = cl C + linc cˆx = cl C + C cˆx C cˆx = cl C C cˆx. Statement 7d follows similarly. According to 26, Section 5.1, we define the weak and strong stationarity conditions for 4 by the KKT-conditions of the TNLP 6 and of the RNLP 5, respectively. Definition 3.1 Let ˆx M be an arbitrary feasible point of the MPCC The point ˆx is called weakly stationary provided there exist Lagrange multipliers ϱ T K bˆx, ν cl C C cˆx, and µ clc C dˆx which solve the system o X = a ˆx + b ˆx ϱ + c ˆx ν + d ˆx µ = cˆx, ν = µ, dˆx The point ˆx is called strongly stationary provided there exist Lagrange multipliers ϱ T K bˆx, ν T C dˆx, and µ T C cˆx which solve system 8.

9 6 Patrick Mehlitz, Gerd Wachsmuth From the theory of finite-dimensional MPCCs we expect any strongly stationary point of 4 to be weakly stationary as well. The following lemma shows that this conjecture is correct. Lemma 3.3 If ˆx M is a strongly stationary point of 4, then it is a weakly stationary point of 4 as well. Proof Since dˆx C cˆx holds, we have C + lindˆx C C cˆx as well. Consequently, T C dˆx = cl C + lindˆx cl C C cˆx is satisfied. Similarly, we can prove T C cˆx = cl C + lincˆx cl C C dˆx. This shows the claim. It has been presented in 26 that the above notion of strong stationarity is an appropriate generalization of the finite-dimensional concept in the case where the cone C is polyhedral. The next lemma shows that the definition of weak stationarity generalizes its finite-dimensional counterpart in a convincing way. We refer the reader to 25, 29 for several stationarity concepts besides strong and weak stationarity for finite-dimensional MPCCs. Furthermore, in 29 the author gives an overview of existing constraint qualifications for such problems. Lemma 3.4 Let X = R n, W = R m, Z = R k, K = R m1,+ {}, and C = R k,+ be given. Here, m = m 1 + m 2 shall hold for m 1, m 2 N. We choose a feasible point ˆx M of the corresponding MPCC 4. Let c 1,..., c k, d 1,..., d k : R n R denote the component functions of c and d, respectively. Moreover, we introduce the index sets I + ˆx := {i {1,..., k} c i ˆx > d i ˆx = } I ˆx := {i {1,..., k} c i ˆx = d i ˆx = } I + ˆx := {i {1,..., k} c i ˆx = d i ˆx < }. The point ˆx is weakly stationary in the sense of Definition 3.1 if and only if there exist multipliers ϱ R m and ν, µ R k which solve the system = aˆx + bˆx ϱ + cˆx ν + dˆx µ l {1,..., m 1 }: ϱ l bˆx ϱ = i I + ˆx: ν i = i I + ˆx: µ i =. 9 Proof Due to the definition of weak stationarity and R k,+ = R k,+, we only have to verify R k,+ cˆx R k,+ cˆx = {ν R k i I + ˆx: ν i = } R k,+ R k,+ dˆx dˆx = {µ R k i I + ˆx: µ i = } the closedness of the sets on the left-hand side is obvious. We just show the first equation, the second one can be handled analogously. This first equality, however, follows from and Hence, the proof is completed. R k,+ cˆx R k,+ cˆx = R k,+ cˆx R k,+ cˆx R k,+ cˆx = {ν R k,+ i I + ˆx: ν i = }. The next theorem states qualification conditions which ensure the different notions of stationarity. Therefore, we need to define H : X W Z Z by x X : Hx := bx, cx, dx.

10 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 7 Theorem 3.1 Let ˆx M be a local minimizer of the MPCC Let the convex cone C W Z Z be defined by C := R K bˆx R C cˆx K C cˆx, dˆx R C dˆx K C dˆx, cˆx, and assume that the following qualification condition holds: H ˆxX C = W Z Z. 1 Then ˆx is a weakly stationary point of Assume that H ˆx is a surjective operator. Then ˆx is a strongly stationary point of 4. Proof We show that condition 1 equals KRZCQ for 6. Due to the fact that ˆx is a local minimizer of 6 as well and the weak stationarity conditions equal the KKT-conditions for that problem, Lemma 2.3 yields the proof for the first statement. Observe that if we define cones C 1 Z and C 2 Z by then KRZCQ for 6 takes the form C 1 := C dˆx C cˆx C 2 := C cˆx C dˆx, H ˆxX R K bˆx R C1 cˆx R C2 dˆx = W Z Z. Hence, we just have to show R C1 cˆx = R C cˆx K C cˆx, dˆx and R C2 dˆx = R C dˆx K C dˆx, cˆx. We only verify the first of these equations since the proof of the second one is similar. By 2d and Lemma 2.2 we easily obtain R C1 cˆx = C dˆx C cˆx + lincˆx = C + lincˆx dˆx C cˆx = C + lincˆx dˆx T C cˆx = R C cˆx dˆx T C cˆx T C cˆx = R C cˆx K C cˆx, dˆx. Hence, the proof of the first statement is completed. The second statement is exactly 26, Proposition 5.2. Remark 3.1 Consider a feasible point ˆx M of the MPCC 4 in finite dimensions as introduced in Lemma 3.4. Then the constraint qualification 1 equals MPCC-MFCQ, i.e. for any ϱ R m and any ν, µ R k we have = bˆx ϱ + cˆx ν + dˆx µ l {1,..., m 1 }: ϱ l bˆx ϱ = i I + ˆx: ν i = i I + ˆx: µ i = = ϱ = ν = µ =. As mentioned in 13, MPCC-MFCQ implies a local optimal solution ˆx of 4 to be M-stationary, i.e. there exist ϱ R m and ν, µ R k which satisfy the conditions in 9 as well as i I ˆx: ν i < µ i > ν i µ i =.

11 8 Patrick Mehlitz, Gerd Wachsmuth The latter stationarity condition is stronger than weak stationarity cf. 13. However, it is not clear how to define the concept of M-stationarity for generalized MPCCs. The surjectivity of H ˆx is a sufficient condition for MPCC-LICQ, i.e. the linear independence of the set { b l ˆx l Jˆx} { c i ˆx i I ˆx I + ˆx } { d i ˆx i I + ˆx I ˆx }. Therein, b 1,..., b m : R n R denote the m component functions of b, while the active index set of b at ˆx is defined by means of Jˆx := {l {1,..., m} b l ˆx = }. Note that MPCC-LICQ possesses the equivalent representation = bˆx ϱ + cˆx ν + dˆx µ l / Jˆx: ϱ l = i I + ˆx: ν i = i I + ˆx: µ i = = ϱ = ν = µ =. According to 28, MPCC-LICQ is still a sufficient condition for a local optimal solution ˆx of 4 to be strongly stationary, and we remark that this also follows from 26, Theorem 5.1. Remark 3.2 From the theory of finite-dimensional MPCCs one could expect that strong stationarity and the KKT-conditions of 4 are equivalent. Unfortunately, it has been presented in 26 that strong stationarity is in general a weaker condition than the KKT-conditions of 4 and without a polyhedricity assumption on the cone C, it may happen that the strong stationarity conditions are too weak to yield a good necessary optimality condition cf. 26, Section 5. On the other hand, a linearization of the original problem may improve the situation as presented by means of two examples in 26, Section 6. Although weak stationarity is a weaker condition than strong stationarity cf. Lemma 3.3 and, hence, weaker than the KKT-conditions of the original MPCC cf. Remark 3.2, it is still useful in order to find possible minimizer candidates of the problem 4. Note that the qualification condition presented in 1 is weaker than the surjectivity of the operator H ˆx which was required in order to guarantee that a local minimizer ˆx M of 4 is strongly stationary. 4 Stationarity conditions for bilevel programming problems In this section, we study the bilevel programming problem F x, y min x,y Gx y K u Ψx 11 where Ψ : X 2 Y is the solution set mapping of the parametric optimization problem fx, y min y gx, y K l. 12 Therein, we assume that the Banach spaces X, Y, W, and Z are reflexive, while K u W and K l Z are nonempty, closed, and convex cones. Moreover, we postulate F : X Y R and G: X W to be continuously Fréchet differentiable while f : X Y R and g : X Y Z have to be twice continuously Fréchet differentiable. At any point x, y X Y which is feasible for the lower level problem 12, i.e., which satisfies gx, y K l, we assume that KRZCQ for the lower level shall hold: g yx, yy R Kl gx, y = Z. Finally, for any x X, the mapping fx, : Y R is assumed to be convex, while gx, : Y Z shall be K l -convex 18, Definition 2.4, i.e. for any y, y Y and any σ, 1, we claim gx, σy + 1 σy σ gx, y 1 σ gx, y K l.

12 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 9 The above assumptions guarantee that the KKT-conditions of 12 are necessary and sufficient for global optimality, that is y Ψx λ K l : f yx, y + g yx, y λ = o Y gx, y K l gx, y, λ =. As a result, it is a common approach to replace problem 11 by its KKT-reformulation F x, y min x,y,λ Gx f yx, y + g yx, y λ = gx, y λ K u o Y K l K l gx, y, λ =, 13 which is an MPCC as presented in Section 3. It has been mentioned in 1 that 11 and 13 are equivalent w.r.t. global optimal solutions in a certain sense and that any local optimal solution of 11 corresponds to a local optimal solution of 13. The converse of the last statement is not true in general but can be verified under additional assumptions in finite-dimensional spaces. However, the corresponding proof heavily relies on upper semicontinuity properties of the Lagrange multiplier mapping of 12, and it is a question of future research if this idea can be generalized to infinite-dimensional spaces. We refer to 5, Lemma 4.44 and Proposition 4.47 for results which ensure the upper semicontinuity of the Lagrange multiplier mapping in the infinite-dimensional situation. Furthermore, one may check 9 for a detailed introduction to the theory of bilevel programming. We adapt Definition 3.1 to problem 13 in order to introduce stationarity concepts for the bilevel programming problem 11. Definition 4.1 Let ˆx, ŷ X Y be a feasible point of the bilevel programming problem The point ˆx, ŷ is called weakly stationary for 11 provided there exist multipliers α K u, β Y, λ K l, ν clk l K l gˆx, ŷ, and µ clk l K l λ which solve the system o X = F xˆx, ŷ + G ˆx α + f 2 yx ˆx, ŷ, β + g 2 yx ˆx, ŷ, β λ + g x ˆx, ŷ ν o Y = F yˆx, ŷ + f yy 2 ˆx, ŷ, β + g yy 2 ˆx, ŷ, β λ + g y ˆx, ŷ ν o Y = f yˆx, ŷ + g yˆx, ŷ λ o Z = g yˆx, ŷβ + µ = Gˆx, α = gˆx, ŷ, λ = gˆx, ŷ, ν = µ, λ The point ˆx, ŷ is called strongly stationary for 11 provided there exist multipliers α K u, β Y, λ K l, ν T K l λ, and µ T K l gˆx, ŷ which satisfy the conditions in 14. Observe that a feasible point ˆx, ŷ X Y of 11 is weakly resp. strongly stationary for 11 w.r.t. Definition 4.1 if and only if there is a lower level multiplier λ Z such that ˆx, ŷ, λ is feasible for 13 and a weakly resp. strongly stationary point of that problem w.r.t. Definition 3.1. It is not difficult to check that the above stationarity concepts generalize the corresponding conditions in finite dimensions stated, e.g., in 11. We omit the proofs here since they follow from Lemma 3.4 and 26. Especially, it is possible to eliminate the multiplier µ from the above stationarity definition in finite dimensions. The same is possible in arbitrary reflexive Banach spaces. The corresponding proofs are straightforward and, hence, omitted as well. Remark 4.1 Let ˆx, ŷ X Y be a feasible point of the bilevel programming problem 11.

13 1 Patrick Mehlitz, Gerd Wachsmuth 1. The point ˆx, ŷ is weakly stationary for 11 if and only if there exist multipliers α K u, β Y, λ K l, and ν cl K l K l gˆx, ŷ which satisfy g yˆx, ŷβ cl K l K l λ as well as the following conditions: o X = F xˆx, ŷ + G ˆx α + f 2 yx ˆx, ŷ, β + g 2 yx ˆx, ŷ, β λ + g x ˆx, ŷ ν o Y = F yˆx, ŷ + f yy 2 ˆx, ŷ, β + g yy 2 ˆx, ŷ, β λ + g y ˆx, ŷ ν o Y = f yˆx, ŷ + g yˆx, ŷ λ = Gˆx, α = gˆx, ŷ, λ = gˆx, ŷ, ν = g yˆx, ŷβ, λ The point ˆx, ŷ X Y is strongly stationary for 11 if and only if there exist multipliers α K u, β Y, λ K l, and ν T K l λ which satisfy g yˆx, ŷβ T Kl gˆx, ŷ and the conditions in 15. The construction of constraint qualifications for 11, which ensure that a local optimal solution of the latter problem is weakly or strongly stationary, turns out to be difficult. Such qualification conditions can be expected to depend on the lower level multiplier λ, which was used for the transformation into 13, since any stationarity condition for 11 is in fact a stationarity condition for the KKT-reformulation 13. Hence, most regularity conditions in literature on finite-dimensional bilevel programming problems just correspond to problem 13 and not to 11 directly. However, if the lower level possesses constraints of special affine type, the situation is more comfortable. Proposition 4.1 Let ˆx, ŷ X Y be a local optimal solution of the bilevel programming problem 11 where the mapping g : X Y Z is given by x, y X : gx, y := g 1 x + By. Here, g 1 : X Z is continuously Fréchet differentiable and B LY, Z is a bounded linear operator. Assume that the linear operator Q LX Y, W Y Z defined by Qd x, d y := G ˆxd x, f yx 2 ˆx, ŷd x, + f yy 2 ˆx, ŷd y,, g 1ˆxd x + Bd y for all d x, d y X Y is surjective. Then ˆx, ŷ is a strongly stationary point for the corresponding bilevel programming problem. Proof First, we find some λ Kl such that ˆx, ŷ, λ is a local optimal solution of the corresponding KKT-reformulation 13. Now we are going to apply the second statement of Theorem 3.1. Hence, we introduce H : X Y Z W Y Z Z by means of Hx, y, z := Gx, f yx, y + B z, g 1 x + By, z for any x, y, z X Y Z. Computing its Fréchet derivative at ˆx, ŷ, λ yields H ˆx, ŷ, λd x, d y, d z = G ˆxd x, f yx 2 ˆx, ŷd x, + f yy 2 ˆx, ŷd y, + B d z, g 1ˆxd x + Bd y, d z for any d x, d y, d z X Y Z. Observe that H ˆx, ŷ, λ does not depend on λ in this special setting. Take an arbitrary point w, y, z, z W Y Z Z, and consider the linear equation H ˆx, ŷ, λd x, d y, d z = w, y, z, z. Obviously, we have d z = z, and the above equation can be reduced to Qd x, d y = w, y B z, z. The latter linear equation possesses a solution since Q is assumed to be surjective. Hence, H ˆx, ŷ, λ is surjective and due to Theorem 3.1 ˆx, ŷ, λ, is strongly stationary for 13, which means by definition that ˆx, ŷ is strongly stationary for the bilevel programming problem 11.

14 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 11 Example 4.1 We fix X = R n, Y = R m, W = R p, Z = R q, K u = R p,+, and K l = R q,+ as well as Gx := Cx c fx, y := 1 2 y Ry + y Px gx, y := Ax + By d for all x, y R n R m and given matrices of appropriate dimension such that R is a symmetric and positively semidefinite matrix. Assume that the matrix C O Q := P R R p+m+q n+m A B possesses full row rank p + m + q note that a necessary condition for that property is p + q n. Then any local optimal solution ˆx, ŷ R n R m of the corresponding bilevel programming problem is strongly stationary, i.e. there exist Lagrange multipliers α R p,+, β R m, λ R q,+, and ν R q which satisfy the following set of conditions: = x F ˆx, ŷ + C α + P β + A ν = y F ˆx, ŷ + Rβ + B ν = Rŷ + Pˆx + B λ = Cˆx c α = Aˆx + Bŷ d λ i I + ˆx, ŷ, λ: ν i = i I + ˆx, ŷ, λ: Bβ i = i I ˆx, ŷ, λ: ν i Bβ i. Therein, the appearing index sets are defined as stated below cf. Lemma 3.4: I + ˆx, ŷ, λ := {i {1,..., q} Aˆx + Bŷ d i < λ i = } I ˆx, ŷ, λ := {i {1,..., q} Aˆx + Bŷ d i = λ i = } I + ˆx, ŷ, λ := {i {1,..., q} Aˆx + Bŷ d i = λ i > }. Note that the constraint qualification, i.e. the rank condition on Q, does not depend on the lower level multiplier λ. We generalize the above example to arbitrary reflexive Banach spaces. Example 4.2 Choose operators A LX, Z, B LY, Z, C LX, W, a continuous, positive semidefinite, symmetric bilinear form R: Y Y R, a continuous bilinear form P : X Y R, and vectors c W as well as d Z such that Gx := Cx c fx, y := 1 Ry, y + Px, y gx, y = Ax + By d 2 holds true for all x, y X Y. If the linear operator Q LX Y, W Y Z defined by Qd x, d y := Cd x, Pd x, + Rd y,, Ad x + Bd y for any d x, d y X Y is surjective, then any local minimizer ˆx, ŷ X Y of the corresponding bilevel programming problem is strongly stationary, i.e. there exist Lagrange multipliers α K u, β Y, λ K l, and ν T K l λ which satisfy Bβ T K l Aˆx + Bŷ d and the following set of conditions: o X o Y o Y = F xˆx, ŷ + C α + P, β + A ν = F yˆx, ŷ + R, β + B ν = Rŷ, + Pˆx, + B λ = Aˆx + Bŷ d, λ = Cˆx c, α = Aˆx + Bŷ d, ν = Bβ, λ. Again we note that the constraint qualification, i.e. the surjectivity of Q, does not depend on the lower level multiplier λ. 16

15 12 Patrick Mehlitz, Gerd Wachsmuth As mentioned above, constraint qualifications for general bilevel programming problems 11 ensuring local optimal solution to be weakly or strongly stationary are likely to depend on the lower level multiplier λ. We underline this statement in the subsequent theorem. Theorem 4.1 Let ˆx, ŷ X Y be a local optimal solution of the bilevel programming problem 11. Then the following statements hold. 1. Let λ Z be chosen such that ˆx, ŷ, λ is a feasible point of 13. Furthermore, suppose that the linear operator Q LX Y, W Y Z defined by Qd x, d y := G ˆxd x, f yx 2 ˆx, ŷd x, + g yx 2 ˆx, ŷd x, λ + f yy 2 ˆx, ŷd y, + g yy 2 ˆx, ŷd y, λ, g ˆx, ŷd x, d y 17 for any d x, d y X Y and the cone C W Y Z given by satisfy the constraint qualification C := R Ku Gˆx { } o Y R Kl gˆx, ŷ K Kl gˆx, ŷ, λ QX Y C = W Y Z. 18 Then ˆx, ŷ is a weakly stationary point of Let λ Z be chosen such that ˆx, ŷ, λ is a feasible point of 13. Furthermore, suppose that the operator Q LX Y, W Y Z defined within 17 is surjective. Then ˆx, ŷ is a strongly stationary point of 11. Proof We start verifying the first statement. From the choice of λ and the definition of weak stationarity for the bilevel programming problem 11 we only have to show that ˆx, ŷ, λ is a weakly stationary point for the MPCC 13. Therefore, we exploit the first statement of Theorem 3.1. In order to do so, we introduce a mapping H : X Y Z W Y Z Z by means of Hx, y, z := Gx, f yx, y + g yx, y z, gx, y, z and a cone C W Y Z Z as mentioned below: C := R Ku Gˆx { o Y } R Kl gˆx, ŷ K Kl gˆx, ŷ, λ R K l λ K K l λ, gˆx, ŷ. From the aforementioned result it is clear that ˆx, ŷ, λ is weakly stationary for 13 provided the constraint qualification H ˆx, ŷ, λx Y Z C = W Y Z Z is satisfied. Hence, we take an arbitrary point w, y, z, z W Y Z Z and show that there is a solution d x, d y, d z X Y Z of the following generalized equation: Observe that H ˆx, ŷ, λd x, d y, d z w, y, z, z C. 19 H ˆx, ŷ, λd x, d y, d z = G ˆxd x, f 2 yx ˆx, ŷd x, + g 2 yx ˆx, ŷd x, λ + f yy 2 ˆx, ŷd y, + g yy 2 ˆx, ŷd y, λ + g y ˆx, ŷ d z, g ˆx, ŷd x, d y, d z holds true. Fixing d z = z and o Z R K l λ K K l λ, gˆx, ŷ, we can reduce 19 to Qd x, d y w, y g yˆx, ŷ z, z C. The latter equation possesses a solution from the assumption of this theorem. Consequently, ˆx, ŷ, λ is weakly stationary for 13, i.e. ˆx, ŷ is weakly stationary for 11. The proof of the second statement is similar to that one of the first assertion: we reduce the surjectivity of H ˆx, ŷ, λ which is a sufficient condition for strong stationarity by Theorem 3.1 to the surjectivity of Q.

16 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 13 Remark 4.2 Observe that the constraint qualification 18 postulated in Theorem 4.1 is stronger than 1 introduced for MPCCs in Theorem 3.1 but much easier to check since the image space has been reduced from W Y Z Z to W Y Z. Within the proof of Theorem 4.1 one can find the operator H ˆx, ŷ, λ and the convex cone C, which can be used to define the weaker constraint qualification H ˆx, ŷ, λx Y Z C = W Y Z Z. 2 It is clear from the proof that 2 is sufficient for the weak stationarity of ˆx, ŷ as well. Observe that the operator Q from 17 is surjective if and only if H ˆx, ŷ, λ is surjective, i.e. the corresponding constraint qualifications, which imply strong stationarity of ˆx, ŷ, are equivalent. Remark 4.3 Recall that in the linear Example 4.2 the constraint qualification implying strong stationarity of any local optimal solution neither depends on the solution nor on a corresponding lower level multiplier. On the other hand, from Theorem 4.1 it is clear that weaker constraint qualifications of KRZCQ-type, which only imply weak stationarity of local optimal solutions, depend on the solution and a corresponding lower level multiplier. Remark 4.4 Let ˆx, ŷ X Y be a local optimal solution of 11 where the corresponding set of lower level multipliers is not a singleton. Then it may happen that the constraint qualifications postulated in Theorem 4.1 hold for some but not all multipliers, i.e. the choice of the multiplier λ in this theorem is of essential importance. Example 4.3 Let X = Y = Z = R 2 and K l = R 2,+ hold, and consider the bilevel programming problem { min x y 2 { 2 y Ψx := Argmin y2 1 2 y1 2 + y 2 x 1, y 2 x 2 }} x,y y whose lower level is convex and regular at any feasible point. Its unique global optimal solution is given by ˆx, ŷ :=,. One may check that the corresponding set of lower level multipliers is given by Λ := conv {, 2, 2, }, i.e. ˆx, ŷ, λ is a global optimal solution of the corresponding KKT-reformulation for any λ Λ. Observe that the matrix Q R 4 4 corresponding to the operator Q from 17 takes the form 2λ 1 Q = and is singular for precisely one multiplier from Λ, namely ˆλ =, 2. Especially, both constraint qualifications introduced within Theorem 4.1 do not hold at ˆx, ŷ, ˆλ but at any other point ˆx, ŷ, λ such that λ Λ \ {ˆλ} is satisfied. Remark 4.5 Let X = R n, Y = R m, W = R p, Z = R q, K u = R p,+, and K l = R q,+ hold, and choose an arbitrary feasible point ˆx, ŷ R n R m of the corresponding bilevel programming problem 11. It is weakly stationary in the sense of Definition 4.1 provided there exist multipliers α R p,+, β R m, λ R q,+, and ν R q which solve the system = x F ˆx, ŷ + Gˆx α + 2 yxfˆx, ŷ β q + λ i 2 yxg i ˆx, ŷ β + x gˆx, ŷ ν, i=1 = y F ˆx, ŷ + 2 yyfˆx, ŷβ q + λ i 2 yyg i ˆx, ŷβ + y gˆx, ŷ ν, i=1 = y fˆx, ŷ + y gˆx, ŷ λ, Gˆx α = gˆx, ŷ λ =, i I + ˆx, ŷ, λ: ν i =, i I + ˆx, ŷ, λ: y g i ˆx, ŷβ =. 21

17 14 Patrick Mehlitz, Gerd Wachsmuth Furthermore, ˆx, ŷ is strongly stationary in the sense of Definition 4.1 provided there exist multipliers α R p,+, β R m, λ R q,+, and ν R q which satisfy 21 as well as i I ˆx, ŷ, λ: ν i y g i ˆx, ŷβ. Let λ R q be chosen such that ˆx, ŷ, λ is a feasible point for 13. Then for α R p, β R m, and ν R q the CQ 18 reads = Gˆx α + 2 yxfˆx, ŷ β q + λ i 2 yxg i ˆx, ŷ β + x gˆx, ŷ ν i=1 = 2 yyfˆx, ŷβ α = q = β = 22 + λ i 2 yyg i ˆx, ŷβ + y gˆx, ŷ ν ν =, i=1 j {1,..., p}: α j Gˆx α = i I + ˆx, ŷ, λ: ν i = while 2 equals the corresponding MPCC-MFCQ condition, which takes the following form: = Gˆx α + 2 yxfˆx, ŷ β q + λ i 2 yxg i ˆx, ŷ β + x gˆx, ŷ ν i=1 = 2 yyfˆx, ŷβ q + λ i 2 yyg i ˆx, ŷβ + y gˆx, ŷ ν i=1 j {1,..., p}: α j Gˆx α = i I + ˆx, ŷ, λ: ν i = i I + ˆx, ŷ, λ: y g i ˆx, ŷβ = Furthermore, the corresponding MPCC-LICQ is given below: = Gˆx α + 2 yxfˆx, ŷ β q + λ i 2 yxg i ˆx, ŷ β + x gˆx, ŷ ν i=1 = 2 yyfˆx, ŷβ q + λ i 2 yyg i ˆx, ŷβ + y gˆx, ŷ ν i=1 j / Jˆx: α j = i I + ˆx, ŷ, λ: ν i = i I + ˆx, ŷ, λ: y g i ˆx, ŷβ = α = = β = ν =. α = = β = ν = Therein, the index sets I + ˆx, ŷ, λ, I ˆx, ŷ, λ, I + ˆx, ŷ, λ, and Jˆx are defined by I + ˆx, ŷ, λ := {i {1,..., q} g i ˆx, ŷ < λ i = } I ˆx, ŷ, λ := {i {1,..., q} g i ˆx, ŷ = λ i = } I + ˆx, ŷ, λ := {i {1,..., q} g i ˆx, ŷ = λ i > } Jˆx := {j {1,..., p} G j ˆx = } where G 1,..., G p : R n R and g 1,..., g q : R n R n R denote the component functions of G and g, respectively. Observe that due to Remark 4.1 we eliminated the multiplier µ from the regularity and stationarity conditions. One may check 11, 3 for more information on stationarity concepts and constraint qualifications for finite-dimensional bilevel programming problems.

18 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 15 5 An application in bilevel optimal control In this section, we want to derive necessary optimality conditions for a certain bilevel optimal control problem, which is given by means of F xt, yt + F 1 t, xt, yt, ut, vt dt ẋt C x xt C u ut = x x = min x,u,y,v y, v Ψx, u 25 where Ψ : W1,2, n T L k 2, T 2 W m 1,2,T Ll 2,T denotes the solution set mapping of the parametric lower level optimal control problem 1 2 yt R yt + 1 yt R y yt + 2Pxt + vt R v vt dt min 2 y,v ẏt A x xt B y yt A u ut B v vt = y y = D u ut + D v vt δt. Here and in what follows, we assume all the constraints appearing in 25, 26, and all derived surrogate problems to hold almost everywhere on, T. Nowadays, many practical problems turn out to have a hierarchical and dynamic structure, which is why their corresponding mathematical models are so-called bilevel optimal control problems. In general, a bilevel optimal control problem could be defined as a bilevel programming problem where at least one decision level is an optimal control problem. Hence, bilevel optimal control combines the difficulties of bilevel programming and infinite-dimensional optimization. There exist several publications introducing applications and numerical approaches to bilevel optimal control, cf. 1, 12, 14, 15, 2, 21, 23, but surprisingly few theoretical results and optimality conditions, which are not based on discretization, cf. 3,4,6,7,27. Here we want to apply the theory of MPCCs in reflexive Banach spaces in order to derive necessary optimality conditions for the bilevel optimal control problem 25. Below we list all the standing assumptions postulated on 25 and its lower level problem The function F : R n R m R is continuously differentiable. 2. The function F 1 :, T R n R m R k R l R is continuously differentiable w.r.t. its last four components and measurable w.r.t. its first component. Furthermore, we assume the existence of constants c, c > such that for any t, T, x R n, y R m, u R k, and v R l the following growth-conditions hold: F 1 t, x, y, u, v c 1 + x y u v 2 2 x,y,u,v F 1 t, x, y, u, v 2 c 1 + x 2 + y 2 + u 2 + v 2. Consequently, F : W n 1,2, T L k 2, T W m 1,2, T L l 2, T R, the upper level objective function, is continuously Fréchet differentiable cf The matrices A x R m n, A u R m k, B y R m m, B v R m l, C x R n n, C u R n k, D u R r k, D v R r l, P R m n, R R m m, R y R m m, and R v R l l are fixed. Furthermore, we assume R, R y as well as R v to be symmetric positive semidefinite and D v to possess full row rank r. 4. The function δ L r 2, T as well as the initial states x R n and y R m are fixed. 5. We choose the state space W 1,2, T, i.e. the set of all absolutely continuous functions possessing a weak derivative which belongs to L 2, T. The latter set is the control space of the above bilevel optimal control problem. Due to these assumptions, the lower level problem 26 is convex w.r.t. the lower level decision variables y and v. It is not difficult to show that the overall problem 25 is of the type presented in Example 4.2. Using the abbreviations introduced in this example, the linear operator B is surjective, 26

19 16 Patrick Mehlitz, Gerd Wachsmuth i.e. KRZCQ is satisfied for the lower level problem at any of its feasible points, since D v possesses full row rank. Hence, the corresponding KKT-conditions are necessary and sufficient for optimality. We state them here and refer the reader to standard literature on optimal control for a detailed derivation of this result cf. 19. Remark 5.1 Let ˆx, û W n 1,2, T L k 2, T be an arbitrary point feasible for the upper level problem of 25. A pair of functions ŷ, ˆv W m 1,2, T L l 2, T is an optimal solution for 26 if and only if it is feasible for 26 and there exist functions ϑ W m 1,2, T and λ L r 2, T such that the following conditions hold for almost every t, T : ϑt = B y ϑt + Pˆxt + R y ŷt ϑt = R ŷt = B v ϑt R vˆvt D v λt λt D u ût + D vˆvt δt λt =. 27 Observe that the above optimality conditions 27 are an equivalent representation of the actual KKTconditions of problem 26, called Pontryagin Maximum Principle. However, using these conditions to replace 25 by a single-level problem leads to a surrogate problem, which is not of the form 13 since we already exploited a reformulation of the lower level KKT-conditions. Hence, by applying the theory of this article it is possible to discuss two different approaches on how to deal with 25: an indirect approach which transforms 25 into an MPCC by means of Remark 5.1, and a direct approach making use of the results presented in Section 4 and Example 4.2, especially. Both approaches turn out to lead to the same necessary optimality conditions and constraint qualifications for 25. Hence, we just present the indirect approach here and leave it to the interested reader to verify the validity of these results evaluating the optimality system 16 for 25. Due to Remark 5.1, we replace 25 by the following single-level optimal control problem with complementarity constraints: F xt, yt + ẋt C x O O ẏt A x B y O xt yt ϑt P R y B y ϑt F 1 t, xt, yt, ut, vt dt C u O A u B v ut = vt O O x x = y y = ϑ ξ = ϑt + R yt = B v ϑt R v vt D v λt = D u ut + D v vt δt λt Du ut + D v vt δt λt dt =. min x,y,ϑ,u,v,λ,ξ 28 Note that the complementarity condition from Remark 5.1 is replaced w.l.o.g. by an integral since the latter construction equals the dual pairing in the space L r 2, T. Furthermore, we added the dummy variable ξ R m for the purpose of constructing a suitable constraint qualification for 28 later on. Now we introduce the reflexive Banach spaces X := W n 1,2, T W m 1,2, T W m 1,2, T L k 2, T L l 2, T L r 2, T R m W := W n 1,2, T W m 1,2, T W m 1,2, T R m L l 2, T Z := L r 2, T

20 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 17 and the cones K := {o W } as well as C := { w L r 2, T wt for a.e. t, T }. 29 We identify the Hilbert space Z with its dual space, and this implies C = C. Let us define a: X R, b: X W, c: X Z, and d: X Z for any p := x, y, ϑ, u, v, λ, ξ X by ap := F xt, yt + F 1 t, xt, yt, ut, vt dt bp := x x C x xτ + C u uτ dτ, y y ϑ ξ A x xτ + B y yτ + A u uτ + B v vτ dτ, Pxτ + R y yτ B y ϑτ dτ, ϑt + R yt, B v ϑ R v v D v λ cp := D u u + D v v δ dp := λ. Then 28 possesses the form 4. Observe that we inserted the boundary conditions on the state functions into the definition of the mapping b. The latter is an affine bounded operator and the same holds true for c and d. Hence, the constraints of 28 are continuously Fréchet differentiable. As mentioned earlier, from the assumptions on F and F 1 the mapping a is continuously Fréchet differentiable as well. In order to rewrite the weak and strong stationarity conditions from Definition 3.1, we have to compute the Fréchet derivatives of a, b, c, and d. Let us fix an arbitrary feasible point p := x, y, ϑ, u, v, λ, ξ X of 28. One may check that for any direction d := d x, d y, d ϑ, d u, d v, d λ, d ξ X the following formulas are correct: a pd = x F xt, yt d x T + y F xt, yt d y T + x F 1 t, xt, yt, ut, vtd x t + y F 1 t, xt, yt, ut, vtd y t + u F 1 t, xt, yt, ut, vtd u t + v F 1 t, xt, yt, ut, vtd v t dt b pd = bd + x, y, o L m 2,T,, o L l 2,T c pd = cd + δ d pd = dd. In order to evaluate the stationarity system 8 for 28, it is necessary to compute the adjoint operators of b p, c p, and d p. We identify the spaces W σ 1,2, T and R σ L σ 2, T for any σ N + with each other since it is possible to define a bijection Φ: W σ 1,2, T R σ L σ 2, T as stated below: g W σ 1,2, T : Φg := g, ġ. The space W1,2, σ T is isomorphic to its dual space, and the corresponding dual pairing or, equivalently, the corresponding inner product of this Hilbert space can be defined as follows: g = g s, g f, h = h s, h f W σ 1,2, T : g, h = g s h s + g f t h f t dt. Forthwith, we keep the above notation, i.e. any element h W σ 1,2, T will be identified with the pair h s, h f := h, ḣ = Φh R σ L σ 2, T. Consequently, we have X = R n L n 2, T R m L m 2, T R m L m 2, T L k 2, T L l 2, T L r 2, T R m W = R n L n 2, T R m L m 2, T R m L m 2, T R m L l 2, T

21 18 Patrick Mehlitz, Gerd Wachsmuth and X = X as well as W = W. Firstly, using integration by parts yields a p = x F xt, yt + x F xt, yt + y F xt, yt + y F xt, yt +, o L m 2,T, x F 1 τ, xτ, yτ, uτ, vτ dτ, x F 1 τ, xτ, yτ, uτ, vτ dτ, y F 1 τ, xτ, yτ, uτ, vτ dτ, y F 1 τ, xτ, yτ, uτ, vτ dτ, u F 1, x, y, u, v, v F 1, x, y, u, v, o L r 2,T, cf. 8 for similar calculations. Now take ϱ := h, k, p, q, s W and d X from above. Then from the definition of the adjoint operator we clearly have d, b p ϱ = b pd, ϱ. In order to find an explicit representation of b p, we evaluate the right-hand side of this equation and apply integration by parts. We present this procedure by means of an easier operator in Appendix 6. Time-consuming calculations lead to the representation b p ϱ = h s h f k s + R q C x h f τ + A x k f τ + P p f τ dτ, C x h f τ + A x k f τ + P p f τ dτ, k f + R q p s + q + p f + q + B y k f τ + R y p f τ dτ, B y k f τ + R y p f τ dτ, B y p f τ + B v sτ dτ, B y p f τ + B v sτ dτ, C u h f A u k f, B v k f R v s, D v s, p s. Next, choose an arbitrary element ν Z. Then it is not difficult to obtain that c p ν =, o L n 2,T,, o L m 2,T,, o L m 2,T, D u ν, D v ν, o L r 2,T, holds true. Similar as above we derive d p µ =, o L n 2,T,, o L m 2,T,, o L m 2,T, o L k 2,T, o L l 2,T, µ, for any µ Z.

22 Weak and strong stationarity in generalized bilevel programming and bilevel optimal control 19 Assume that ϱ := h, k, p, q, s W, ν Z, and µ Z solve the system 8. Then, due to the above calculations, the following conditions are satisfied for almost every t, T : = x F xt, yt + h s + x F 1 τ, xτ, yτ, uτ, vτ C x h f τ A x k f τ P p f τ dτ, = x F xt, yt + h f t + x F 1 τ, xτ, yτ, uτ, vτ C x h f τ A x k f τ P p f τ dτ, t = y F xt, yt + k s + R q + y F 1 τ, xτ, yτ, uτ, vτ B y k f τ R y p f τ dτ, = y F xt, yt + k f t + R q + y F 1 τ, xτ, yτ, uτ, vτ B y k f τ R y p f τ dτ, t = p s + q + = p f t + q + B y p f τ + B v sτ dτ, t B y p f τ + B v sτ dτ, 3a 3b 3c 3d 3e 3f = u F 1 t, xt, yt, ut, vt C u h f t A u k f t + D u νt, 3g = v F 1 t, xt, yt, ut, vt B v k f t + D v νt R v µt, 3h = D v st + µt, = p s. From 3b, 3d, and 3f we see that the functions h f, k f, and p f are elements of W1,2, σ T for appropriate σ N + again. Further, the equations 3a 3f and 3j yield that these functions are solutions of the boundary value problem the differential equation shall hold for almost every t, T 3i 3j ḣ f t = x F 1 t, xt, yt, ut, vt C x h f t A x k f t P p f t, k f t = y F 1 t, xt, yt, ut, vt B y k f t R y p f t, ṗ f t = B y p f t + B v st, h f T = x F xt, yt, k f T = y F xt, yt R q, p f =, p f T = q. 31 Moreover, 3g and 3h provide a linearized maximum condition cf. 34. Together with 31 and the equation D v s = µ this system is equivalent to 3 and, hence, to o X = a p + b p ϱ + c p ν + d p µ. It remains to evaluate the sign conditions and complementarity conditions on the multipliers ϱ, ν, and µ from Definition 3.1. Let us introduce three sets I + i, p, I i, p, and I + i, p by means of I + i, p := {t, T c i pt < d i pt = } I i, p := {t, T c i pt = d i pt = } I + i, p := {t, T c i pt = d i pt > } for any i {1,..., r}. Therein, c 1 p,..., c r p, d 1 p,..., d r p L 2, T denote the component mappings of cp and dp, respectively.

23 2 Patrick Mehlitz, Gerd Wachsmuth Lemma 5.1 Let p := x, y, ϑ, u, v, λ, ξ X be a feasible point of 28. Then we have cl C C cp cp = { ν L r 2, T i {1,..., r}: νi t = for a.e. t I + i, p } cl C C dp dp = { µ L r 2, T i {1,..., r}: µi t = for a.e. t I + i, p } as well as { K C dp, cp = ν L r 2, T i {1,..., r}: ν i t = for a.e. t I + i, p ν i t for a.e. t I i, p { K C cp, dp = µ L r 2, T i {1,..., r}: µ i t = for a.e. t I + i, p µ i t for a.e. t I i, p } }. Proof Since cpt holds for almost every t, T, we obtain { C cp = C cp r } T = ν C ν i tc i pt dt = Hence, it is easy to see that = { ν C i {1,..., r}: νi t = for a.e. t I + i, p }. C C cp = { ν L r 2, T i {1,..., r}: νi t for a.e. t I + i, p } holds true and this set is closed. This yields the first assertion. It is well-known that the tangent cone T C dp is given by means of T C dp = { ν L r 2, T i {1,..., r}: νi t for a.e. t I + i, p I i, p }, i=1 see 5, Example This implies the third assertion. The second and fourth assertion follow analogously. In the following lemma, we collect all the above results to introduce the notions of weak and strong stationarity in the sense of Definition 3.1 applied to 28. Lemma 5.2 Let p := x, y, ϑ, u, v, λ, ξ X be an arbitrary feasible point of The point p is weakly stationary for 28 if and only if there exist φ x W n 1,2, T, φ y, φ ϑ W m 1,2, T, ν L r 2, T, and β L l 2, T as well as a vector η R m which satisfy the following conditions for almost every t, T : a Adjoint condition φ x t = x F 1 t, xt, yt, ut, vt C x φ x t A x φ y t P φ ϑ t φ y t = y F 1 t, xt, yt, ut, vt B y φ y t R y φ ϑ t φ ϑ t = B y φ ϑ t B v βt, 32 b Transversality condition φ x T = x F xt, yt φ y T = y F xt, yt + R η φ ϑ = φ ϑ T = η, 33 c Linearized maximum condition = u F 1 t, xt, yt, ut, vt C u φ x t A u φ y t + D u νt = v F 1 t, xt, yt, ut, vt B v φ y t + D v νt + R v D v βt, 34