Bilevel optimal control: existence results and stationarity conditions

TL;DR

This paper reviews bilevel optimal control problems, focusing on existence results and stationarity conditions, especially for PDE-based models with parameter reconstruction, providing theoretical insights and solution approaches.

Contribution

It offers a comprehensive review of bilevel optimal control, introduces new existence results, and derives necessary optimality conditions for PDE-based problems with parameter reconstruction.

Findings

Existence of solutions for specific bilevel control problems.

Derivation of necessary optimality conditions using the optimal value function.

Application of relaxation approaches to establish stationarity conditions.

Abstract

The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.

Equations

$$x_k\rightharpoonup \bar{x}⟹j(\bar{x})\le \underset{k\to \infty }{\mathrm{liminf}}j(x_k)\left(x_k\rightharpoonup \bar{x}⟹j(\bar{x})\ge \underset{k\to \infty }{\mathrm{limsup}}j(x_k)\right)$$

$$A^{\circ }:=\left\{x^{\star }\in {\mathcal{X}}^{\star }|\forall x\in A:{⟨x^{\star },x⟩}_{\mathcal{X}}\le 0\right\}$$

$$\begin{array}{rl}F(x,z)& \to \underset{x,z}{\min }\\ x& \in X_{\text{ad}}\\ z& \in \Psi (x),\end{array}$$

$$\begin{array}{rl}f(x,z)& \to \underset{z}{\min }\\ z& \in \Gamma (x).\end{array}$$

$$\forall x\in X_{\text{ad}}:\phi (x):=\underset{z}{\inf }\{f(x,z)|z\in \Gamma (x)\}.$$

$$\forall x\in \mathcal{X}:\Gamma (x):=\{z\in \mathcal{Z}|g(x,z)\in C\}.$$

$$g_z^{\prime }(\bar{x},\bar{z})\mathcal{Z}-\mathrm{cone}(C-\{g(\bar{x},\bar{z})\})=\mathcal{W}$$

$$\underset{k\to \infty }{\mathrm{limsup}}\phi (x_k)\le \underset{k\to \infty }{\mathrm{limsup}}f(x_k,z_k)\le f(\bar{x},\bar{z})=\phi (\bar{x}),$$

$$\phi (\bar{x})\le f(\bar{x},\bar{z})\le \underset{k\to \infty }{\mathrm{liminf}}f(x_k,z_k)=\underset{k\to \infty }{\mathrm{liminf}}\phi (x_k)\le \underset{k\to \infty }{\mathrm{limsup}}\phi (x_k)\le \phi (\bar{x}),$$

$$\frac{1}{2}{\Vert y-{\sum }_{i=1}^n{x}_i^w{f}^i\Vert }_{L^2(\Omega )}^2$$

$$-\Delta y$$

$$u_a\le u$$

$$\begin{array}{rl}\frac{1}{2}{\Vert y-y_{\text{o}}\Vert }_{L^2(\Omega )}^2+\frac{1}{2}{\Vert u-u_{\text{o}}\Vert }_{L^2(\Omega )}^2& \to \underset{x,y,u}{\min }\\ (x^w,x^s)& \in X_{\text{ad}}\\ (y,u)& \in \Psi (x^w,x^s).\end{array}$$

$$j(\xi ,z)$$

$$g(\xi ,z)$$

$$\frac{1}{2}{\Vert x-x_{\text{d}}\Vert }_{L^2(I;{\mathbb{R}}^n)}^2+\frac{\sigma }{2}{\Vert u\Vert }_{L^2(I;{\mathbb{R}}^p)}^2+J(x(T),y)$$

$$\dot{x}-Ax-Bu$$

$$x(0)$$

$$u_a\le u$$

$$y$$

$$F(x,\psi (x))$$

$$x$$

$$F(x,z)$$

$$x$$

$$f(x,z)-\phi (x)$$

$$z$$

$$\forall z,z^{\prime }\in \mathcal{Z}\forall \gamma \in [0,1]:g(x,\gamma z+(1-\gamma )z^{\prime })-\gamma g(x,z)-(1-\gamma )g(x,z^{\prime })\in C$$

$$0=f_z^{\prime }(x,z)+g_z^{\prime }(x,z{)}^{\star }\lambda ,$$

$$\lambda \in C^{\circ },$$

$$0={⟨\lambda ,g(x,z)⟩}_{\mathcal{W}}.$$

$$\begin{array}{rl}F(x,z)& \to \underset{x,z,\lambda }{\min }\\ x& \in X_{\text{ad}}\\ f_z^{\prime }(x,z)+g_z^{\prime }(x,z{)}^{\star }\lambda & =0\\ g(x,z)& \in C\\ \lambda & \in C^{\circ }\\ {⟨\lambda ,g(x,z)⟩}_{\mathcal{W}}& =0.\end{array}$$

$$\frac{1}{2}{\Vert y-{\sum }_{i=1}^n{x}_i{f}^i\Vert }_{L^2(\Omega )}^2+\frac{\sigma }{2}{\Vert u\Vert }_{L^2(\Omega )}^2$$

$$-\Delta y$$

$$\frac{1}{2}{\Vert y-y_{\text{o}}\Vert }_{L^2(\Omega )}^2+\frac{1}{2}{\Vert u-u_{\text{o}}\Vert }_{L^2(\Omega )}^2$$

$$x$$

$$(y,u)$$

$$\begin{array}{rl}\frac{1}{2}{\Vert y-y_{\text{o}}\Vert }_{L^2(\Omega )}^2+\frac{1}{2}{\Vert u-u_{\text{o}}\Vert }_{L^2(\Omega )}^2& \to \underset{x,y,u,p}{\min }\\ x& \in X_{\text{ad}}\\ y-{\sum }_{i=1}^n{x}_i{f}^i-\Delta p& =0\\ \sigma u-p& =0\\ -\Delta y-u& =0,\end{array}$$

$$\displaystyle 0\,=\,\bar{z}-\bigl{(}\left\langle\bar{\mu},f^{i}\right\rangle_{L^{2}(\Omega)}\bigr{)}_{i=1}^{n},$$

Full text

1 What is bilevel optimal control?

A bilevel programming problem is a hierarchical optimization problem of two decision makers where the objective functional as well as the feasible set of the so-called upper level decision maker (or leader) depend implicitly on the solution set of a second parametric optimization problem which will be called lower level (or follower’s) problem. Both decision makers act as follows: First, the leader chooses an instance from his feasible set which then serves as the parameter in the follower’s problem. Thus, the follower is in position to solve his problem and passes an optimal solution back to the leader who now may compute the associated value of the objective functional. As soon as the lower level solution set is not a singleton for at least one value of the upper level variable, problems of this type may be ill-posed which is why different solution concepts including the so-called optimistic and pessimistic approach have been developed. Bilevel programming problems generally suffer from inherent lacks of convexity, regularity, and smoothness which makes them theoretically challenging. The overall concept of bilevel optimization dates back to vonStackelberg1934 where this problem class is introduced in the context of economical game theory. More than 80 years later, bilevel programming is one of the hottest topics in mathematical optimization since numerous real-world applications can be transferred into models of bilevel structure. A detailed introduction to bilevel programming can be found in the monographs Bard1998 ; Dempe2002 ; DempeKalashnikovPerezValdesKalashnykova2015 ; ShimizuIshizukaBard1997 while a satisfying overview of existing literature is given in Dempe2018 where more than 1350 published books, PhD-theses, and research articles are listed.

Optimal control of ordinary or partial differential equations (ODEs and PDEs, respectively) describes the task of identifying input quantities which control the state function of the underlying differential equation such that a given cost functional is minimized, see HinzePinnauUlbrichUlbrich2009 ; LewisVrabieSyrmos2012 ; Troeltzsch2009 ; Troutman1996 for an introduction to this topic. Noting that the decision variables are elements of suitable function spaces, optimal control is a particular field of programming in (infinite-dimensional) Banach spaces, see BonnansShapiro2000 .

In bilevel optimal control, bilevel programming problems are considered where at least one decision maker has to solve an optimal control problem. Thus, we are facing the intrinsic difficulties of bilevel optimization and optimal control when investigating this problem class. Naturally, one may subdivide bilevel optimal control problems into three subclasses depending on which decision maker has to perform optimal control. Each of these problem classes appears in practice and has to be tackled with different techniques in order to infer optimality conditions or solution algorithms.

The situation where only the upper level decision maker has to solve an optimal control problem of ordinary differential equations while the lower level problem explicitly depends on the terminal state of the leader’s state variable has been considered in BenitaDempeMehlitz2016 ; BenitaMehlitz2016 . Problems of this type arise from the topic of gas balancing in energy networks, see KalashnikovBenitaMehlitz2015 , and can be investigated by combining tools from finite-dimensional parametric optimization and standard optimal control. The situation where parameters within an optimal control problem have to be estimated or reconstructed by certain measurements is a typical example of a bilevel optimal control problem where only the lower level decision maker has to solve an optimal control problem. This particular instance of bilevel optimal control may therefore be also called inverse optimal control. In AlbrechtLeiboldUlbrich2012 ; AnbrechtPassenbergSobotkaPeerBussUlbrich2010 ; AlbrechtUlbrich2017 ; Hatz2014 ; MombaurTruongLaumond2010 , inverse optimal control problems of ODEs are considered in the context of human locomotion. Some more theoretical results for such problems are presented in HatzSchloederBock2012 ; Ye1995 ; Ye1997 . First steps regarding the inverse optimal control of PDEs have been done recently in the papers DempeHarderMehlitzWachsmuth2019 ; HarderWachsmuth2019 ; HollerKunischBarnard2018 . The paper PalagachevGerdts2017 deals with the scheduling of multiple agents which are controlled at the lower level stage. In FischLenzHolzapfelSachs2012 , the authors discuss a bilevel optimal control problem where airplanes are controlled at multiple lower levels in order to increase the fairness in air racing. Finally, it is possible that leader and follower have to solve an optimal control problem. This setting has been discussed theoretically in Carlson2013 ; Mehlitz2016 ; MehlitzWachsmuth2016 ; PalagachevGerdts2016 . Underlying applications arise e.g. when time-dependent coupling of container crane movements is under consideration, see Knauer2012 ; KnauerBueskens2010 .

The optimal control of (quasi-) variational inequalities ((Q)VIs) seems to be closely related to the subject of bilevel optimal control since the underlying variational problem, which assigns to each control the uniquely determined state function, can be modeled as a parametric optimization problem in function spaces. Those problems are of hierarchical structure, but neither leader nor follower has to solve an optimal control problem in the classical meaning. In the seminal work Mignot1976 , Mignot shows that the control-to-state map of an elliptic VI in the Sobolev space $H_{0}^{1}(\Omega)$ is directionally differentiable, and (in the absence of control constraints) this leads to an optimality system of strong-stationarity-type. If control constraints are present, one typically uses a regularization approach for the derivation of optimality conditions. This idea dates back to Barbu1984 and we refer to SchielaWachsmuth2013 for a modern treatment. Finally, we would like to mention that a comparison of several optimality systems and further references regarding this topic can be found in HarderWachsmuth2018a .

2 Notation and preliminaries

Let us briefly recall some essentials of functional analysis we are going to exploit. For a (real) Banach space $\mathcal{X}$, $\left\|\cdot\right\|_{\mathcal{X}}\colon\mathcal{X}\to\mathbb{R}$ denotes its norm. Furthermore, $\mathcal{X}^{\star}$ represents the topological dual of $\mathcal{X}$. We use $\left\langle\cdot,\cdot\right\rangle_{\mathcal{X}}\colon\mathcal{X}^{\star}\times\mathcal{X}\to\mathbb{R}$ in order to denote the associated dual pairing. For a sequence $\{x_{k}\}_{k\in\mathbb{N}}\subset\mathcal{X}$ and some point $\bar{x}\in\mathcal{X}$, strong and weak convergence of $\{x_{k}\}_{k\in\mathbb{N}}$ to $\bar{x}$ will be represented by $x_{k}\to\bar{x}$ and $x_{k}\rightharpoonup\bar{x}$, respectively. Recall that in a finite-dimensional Banach space $\mathcal{X}$, the concepts of strong and weak convergence coincide. A functional $J\colon\mathcal{X}\to\mathbb{R}$ is said to be weakly sequentially lower (upper) semicontinuous at $\bar{x}$, whenever

[TABLE]

holds for all sequences $\{x_{k}\}_{k\in\mathbb{N}}\subset X$. We say that $j$ is weakly sequentially lower (upper) semicontinuous if it possesses this property at each point from $\mathcal{X}$. It is well known that convex and continuous functionals are weakly sequentially lower semicontinuous. If the canonical embedding $\mathcal{X}\ni x\mapsto\left\langle\cdot,x\right\rangle_{\mathcal{X}}\in\mathcal{X}^{\star\star}$ is an isomorphism, then $\mathcal{X}$ is said to be reflexive. The particular Banach space $\mathbb{R}^{n}$ is equipped with the Euclidean norm $\left|\cdot\right|_{2}$. Furthermore, we use $x\cdot y$ to represent the Euclidean inner product in $\mathbb{R}^{n}$.

A set $A\subset\mathcal{X}$ is said to be weakly sequentially closed whenever the weak limits of all weakly convergent sequences from $A$ belong to $A$ as well. We note that closed and convex sets are weakly sequentially closed. We call $A$ weakly sequentially compact whenever each sequence from $A$ possesses a weakly convergent subsequence whose limit belongs to $A$. Each bounded, closed, and convex subset of a reflexive Banach space is weakly sequentially compact.

For a second Banach space $\mathcal{Y}$, $\mathbb{L}\left[\mathcal{X},\mathcal{Y}\right]$ is used to denote the Banach space of all bounded linear operators mapping from $\mathcal{X}$ to $\mathcal{Y}$. For $\mathtt{A}\in\mathbb{L}\left[\mathcal{X},\mathcal{Y}\right]$, $\mathtt{A}^{\star}\in\mathbb{L}\left[\mathcal{Y}^{\star},\mathcal{X}^{\star}\right]$ denotes its adjoint. If $\mathcal{X}\subset\mathcal{Y}$ holds while the associated identity in $\mathbb{L}\left[\mathcal{X},\mathcal{Y}\right]$ is continuous, then $\mathcal{X}$ is said to be continuously embedded into $\mathcal{Y}$ which will be denoted by $\mathcal{X}\hookrightarrow\mathcal{Y}$. Whenever the identity is compact, the embedding $\mathcal{X}\hookrightarrow\mathcal{Y}$ is called compact. For a set-valued mapping $\Gamma\colon\mathcal{X}\rightrightarrows\mathcal{Y}$, $\operatorname{gph}\Gamma:=\{(x,y)\in\mathcal{X}\times\mathcal{Y}\,|\,y\in\Gamma(x)\}$ and $\operatorname{dom}\Gamma:=\{x\in\mathcal{X}\,|\,\Gamma(x)\neq\varnothing\}$ represent the graph and the domain of $\Gamma$, respectively.

Let $A\subset\mathcal{X}$ be nonempty and convex. Then, the closed, convex cone

[TABLE]

is called the polar cone of $A$. For a fixed point $\bar{x}\in A$, $\mathcal{N}_{A}(\bar{x}):=(A-\{\bar{x}\})^{\circ}$ is referred to as the normal cone (in the sense of convex analysis) to $A$ at $\bar{x}$. For the purpose of completeness, let us set $\mathcal{N}_{A}(\hat{x}):=\varnothing$ for all $\hat{x}\in\mathcal{X}\setminus A$. Note that whenever $C\subset\mathcal{X}$ is a closed, convex cone satisfying $\bar{x}\in C$, then we have the relation $\mathcal{N}_{C}(\bar{x})=C^{\circ}\cap\{x^{\star}\in\mathcal{X}^{\star}\,|\,\langle x^{\star},x\rangle_{\mathcal{X}}=0\}$.

Detailed information on the function spaces we are going to exploit can be found in the monograph AdamsFournier2003 .

3 Bilevel programming in Banach spaces

Let us consider the bilevel programming problem

[TABLE]

where $\Psi\colon\mathcal{X}\rightrightarrows\mathcal{Z}$ is the solution mapping of the parametric optimization problem

[TABLE]

Note that we minimize the objective functional in (BPP) w.r.t. both variables which is related to the so-called optimistic approach of bilevel programming. In this section, we first want to discuss the existence of optimal solutions associated with (BPP). Afterwards, we briefly discuss possible approaches which can be used to infer optimality conditions for this problem class.

3.1 Existence theory

In this section, we aim to characterize situations where (BPP) possesses optimal solutions. Noting that compact sets are generally rare in infinite-dimensional spaces, one cannot rely on classical existence results from bilevel programming. Indeed, compactness assumptions on the feasible sets have to be relaxed in order to guarantee applicability of possible results. As a consequence, we need to demand more restrictive properties than (lower semi-) continuity of the appearing objective functionals in order to balance things in a reasonable way. One may check e.g. Jahn1996 for a detailed discussion of existence theory for optimization problems in Banach spaces. Particularly, it is presented that each weakly sequentially lower semicontinuous functional achieves its minimum over a nonempty and weakly sequentially compact set. The above remarks justify the subsequently stated general assumptions of this section.

Assumption 3.1

We consider Banach spaces $\mathcal{X}$ and $\mathcal{Z}$. The objective functionals $F,f\colon\mathcal{X}\times\mathcal{Z}\to\mathbb{R}$ are weakly sequentially lower semicontinuous. The set $X_{\textup{ad}}\subset\mathcal{X}$ is assumed to be nonempty and weakly sequentially compact. Furthermore, $\Gamma\colon\mathcal{X}\rightrightarrows\mathcal{Z}$ is a set-valued mapping with $X_{\textup{ad}}\subset\operatorname{dom}\Gamma$ such that $(X_{\textup{ad}}\times\mathcal{Z})\cap\operatorname{gph}\Gamma$ is weakly sequentially compact.

In the setting where $\mathcal{X}$ and $\mathcal{Z}$ are finite-dimensional, e.g. instances of $\mathbb{R}^{n}$, the above assumptions reduce to the lower semicontinuity of the objective functionals as well as some compactness assumptions on $X_{\textup{ad}}$ and $\operatorname{gph}\Gamma$ which is rather standard in bilevel programming, see e.g. Dempe2002 . For our upcoming analysis, we will exploit the function $\varphi\colon X_{\textup{ad}}\to\overline{\mathbb{R}}$ defined by

[TABLE]

By definition, $\varphi$ assigns to each parameter $x\in X_{\textup{ad}}$ the optimal function value of the lower level problem (1). Assumption 3.1 guarantees that the infimal value $\varphi(x)$ is actually attained (i.e. $\Psi(x)\neq\varnothing$) since for all $x\in X_{\text{ad}}$, $f(x,\cdot)\colon\mathcal{Z}\to\mathbb{R}$ is weakly sequentially lower semicontinuous while $\Gamma(x)$ is nonempty and weakly sequentially compact.

Below, we need to study the (upper) semicontinuity properties of $\varphi$. In order to do that, we need to address some continuity properties of the mapping $\Gamma$, see HerzogSchmidt2012 .

Definition 3.2

Fix $(\bar{x},\bar{z})\in\operatorname{gph}\Gamma$. Then, $\Gamma$ is called inner semicontinuous (weakly-weakly inner semicontinuous) at $(\bar{x},\bar{z})$ if for each sequence $\{x_{k}\}_{k\in\mathbb{N}}\subset\operatorname{dom}\Gamma$ satisfying $x_{k}\to\bar{x}$ ($x_{k}\rightharpoonup\bar{x}$), there exists a sequence $\{z_{k}\}_{k\in\mathbb{N}}\subset\mathcal{Z}$ satisfying $z_{k}\in\Gamma(x_{k})$ for all $k\in\mathbb{N}$ as well as $z_{k}\to\bar{z}$ ($z_{k}\rightharpoonup\bar{z}$).

It needs to be noted that the concepts of inner and lower semicontinuity of set-valued mappings, see BankGuddatKlatteKummerTammer1983 , are closely related. Particularly, the lower semicontinuity of $\Gamma$ at some point $\bar{x}\in\operatorname{dom}\Gamma$ is equivalent to its inner semicontinuity at all points $(\bar{x},z)$ with $z\in\Gamma(\bar{x})$.

In the particular situation where the mapping $\Gamma$ is characterized via smooth generalized inequalities, there is an easy criterion which is sufficient for inner semicontinuity.

Remark 3.3

We assume that there exists a continuously Fréchet differentiable function $g\colon\mathcal{X}\times\mathcal{Z}\to\mathcal{W}$, where $\mathcal{W}$ is a Banach space, and some nonempty, closed, convex set $C\subset\mathcal{W}$ such that $\Gamma$ is given by

[TABLE]

For fixed $\bar{z}\in\Gamma(\bar{x})$, we assume that the condition

[TABLE]

is valid. Then, $\Gamma$ is inner semicontinuous at $(\bar{x},\bar{z})$, see e.g. (BonnansShapiro2000, , Section 2.3.3). We note that (3) often is referred to as Robinson’s constraint qualification, see Robinson1976 , or Kurcyusz–Zowe constraint qualification, see ZoweKurcyusz1979 . In the setting of finite-dimensional nonlinear parametric optimization, this condition simply reduces to the Mangasarian–Fromovitz constraint qualification, see BonnansShapiro2000 for details. Let us note that (3) trivially holds whenever the operator $g^{\prime}_{z}(\bar{x},\bar{z})$ is surjective.

We note that weakly-weakly inner semicontinuity of $\Gamma$ is inherent whenever this map is actually constant. A nontrivial situation is described in the following example.

Example 1

Terminal state dependence of lower level For some time interval $I:=(0,T)$ and some natural number $n$, we consider the Hilbert space $\mathcal{X}:=H^{1}(I;\mathbb{R}^{n})$. Clearly, the embedding $\mathcal{X}\hookrightarrow C(\overline{I};\mathbb{R}^{n})$ is compact, see AdamsFournier2003 . This means that the evaluation operator $\mathcal{X}\ni x\mapsto x(T)\in\mathbb{R}^{n}$ is well-defined and compact as well.

For some set-valued mapping $\Upsilon\colon\mathbb{R}^{n}\rightrightarrows\mathcal{Z}$, we define $\Gamma(x):=\Upsilon(x(T))$ for all $x\in\mathcal{X}$. The above observation implies that $\Gamma$ is weakly-weakly inner semicontinuous at $(\bar{x},\bar{z})\in\operatorname{gph}\Gamma$ whenever $\Upsilon$ is inner semicontinuous at $(\bar{x}(T),\bar{z})$ and the latter can be guaranteed via standard assumptions, see e.g. Remark 3.3.

The setting in this example reflects the situation of time-dependent coupling between upper and lower level, see (Knauer2012, , Section 5), or a finite-dimensional lower level problem depending only on the terminal value of the leader’s state variable, see BenitaDempeMehlitz2016 ; BenitaMehlitz2016 ; KalashnikovBenitaMehlitz2015 .

It needs to be noted that the analysis of the above situation can be extended to cases where $\mathcal{X}$ is a function space over some domain $\Omega\subset\mathbb{R}^{d}$ which is embedded compactly into $C(\overline{\Omega})$, and the function of interest is evaluated at finitely many points from $\overline{\Omega}$. This applies to the setting $\mathcal{X}:=H^{2}(\Omega)$ where $\Omega$ is a bounded Lipschitz domain, see AdamsFournier2003 .

In the following lemma, which is inspired by (HerzogSchmidt2012, , Theorem 2.5), we study upper semicontinuity properties of the function $\varphi$. This will be useful in order to infer closedness properties of the feasible set associated with (BPP).

Lemma 3.4

Fix some point $\bar{x}\in X_{\textup{ad}}$ and $\bar{z}\in\Psi(\bar{x})$.

Assume that $f$ is weakly sequentially upper semicontinuous at $(\bar{x},\bar{z})$ while $\Gamma$ is weakly-weakly inner semicontinuous at $(\bar{x},\bar{z})$. Then, $\varphi$ is weakly sequentially upper semicontinuous at $\bar{x}$. 2. 2.

Let $\mathcal{X}$ be finite-dimensional. Assume that $f$ is upper semicontinuous at $(\bar{x},\bar{z})$ while $\Gamma$ is inner semicontinuous at $(\bar{x},\bar{z})$. Then, $\varphi$ is upper semicontinuous at $\bar{x}$.

Proof

We only verify the first statement of the lemma. The second one can be shown using analogous arguments.

Let $\{x_{k}\}_{k\in\mathbb{N}}\subset X_{\textup{ad}}$ be a sequence satisfying $x_{k}\rightharpoonup\bar{x}$. Exploiting the weakly-weakly inner semicontinuity of $\Gamma$ at $(\bar{x},\bar{z})$, we find a sequence $\{z_{k}\}_{k\in\mathbb{N}}\subset\mathcal{Z}$ satisfying $z_{k}\in\Gamma(x_{k})$ for all $k\in\mathbb{N}$ and $z_{k}\rightharpoonup\bar{z}$. By definition, $\varphi(x_{k})\leq f(x_{k},z_{k})$ holds for all $k\in\mathbb{N}$. Now, the weak sequential upper semicontinuity of $f$ at $(\bar{x},\bar{z})$ yields

[TABLE]

and this shows the claim.

Now, we exploit the above lemma in order to infer the existence of optimal solutions to (BPP).

Theorem 3.5

In each of the settings described below, (BPP) possesses an optimal solution.

The mapping $f$ is weakly sequentially upper semicontinuous on $X_{\textup{ad}}\times\mathcal{Z}$ while $\Gamma$ is weakly-weakly inner semicontinuous on $X_{\textup{ad}}\times\mathcal{Z}$. 2. 2.

The Banach space $\mathcal{X}$ is finite-dimensional. The mapping $f$ is upper semicontinuous on $X_{\textup{ad}}\times\mathcal{Z}$ while $\Gamma$ is inner semicontinuous on $X_{\textup{ad}}\times\mathcal{Z}$.

Proof

Again, we only show the theorem’s first assertion.

For the proof, we just need to verify that the feasible set $(X_{\textup{ad}}\times\mathcal{Z})\cap\operatorname{gph}\Psi$ of (BPP) is nonempty and weakly sequentially compact since the objective $F$ is supposed to be weakly sequentially lower semicontinuous. Noting that $\Psi(x)\neq\varnothing$ holds true for all $x\in X_{\textup{ad}}$, the nonemptiness of $(X_{\textup{ad}}\times\mathcal{Z})\cap\operatorname{gph}\Psi$ is obvious.

Let $\{(x_{k},z_{k})\}_{k\in\mathbb{N}}\subset(X_{\textup{ad}}\times\mathcal{Z})\cap\operatorname{gph}\Psi$ be an arbitrary sequence. Clearly, we have $\{(x_{k},z_{k})\}_{k\in\mathbb{N}}\subset(X_{\textup{ad}}\times\mathcal{Z})\cap\operatorname{gph}\Gamma$ and by Assumption 3.1, there exists a subsequence (without relabeling) converging weakly to $(\bar{x},\bar{z})\in(X_{\textup{ad}}\times\mathcal{Z})\cap\operatorname{gph}\Gamma$. Now, the definition of the function $\varphi$ and Lemma 3.4 yield

[TABLE]

which shows $\varphi(\bar{x})=f(\bar{x},\bar{z})$, i.e. $\bar{z}\in\Psi(\bar{x})$ follows. This yields that the point $(\bar{x},\bar{z})$ belongs to $(X_{\textup{ad}}\times\mathcal{Z})\cap\operatorname{gph}\Psi$, and this shows the claim.

Let us apply the above theory to some example problems from bilevel optimal control.

Example 2

Inverse nonregularized control of Poisson’s equation Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with Lipschitz boundary $\operatorname{bd}\Omega$. For fixed parameters $x^{w}\in\mathbb{R}^{n}$ and $x^{s}\in L^{2}(\Omega)$, we consider the optimal control of Poisson’s equation

[TABLE]

where $f^{1},\ldots,f^{n}\in L^{2}(\Omega)$ are fixed form functions and $u_{a},u_{b}\in L^{2}(\Omega)$ are given functions satisfying $u_{a}<u_{b}$ almost everywhere on $\Omega$. The variables $y$ and $u$ are chosen from the respective spaces $H^{1}_{0}(\Omega)$ and $L^{2}(\Omega)$. The underlying PDE has to be understand in weak sense in $H^{-1}(\Omega):=H^{1}_{0}(\Omega)^{\star}$. In this regard, the source term $x^{s}+u$ from $L^{2}(\Omega)$ is embedded into $H^{-1}(\Omega)$, implicitly. Noting that no regularization term w.r.t. the control appears in the objective functional, optimal controls are promoted which take values only at the lower and upper bound $u_{a}$ and $u_{b}$, and such controls are referred to as bang-bang, see Troeltzsch2009 .

Let $\Psi\colon\mathbb{R}^{n}\times L^{2}(\Omega)\rightrightarrows H^{1}_{0}(\Omega)\times L^{2}(\Omega)$ be the solution map associated with the above optimal control problem. In the superordinate upper level problem, we aim to identify the lower level desired state via correct choice of the weights $x^{w}\in\mathbb{R}^{n}$ and constant source $x^{s}\in L^{2}(\Omega)$ from a nonempty, closed, convex, and bounded set $X_{\textup{ad}}\subset\mathbb{R}^{n}\times L^{2}(\Omega)$ such that a resulting optimal solution is close to observed data functions $y_{\textup{o}},u_{\textup{o}}\in L^{2}(\Omega)$. A suitable model for this program is given by

[TABLE]

Due to continuity and convexity of the appearing objective functionals, they are weakly sequentially lower semicontinuous. Furthermore, the compactness of $H^{1}_{0}(\Omega)\hookrightarrow L^{2}(\Omega)$ even guarantees that the objective of the lower level problem is weakly sequentially continuous. The set $X_{\textup{ad}}$ is nonempty and weakly sequentially compact by assumption. Exploiting the linearity and continuity of the solution operator $(-\Delta)^{-1}$ of Poisson’s equation, it is not difficult to see that the graph of the lower level feasible set mapping $\Gamma$ is convex and closed. The boundedness of $X_{\textup{ad}}$ ensures the boundedness of $(X_{\textup{ad}}\times H^{1}_{0}(\Omega)\times L^{2}(\Omega))\cap\operatorname{gph}\Gamma$, and it is not difficult to see that this set is weakly sequentially compact as well. Using the properties of $(-\Delta)^{-1}$, it is easy to see that $\Gamma$ is weakly-weakly inner semicontinuous at all points of its graph. Now, Theorem 3.5 yields the existence of a solution to the bilevel optimal control problem under consideration.

Example 3

Optimal control of ODEs with terminal penalty cost For a fixed given vector $\xi\in\mathbb{R}^{n}$ of parameters, we consider the parametric optimization problem

[TABLE]

where $j\colon\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}$ is continuous and $g\colon\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{k}$ is continuously differentiable. Furthermore, we assume that $\Upsilon(\xi):=\{y\in\mathbb{R}^{m}\,|\,g(\xi,y)\leq 0\}$ is nonempty for each $\xi\in\mathbb{R}^{n}$, that $\bigcup_{\xi\in\mathbb{R}^{n}}\Upsilon(\xi)$ is bounded, and that the Mangasarian–Fromovitz constraint qualification holds at all feasible points associated with (4).

The associated upper level problem shall be given by

[TABLE]

where $I:=(0,T)$ is a time interval, $x_{\textup{d}}\in L^{2}(I;\mathbb{R}^{n})$ is a desired state, $\sigma\geq 0$ is a regularization parameter, $J\colon\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}$ is lower semicontinuous, $A\in\mathbb{R}^{n\times n}$ as well as $B\in\mathbb{R}^{n\times p}$ are fixed matrices, $u_{a},u_{b}\in L^{2}(I;\mathbb{R}^{p})$ are fixed functions satisfying $u_{a}<u_{b}$ almost everywhere on $I$, and $\Psi\colon H^{1}(I;\mathbb{R}^{n})\rightrightarrows\mathbb{R}^{m}$ assigns to each $x\in H^{1}(I;\mathbb{R}^{n})$ the solution set of (4) for the fixed parameter $\xi:=x(T)$. The controls in (5) are chosen from $L^{2}(I;\mathbb{R}^{p})$.

Problem (5) describes the situation where an ODE system has to be controlled in such a way that certain penalty cost resulting from the terminal value of the state function as well as the distance to a desirable state are minimized with minimal control effort. Optimization problems of this kind arise in the context of gas balancing in energy networks and were studied in BenitaDempeMehlitz2016 ; BenitaMehlitz2016 ; KalashnikovBenitaMehlitz2015 . Invoking Remark 3.3, the subsequently stated example, and Theorem 3.5, we obtain the existence of an optimal solution associated with (5).

Another typical situation arises when the lower level problem (1) is uniquely solvable for each upper level feasible point.

Theorem 3.6

Assume that there exists a map $\psi\colon X_{\textup{ad}}\to\mathcal{Z}$ sending weakly convergent sequences from $X_{\textup{ad}}$ to weakly convergent sequences in $\mathcal{Z}$ such that $\Psi(x)=\{\psi(x)\}$ holds for all $x\in X_{\textup{ad}}$. Then, (BPP) possesses an optimal solution.

Proof

The assumptions of the theorem guarantee that (BPP) is equivalent to

[TABLE]

Furthermore, $X_{\textup{ad}}\ni x\mapsto F(x,\psi(x))\in\mathbb{R}$ is weakly sequentially lower semicontinuous on $X_{\textup{ad}}$ since $F$ is possesses this property on $\mathcal{X}\times\mathcal{Z}$ while $\psi$ preserves weak convergence of sequences from $X_{\textup{ad}}$. Thus, the above problem possesses an optimal solution $\bar{x}$, i.e. (BPP) possesses the optimal solution $(\bar{x},\psi(\bar{x}))$.

The above theorem particularly applies to situations where the upper level variable comes from a finite-dimensional Banach space while the solution operator associated to the lower level problem is continuous. This setting has been discussed in DempeHarderMehlitzWachsmuth2019 ; HarderWachsmuth2019 and will be of interest in Section 4.

3.2 How to derive necessary optimality conditions in bilevel optimal control

In order to derive necessary optimality conditions for bilevel programming problems, one generally aims to transfer the hierarchical model into a single-level program first. Therefore, three major approaches are suggested in the literature. First, whenever the lower level problem possesses a uniquely determined solution for each fixed value of the upper level problem, one could use the associated solution operator to eliminate the lower level variable from the model. This approach has been used in HarderWachsmuth2019 ; Mehlitz2017 in order to derive necessary optimality conditions for bilevel optimal control problems. Second, it is possible to exploit the optimal value function from (2) in order to replace (BPP) equivalently by the so-called optimal value reformulation

[TABLE]

In BenitaDempeMehlitz2016 ; BenitaMehlitz2016 ; DempeHarderMehlitzWachsmuth2019 ; Ye1995 ; Ye1997 , the authors exploited this idea to infer optimality conditions in the context of bilevel optimal control. We will demonstrate in Section 4, how a relaxation method can be combined with the optimal value approach in order to obtain a satisfactory stationarity condition for a particular problem class from inverse optimal control. Finally, as long as the lower level problem is convex w.r.t. $z$ and regular in the sense that a constraint qualification is satisfied at each feasible point, it is possible to replace the implicit constraint $z\in\Psi(x)$ by suitable necessary and sufficient optimality conditions of Karush–Kuhn–Tucker (KKT) type. In the context of bilevel optimal control, this approach has been discussed in MehlitzWachsmuth2016 . In this section, we will briefly sketch this last approach. Therefore, we have to fix some assumptions first.

Assumption 3.7

We assume that the mapping $\Gamma$ is given as stated in Remark 3.3 where $C$ is a cone. Furthermore, we suppose that $f(x,\cdot)\colon\mathcal{Z}\to\mathbb{R}$ is convex and that $g(x,\cdot)\colon\mathcal{Z}\to\mathcal{W}$ is $C$-convex for each $x\in X_{\text{ad}}$. The latter means that

[TABLE]

holds true.

Du to the postulated assumptions, for fixed $x\in X_{\textup{ad}}$, $z\in\Psi(x)$ holds true if and only if there exists a Lagrange multiplier $\lambda\in\mathcal{W}^{\star}$ which solves the associated lower level KKT system which is given as stated below:

[TABLE]

Here, it was essential that the lower level problem is convex w.r.t. $z$ while Robinson’s constraint qualification (3) is valid at all lower level feasible points. Due to the above arguments, it is now reasonable to investigate the so-called KKT reformulation associated with (BPP) which is given as stated below:

[TABLE]

Let us note that the lower level Lagrange multiplier $\lambda$ plays the role of a variable in (KKT). This may cause that the problems (BPP) and (KKT) are not equivalent w.r.t. local minimizers as soon as $\lambda$ is not uniquely determined for each $x\in X_{\textup{ad}}$ where $\Psi(x)\neq\varnothing$ holds, see Mehlitz2017:2 . As reported in DempeDutta2012 , this phenomenon is already present in standard finite-dimensional bilevel programming.

In the situation where $C=\{0\}$ holds, the final two constraints in (KKT) are trivial and can be omitted. Then, (KKT) reduces to a standard nonlinear program in Banach spaces which can be tackled via classical arguments. Related considerations can be found in HollerKunischBarnard2018 . The subsequently stated example visualizes this approach.

Example 4

Inverse control of Poisson’s equation For a bounded domain $\Omega\subset\mathbb{R}^{d}$ and a parameter vector $x\in\mathbb{R}^{n}$, we consider the parametric optimal control problem

[TABLE]

where $f^{1},\ldots,f^{n}\in L^{2}(\Omega)$ are given form functions and $\sigma>0$ is a regularization parameter. For observations $y_{\textup{o}},u_{\textup{o}}\in L^{2}(\Omega)$ and a nonempty, convex, compact set $X_{\textup{ad}}\subset\mathbb{R}^{n}$, we consider the superordinate inverse optimal control problem

[TABLE]

where $\Psi\colon\mathbb{R}^{n}\rightrightarrows H^{1}_{0}(\Omega)\times L^{2}(\Omega)$ represents the solution mapping of the aforementioned parametric optimal control problem. We can use Theorem 3.6 in order to infer the existence of an optimal solution associated with this bilevel optimal control problem.

Noting that $-\Delta\colon H^{1}_{0}(\Omega)\to H^{-1}(\Omega)$ provides an isomorphism, the associated KKT reformulation, given by

[TABLE]

is equivalent to the original hierarchical model. One can easily check that Robinson’s constraint qualification is valid at each feasible point of this program which means that its KKT conditions provide a necessary optimality condition for the underlying inverse optimal control problem. Thus, whenever $(\bar{x},\bar{y},\bar{u})\in\mathbb{R}^{n}\times H^{1}_{0}(\Omega)\times L^{2}(\Omega)$ is a locally optimal solution of (6), then we find multipliers $\bar{z}\in\mathbb{R}^{n}$, $\bar{p},\bar{\mu},\bar{\rho}\in H^{1}_{0}(\Omega)$, and $\bar{w}\in L^{2}(\Omega)$ which satisfy

[TABLE]

In case where $C$ is a non-trivial cone, the final three constraints of (KKT) form a so-called system of complementarity constraints, i.e. this program is a mathematical program with complementarity constrains (MPCC) in Banach spaces. As shown in MehlitzWachsmuth2016 , this results in the violation of Robinson’s constraint qualification at all feasible points of (KKT) and, consequently, the KKT conditions of (KKT) may turn out to be too restrictive in order to yield an applicable necessary optimality condition. Instead, weaker problem-tailored stationarity notions and constraint qualifications need to be introduced which respect the specific variational structure, see Mehlitz2017:2 ; MehlitzWachsmuth2016 ; Wachsmuth2015 ; Wachsmuth2017 . In bilevel optimal control, complementarity constraints are typically induced by the cone of nonnegative functions in a suitable function space, e.g. $L^{2}(\Omega)$, $H^{1}_{0}(\Omega)$, or $H^{1}(\Omega)$. Respective considerations can be found in ClasonDengMehlitzPruefert2019 ; GuoYe2016 ; HarderWachsmuth2018a ; HarderWachsmuth2018b ; MehlitzWachsmuth2018 ; MehlitzWachsmuth2019 .

4 Stationarity conditions in inverse optimal control

In this section, we demonstrate by means of a specific class of parameter reconstruction problems how stationarity conditions in bilevel optimal control can be derived.

For a bounded domain $\Omega\subset\mathbb{R}^{d}$ and a parameter $x\in\mathbb{R}^{n}_{+}$, where $\mathbb{R}^{n}_{+}$ denotes the nonnegative orthant in $\mathbb{R}^{n}$, we study the parametric optimal control problem

[TABLE]

as well as the superordinated bilevel optimal control problem

[TABLE]

where $\Psi\colon\mathbb{R}^{n}\rightrightarrows\mathcal{Y}\times L^{2}(\Omega)$ denotes the solution set mapping of (P$(x)$). In (P$(x)$), the state equation $\mathtt{A}y-\mathtt{B}u=0$ couples the control $u\in L^{2}(\Omega)$ and the state $y\in\mathcal{Y}$. In this regard, $\mathtt{A}$ can be interpreted as a differential operator. Noting that (IOC) is motivated by underlying applications from parameter reconstruction, it is an inverse optimal control program.

Assumption 4.1

We assume that $\mathcal{Y}$ and $\mathcal{W}$ are reflexive Banach spaces. The functional $F\colon\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)\to\mathbb{R}$ is supposed to be continuously Fréchet differentiable and convex. Let $X_{\textup{ad}}\subset\mathbb{R}^{n}_{+}$ be nonempty and compact. The functional $j\colon\mathcal{Y}\to\mathbb{R}^{n}$ is assumed to be twice continuously Fréchet differentiable and its $n$ component functions are supposed to be convex. Moreover, we assume that the mapping $j$ satisfies $j(\mathcal{Y})\subset\mathbb{R}^{n}_{+}$. Furthermore, $\sigma>0$ is fixed. Let linear operators $\mathtt{A}\in\mathbb{L}\left[\mathcal{Y},\mathcal{W}\right]$ as well as $\mathtt{B}\in\mathbb{L}\left[L^{2}(\Omega),\mathcal{W}\right]$ be chosen such that $\mathtt{A}$ is continuously invertible while $\mathtt{B}$ is compact. Finally, we assume that $u_{a},u_{b}\colon\Omega\to\overline{\mathbb{R}}$ are measurable functions such that

[TABLE]

is nonempty.

Below, we present two illustrative examples where all these assumptions hold.

Example 5

Weighted lower level target-type objectives We choose $\mathcal{Y}:=H^{1}_{0}(\Omega)$, $\mathcal{W}:=H^{-1}(\Omega)$, as well as $\mathtt{A}:=-\Delta$ while $\mathtt{B}$ represents the compact embedding $L^{2}(\Omega)\hookrightarrow H^{-1}(\Omega)$. For fixed functions $y^{1}_{\textup{d}},\ldots,y^{n}_{\textup{d}}\in L^{2}(\Omega)$, the lower level objective function is defined by

[TABLE]

The upper level feasible set is given by the standard simplex

[TABLE]

Such bilevel optimal control problems, where the precise form of the lower level target-type objective mapping has to be reconstructed, have been studied in HarderWachsmuth2019 .

Example 6

Optimal measuring Let $\Omega\subset\mathbb{R}^{d}$, $d\in\{2,3\}$, be a bounded Lipschitz domain. We fix $p\in(3,6)$ as in (JerisonKenig1995, , Theorem 0.5). Let us set $\mathcal{Y}:=W^{1,p}_{0}(\Omega)$ and $\mathcal{W}:=W^{-1,p}(\Omega)$. Again, we fix $\mathtt{A}:=-\Delta$, and $\mathtt{B}$ represents the embedding $L^{2}(\Omega)\hookrightarrow W^{-1,p}(\Omega):=W_{0}^{1,p^{\prime}}(\Omega)^{\star}$ where $p^{\prime}=p/(p-1)$ is the conjugate coefficient associated with $p$. According to (JerisonKenig1995, , Theorem 0.5), $\mathtt{A}$ is continuously invertible. Due to the Rellich–Kondrachov theorem, the embedding from $W_{0}^{1,p^{\prime}}(\Omega)$ to $L^{2}(\Omega)$ is compact. Since $\mathtt{B}$ is the adjoint of this embedding, Schauder’s theorem implies the compactness of $\mathtt{B}$. Furthermore, we note that the embedding $W^{1,p}_{0}(\Omega)\hookrightarrow C(\overline{\Omega})$ is compact in this setting.

Let $\omega^{1},\ldots,\omega^{n}\in\overline{\Omega}$ be fixed points. We consider the lower level objective function given by

[TABLE]

where $y_{\textup{d}}\in C(\overline{\Omega})$ is a given desired state. Noting that the state space $W^{1,p}_{0}(\Omega)$ is continuously embedded into $C(\overline{\Omega})$, this functional is well-defined. At the upper level stage, we minimize

[TABLE]

where $x$ comes from the standard simplex given in (7). The associated bilevel optimal control problem optimizes the measurement of the distance between the actual state and the desired state by reduction to pointwise evaluations.

4.1 The lower level problem

For brevity, we denote by $f\colon\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)\to\mathbb{R}$ the objective functional of (P$(x)$). By construction, the map $f(x,\cdot,\cdot)\colon\mathcal{Y}\times L^{2}(\Omega)\to\mathbb{R}$ is convex for each $x\in\mathbb{R}^{n}_{+}$.

Lemma 4.2

For each $x\in\mathbb{R}^{n}_{+}$, (P$(x)$) possesses a unique optimal solution.

Proof

Noting that $\mathtt{A}$ is an isomorphism, we may consider the state-reduced problem

[TABLE]

where $\mathtt{S}:=\mathtt{A}^{-1}\circ\mathtt{B}\in\mathbb{L}\left[L^{2}(\Omega),\mathcal{Y}\right]$ is the solution operator of the constraining PDE. Due to the above considerations, the linearity of $\mathtt{S}$, and the continuity of all appearing functions, the objective functional of (8) is convex and continuous. Observing that $x\cdot j(\mathtt{S}u)\geq 0$ holds for all $u\in L^{2}(\Omega)$ while $L^{2}(\Omega)\ni u\mapsto\tfrac{\sigma}{2}\left\|u\right\|_{L^{2}(\Omega)}^{2}\in\mathbb{R}$ is coercive, the objective of (8) is already strongly convex w.r.t. $u$. Since $U_{\textup{ad}}$ is weakly sequentially closed, (8) needs to possess a unique solution $\bar{u}$. Consequently, $(\mathtt{S}\bar{u},\bar{u})$ is the uniquely determined solution of (P$(x)$).

Observing that the lower level problem (P$(x)$) is regular in the sense that Robinson’s constraint qualification is valid at all feasible points, see Remark 3.3, its uniquely determined solution for the fixed parameter $x\in\mathbb{R}^{n}_{+}$ is characterized by the associated KKT system

[TABLE]

where $p\in\mathcal{W}^{\star}$ and $\lambda\in L^{2}(\Omega)$ are the Lagrange multipliers.

The finding of Lemma 4.2 allows us to introduce mappings $\psi^{y}\colon\mathbb{R}^{n}_{+}\to\mathcal{Y}$ and $\psi^{u}\colon\mathbb{R}^{n}_{+}\to L^{2}(\Omega)$ by $\Psi(x)=\{(\psi^{y}(x),\psi^{u}(x))\}$ for all $x\in\mathbb{R}^{n}_{+}$. Since $\mathtt{A}^{\star}$ is continuously invertible, $p$ is uniquely determined by (9a) and, consequently, the uniqueness of $\lambda$ follows from (9b). This gives rise to the mappings $\phi^{p}\colon\mathbb{R}^{n}_{+}\to\mathcal{W}^{\star}$ and $\phi^{\lambda}\colon\mathbb{R}^{n}_{+}\to L^{2}(\Omega)$ that assign to each $x\in\mathbb{R}^{n}_{+}$ the lower level Lagrange multipliers $p$ and $\lambda$ which characterize the unique minimizer $(\psi^{y}(x),\psi^{u}(x))$, respectively.

Before we continue, we give an auxiliary result on $j$.

Lemma 4.3

Let $\hat{X}\subset\mathbb{R}^{n}_{+}$ be compact. Then, there exists a constant $C>0$, such that

[TABLE]

with $y_{i}:=\psi^{y}(x_{i})$, $i=1,2$, holds for all $x_{1},x_{2}\in\hat{X}$.

Proof

Since $\hat{X}$ is assumed to be compact, the points $y_{2}+(1-t)\,y_{1}$, $t\in[0,1]$, belong to the compact set $\hat{\mathcal{Y}}:=\operatorname{cl}\operatorname{conv}\{\psi^{y}(\hat{x})\,|\,\hat{x}\in\hat{X}\}$. Since $j^{\prime\prime}$ is continuous, we have $C:=\sup_{\hat{y}\in\hat{\mathcal{Y}}}\left\|j^{\prime\prime}(\hat{y})\right\|<\infty$. Now, the Taylor estimate follows from (Cartan1967, , Theorem 5.6.1) and the Lipschitz estimate on $j^{\prime}$ is clear.

Below, we want to study the continuity properties of the mappings $\psi^{y}$ and $\psi^{u}$ as well as $\phi^{p}$ and $\phi^{\lambda}$.

Lemma 4.4

There are continuous functions $C_{y},C_{u}\colon\mathbb{R}^{n}_{+}\to\mathbb{R}$ such that the following estimates hold:

[TABLE]

Particularly, $\psi^{y}$ and $\psi^{u}$ are Lipschitz continuous on $X_{\textup{ad}}$. Additionally, $\phi^{p}$ and $\phi^{\lambda}$ are continuous on $\mathbb{R}^{n}_{+}$ and Lipschitz continuous on $X_{\textup{ad}}$.

Proof

Fix $x_{1},x_{2}\in\mathbb{R}^{n}_{+}$ arbitrarily and set $y_{i}:=\psi^{y}(x_{i})$ as well as $u_{i}:=\psi^{u}(x_{i})$ for $i=1,2$. Furthermore, let $p_{i}\in\mathcal{W}^{\star}$ and $\lambda_{i}\in L^{2}(\Omega)$ be the multipliers which solve (9) for $i=1,2$. Testing the associated condition (9b) with $u_{2}-u_{1}$ and exploiting (9c), we have

[TABLE]

Adding up these inequalities yields

[TABLE]

Next, we rearrange this inequality and exploit (9a), $y_{i}=(\mathtt{A}^{-1}\circ\mathtt{B})u_{i}$, $i=1,2$, as well as the convexity of the mapping $\mathcal{Y}\ni y\mapsto x_{2}\cdot j(y)\in\mathbb{R}$ in order to obtain

[TABLE]

for some constant $C>0$ which does not depend on $x_{i}$, $y_{i}$, and $u_{i}$, $i=1,2$. This way, we have

[TABLE]

which yields the estimate

[TABLE]

As a consequence, the map $\psi^{u}$ is continuous everywhere on $\mathbb{R}^{n}_{+}$. Observing that $\psi^{y}=\mathtt{A}^{-1}\circ\mathtt{B}\circ\psi^{u}$ holds, $\psi^{y}$ is continuous on $\mathbb{R}^{n}_{+}$ as well. Recalling that $j$ is continuously Fréchet differentiable, the desired estimates follow by setting

[TABLE]

This completes the proof for $\psi^{y}$ and $\psi^{u}$.

The continuity of $\phi^{p}$ and $\phi^{\lambda}$ on $\mathbb{R}^{n}_{+}$ follows easily by continuity of $\psi^{y}$ and $\psi^{u}$ exploiting (9a), (9b), and the continuity of $j^{\prime}$. Since the map $j^{\prime}\circ\psi^{y}\colon\mathbb{R}^{n}_{+}\to\mathbb{L}\left[\mathcal{Y},\mathbb{R}^{n}\right]$ is continuous on $\mathbb{R}^{n}_{+}$ and, by Lemma 4.3, Lipschitz on the compact set $X_{\textup{ad}}$, we obtain

[TABLE]

for all $x_{1},x_{2}\in X_{\textup{ad}}$ and some constants $\hat{C},C>0$. The Lipschitz continuity of $\phi^{\lambda}$ on $X_{\textup{ad}}$ now follows from (9b).

We introduce a function $\varphi\colon\mathbb{R}^{n}_{+}\to\mathbb{R}$ by means of

[TABLE]

Due to the above lemma, $\varphi$ is continuous and equals the optimal value function associated with (P$(x)$). Observing that the function $f$ is affine w.r.t. the parameter $x$, it is easy to see that $\varphi$ is concave, see (FiaccoKyparisis1986, , Proposition 3.5) as well.

Next, we are going to study the differentiability of $\varphi$. We are facing the problem that $\varphi$ is only defined on the closed set $\mathbb{R}^{n}_{+}$. In fact, for $x\in\mathbb{R}^{n}\setminus\mathbb{R}^{n}_{+}$ the objective function of (P$(x)$) might be non-convex or unbounded from below and (P$(x)$) might fail to possess a minimizer. To circumvent this difficulty, we first prove that $\varphi$ admits a first-order Taylor expansion, and then we extend $\varphi$ to a continuously differentiable function via Whitney’s extension theorem.

Lemma 4.5

We define the function $\varphi^{\prime}\colon\mathbb{R}^{n}_{+}\to\mathbb{R}^{n}$ via

[TABLE]

Then, $\varphi^{\prime}$ is continuous and for every compact subset $\hat{X}\subset\mathbb{R}^{n}_{+}$ there exists a constant $C>0$ such that the Taylor-like estimate

[TABLE]

holds.

Proof

The continuity of $\varphi^{\prime}$ follows from Lemma 4.4. Now, let $\hat{X}\subset\mathbb{R}^{n}_{+}$ be compact. For arbitrary $\bar{x},x\in\hat{X}$, we define $\bar{y}:=\psi^{y}(\bar{x})$, $\bar{u}:=\psi^{u}(\bar{x})$, $y:=\psi^{y}(x)$, and $u:=\psi^{u}(x)$. Then, we have

[TABLE]

Next, we are going to employ the optimality condition (9). To this end, we denote the multipliers at the solution $(y,u)$ for the parameter $x$ by $p\in\mathcal{W}^{\star}$ and $\lambda\in L^{2}(\Omega)$. Now, (9) implies

[TABLE]

and

[TABLE]

If we denote by $\bar{\lambda}$ the multiplier associated to the parameter $\bar{x}$, (9c) implies

[TABLE]

By combining the above estimates, we get

[TABLE]

Now, the claim follows from Lemmas 4.3 and 4.4.

Next, we employ Whitney’s extension theorem to extend $\varphi$ to all of $\mathbb{R}^{n}$.

Lemma 4.6

There exists a continuously differentiable function $\hat{\varphi}\colon\mathbb{R}^{n}\to\mathbb{R}$ such that $\hat{\varphi}(x)=\varphi(x)$ and $\hat{\varphi}^{\prime}(x)=\varphi^{\prime}(x)$ for all $x\in X_{\textup{ad}}$. Here, $\varphi^{\prime}$ is the function defined in Lemma 4.5.

Proof

In order to apply Whitney’s extension theorem, see (Hoermander2003, , Theorem 2.3.6), we have to show that the function $\eta\colon X_{\textup{ad}}\times X_{\textup{ad}}\to\mathbb{R}$, defined via

[TABLE]

for $x,y\in X_{\textup{ad}}$, is continuous on $X_{\textup{ad}}\times X_{\textup{ad}}$. It is clear that this function is continuous at $(x,y)\in X_{\textup{ad}}\times X_{\textup{ad}}$ for $x\neq y$. Hence, it remains to show

[TABLE]

for all $a\in X_{\textup{ad}}$, but this follows from Lemma 4.5 since $X_{\textup{ad}}$ is compact.

Note that we extended $\varphi$ from $X_{\textup{ad}}$ to $\mathbb{R}^{n}$ in Lemma 4.6. Technically, this means that the extended function $\hat{\varphi}$ may possess different values than the origin optimal value function $\varphi$ on $\mathbb{R}^{n}_{+}\setminus X_{\textup{ad}}$. This, however, does not cause any trouble in the subsequent considerations since we focus on parameters from $X_{\textup{ad}}$ only.

4.2 The optimal value reformulation and its relaxation

Based on Lemma 4.4, the following result follows from Theorem 3.6 while noting that the upper level variables are chosen from a finite-dimensional Banach space.

Theorem 4.7

Problem (IOC) possesses an optimal solution.

Our aim is to characterize the local minimizers of (IOC) by means of necessary optimality conditions. In order to do so, we want to exploit the continuously differentiable extension $\hat{\varphi}\colon\mathbb{R}^{n}\to\mathbb{R}$ of the optimal value function $\varphi$ associated with (P$(x)$), see Lemma 4.6. Observing that $\hat{\varphi}(x)=\varphi(x)$ holds true for all $x\in X_{\textup{ad}}$, (IOC) can be transferred into the equivalent problem

[TABLE]

where $f$ still denotes the objective of the lower level problem (P$(x)$). This is a single-level optimization problem with continuously Fréchet differentiable data functions, see Lemma 4.6. However, it is easy to check that Robinson’s constraint qualification does not hold at the feasible points of (OVR), see e.g. (DempeHarderMehlitzWachsmuth2019, , Lemma 5.1). This failure is mainly caused by the fact that $f(x,y,u)-\hat{\varphi}(x)\leq 0$ is in fact an equality constraint by definition of the optimal value function. Due to this lack of regularity, one cannot expect that the classical KKT conditions provide a necessary optimality condition for (OVR). Furthermore, the nonlinearity of $f$ provokes that the smooth mapping $\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)\ni(x,y,u)\mapsto f(x,y,u)-\hat{\varphi}(x)\in\mathbb{R}$ may not serve as an exact penalty function around local minimizers of (OVR). Thus, approaches related to partial penalization w.r.t. the constraint $f(x,y,u)-\hat{\varphi}(x)\leq 0$, see e.g. BenitaDempeMehlitz2016 ; BenitaMehlitz2016 ; Ye1995 ; Ye1997 , do not seem to be promising here.

In order to overcome these difficulties, we are going to relax this critical constraint. More precisely, for a sequence $\{\varepsilon_{k}\}_{k\in\mathbb{N}}\subset\mathbb{R}$ of positive relaxation parameters satisfying $\varepsilon_{k}\downarrow 0$, we investigate the programs

[TABLE]

One can easily check that this relaxation provokes regularity of all feasible points. A formal proof of this result parallels the one of (DempeHarderMehlitzWachsmuth2019, , Lemma 5.2).

We first want to state an existence result for (OVR$(\varepsilon_{k})$).

Lemma 4.8

For each $k\in\mathbb{N}$, (OVR$(\varepsilon_{k})$) possesses an optimal solution.

Proof

Let $\{(x_{l},y_{l},u_{l})\}_{l\in\mathbb{N}}\subset\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)$ be a minimizing sequence of (OVR$(\varepsilon_{k})$), i.e. a sequence of feasible points whose associated objective values tend to the infimal value $\alpha\in\overline{\mathbb{R}}$ of (OVR$(\varepsilon_{k})$). The compactness of $X_{\textup{ad}}$ implies that $\{x_{l}\}_{l\in\mathbb{N}}$ is bounded and, thus, converges along a subsequence (without relabeling) to $\bar{x}\in X_{\textup{ad}}$. By feasibility, we have

[TABLE]

for each $l\in\mathbb{N}$. By boundedness of $\{x_{l}\}_{l\in\mathbb{N}}$ and continuity of $\hat{\varphi}$ on $X_{\textup{ad}}$, we obtain boundedness of $\{u_{l}\}_{l\in\mathbb{N}}$. Consequently, the latter converges weakly (along a subsequence without relabeling) to $\bar{u}\in L^{2}(\Omega)$ which belongs to the weakly sequentially closed set $U_{\textup{ad}}$. Since $\mathtt{B}$ is compact, we have $y_{l}\to\bar{y}$ in $\mathcal{Y}$ by validity of the state equation. Here, we used $\bar{y}:=(\mathtt{A}^{-1}\circ\mathtt{B})\bar{u}$.

Recall that $j$ and $\hat{\varphi}$ are continuous functions. Exploiting the weak sequential lower semicontinuity of (squared) norms, we obtain

[TABLE]

i.e. $(\bar{x},\bar{y},\bar{u})$ is feasible to (OVR$(\varepsilon_{k})$). Finally, the weak sequential lower semicontinuity of $F$ yields

[TABLE]

i.e. $(\bar{x},\bar{y},\bar{u})$ is a global minimizer of (OVR$(\varepsilon_{k})$).

Next, we investigate the behavior of a sequence $\{(\bar{x}_{k},\bar{y}_{k},\bar{u}_{k})\}_{k\in\mathbb{N}}$ of global minimizers associated with (OVR$(\varepsilon_{k})$) as $k\to\infty$.

Theorem 4.9

For each $k\in\mathbb{N}$, let $(\bar{x}_{k},\bar{y}_{k},\bar{u}_{k})\in\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)$ be a global minimizer of (OVR$(\varepsilon_{k})$). Then, the sequence $\{(\bar{x}_{k},\bar{y}_{k},\bar{u}_{k})\}_{k\in\mathbb{N}}$ possesses a subsequence (without relabeling) such that the convergences $\bar{x}_{k}\to\bar{x}$, $\bar{y}_{k}\to\bar{y}$, and $\bar{u}_{k}\to\bar{u}$ hold where $(\bar{x},\bar{y},\bar{u})$ is a global minimizer of (OVR) and, thus, of (IOC).

Proof

Due to compactness of $X_{\textup{ad}}$, the sequence $\{\bar{x}_{k}\}_{k\in\mathbb{N}}$ is bounded and converges along a subsequence (without relabeling) to some $\bar{x}\in X_{\textup{ad}}$. We set $\bar{y}:=\psi^{y}(\bar{x})$ and $\bar{u}:=\psi^{u}(\bar{x})$.

Let us set $y_{k}:=\psi^{y}(\bar{x}_{k})$ and $u_{k}:=\psi^{u}(\bar{x}_{k})$ for each $k\in\mathbb{N}$. Due to the componentwise convexity and differentiability of the mapping $j$, we obtain

[TABLE]

Here, we used feasibility of $(\bar{y}_{k},\bar{u}_{k})$ and optimality of $(y_{k},u_{k})$ for (P$(x)$) where $x:=\bar{x}_{k}$ holds.

The above considerations as well as the continuity of $\psi^{u}$ yield

[TABLE]

i.e. $\bar{u}_{k}\to\bar{u}$ follows. Due to $\bar{y}_{k}=(\mathtt{A}^{-1}\circ\mathtt{B})\bar{u}_{k}$ and $\bar{y}=(\mathtt{A}^{-1}\circ\mathtt{B})\bar{u}$, we also have $\bar{y}_{k}\to\bar{y}$.

Since each feasible point $(x,y,u)\in\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)$ of (OVR) is a feasible point of (OVR$(\varepsilon_{k})$) for arbitrary $k\in\mathbb{N}$, we have $F(\bar{x}_{k},\bar{y}_{k},\bar{u}_{k})\leq F(x,y,u)$ for all $k\in\mathbb{N}$. Taking the limit $k\to\infty$ while observing that $F$ is (weakly sequentially) lower semicontinuous, we have $F(\bar{x},\bar{y},\bar{u})\leq F(x,y,u)$, i.e. $(\bar{x},\bar{y},\bar{u})$ is a global minimizer of (OVR) and, thus, of (IOC).

Clearly, the above theorem is of limited use for the numerical treatment of (IOC) since the programs (OVR$(\varepsilon_{k})$) are nonconvex optimal control problems whose constraints comprise the implicitly known function $\hat{\varphi}$. However, we can exploit Theorem 4.9 for the derivation of a necessary optimality condition for (IOC).

4.3 Derivation of stationarity conditions

For the derivation of a necessary optimality condition which characterizes the local minimizers of (IOC), we will exploit the relaxation approach described in Section 4.2. Combining the KKT systems of (OVR$(\varepsilon_{k})$) and (P$(x)$) will lead to a stationarity system for global minimizers of (IOC). Afterwards, this result can be extended to all local minimizers of (IOC).

As already mentioned in Section 4.2, we cannot rely on the KKT conditions of (OVR) to be applicable necessary optimality conditions for (IOC). In order to derive a reasonable stationarity system, we first observe that for given $x\in X_{\textup{ad}}$, we can characterize $(\psi^{y}(x),\psi^{u}(x))$ to be the uniquely determined solution of the KKT system (9) associated with (P$(x)$). Plugging this system into the constraints of (IOC) in order to eliminate the implicit constraint $(y,u)\in\Psi(x)$, we arrive at the associated KKT reformulation

[TABLE]

where $\mathcal{N}_{U_{\textup{ad}}}\colon L^{2}(\Omega)\rightrightarrows L^{2}(\Omega)$ denotes the normal cone mapping associated with $U_{\textup{ad}}$. A simple calculation shows

[TABLE]

In order to infer a stationarity system for (IOC), we compute the roots of the partial derivatives of the MPCC-Lagrangian associated with (10). The properties of the multipliers which address the equilibrium condition $(u,\lambda)\in\operatorname{gph}\mathcal{N}_{U_{\textup{ad}}}$ are motivated by the pointwise structure of this set and the theory on finite-dimensional complementarity problems.

Definition 4.10

A feasible point $(\bar{x},\bar{y},\bar{u})\in\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)$ of (IOC) is said to be weakly stationary (W-stationary) whenever there exist multipliers $\bar{z}\in\mathbb{R}^{n}$, $\bar{\mu}\in\mathcal{Y}$, $\bar{p},\bar{\rho}\in\mathcal{W}^{\star}$, and $\bar{\lambda},\bar{w},\bar{\xi}\in L^{2}(\Omega)$ which satisfy

[TABLE]

Whenever these multipliers additionally satisfy

[TABLE]

$(\bar{x},\bar{y},\bar{u})$* is called Clarke-stationary (C-stationary). If (12) can be strengthened to*

[TABLE]

then $(\bar{x},\bar{y},\bar{u})$ is referred to as strongly stationary (S-stationary). Here, the measurable sets $I^{a+}(\bar{u})$ and $I^{b-}(\bar{u})$ are given by

[TABLE]

Note that all subsets of $\Omega$ appearing above are well-defined up to subsets of $\Omega$ possessing measure zero.

Observe that the conditions (11f) - (11i) just provide the KKT system (9) of (P$(x)$) for $x:=\bar{x}$ which characterizes the associated lower level Lagrange multipliers $\bar{p}$ and $\bar{\lambda}$. This way, a feasible point of (10) is fixed and the actual respective W-, C-, and S-stationarity conditions can be inferred.

In line with the results from DempeHarderMehlitzWachsmuth2019 ; HarderWachsmuth2019 , we are going to show that the local minimizers of (IOC) are C-stationary. In order to do that, we choose an arbitrary sequence $\{\varepsilon_{k}\}_{k\in\mathbb{N}}\subset\mathbb{R}$ of positive penalty parameters tending to zero as $k\to\infty$. Due to Lemma 4.8, the program (OVR$(\varepsilon_{k})$) possesses a global minimizer $(\bar{x}_{k},\bar{y}_{k},\bar{u}_{k})\in\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)$. As we mentioned in Section 4.2, (OVR$(\varepsilon_{k})$) is regular as well as smooth at this point and, thus, we find multipliers $z_{k}\in\mathbb{R}^{n}$, $\alpha_{k}\in\mathbb{R}$, $p_{k}\in\mathcal{W}^{\star}$, and $\lambda_{k}\in L^{2}(\Omega)$ which solve the associated KKT system

[TABLE]

Furthermore, an evaluation of the lower level KKT system (9) yields

[TABLE]

Recall that $\phi^{p}\colon\mathbb{R}^{n}_{+}\to\mathcal{W}^{\star}$ and $\phi^{\lambda}\colon\mathbb{R}^{n}_{+}\to L^{2}(\Omega)$ denote the Lagrange multiplier mappings associated with the lower level problem (P$(x)$) which are continuous due to Lemma 4.4.

Due to Theorem 4.9, we may assume that we have $\bar{x}_{k}\to\bar{x}$, $\bar{y}_{k}\to\bar{y}$, and $\bar{u}_{k}\to\bar{u}$ where $(\bar{x},\bar{y},\bar{u})\in\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)$ is a global minimizer of (IOC).

Summarizing all these assumptions, we obtain the following results.

Lemma 4.11

There exist $\bar{z}\in\mathbb{R}^{n}$, $\bar{\mu}\in\mathcal{Y}$, $\bar{\rho}\in\mathcal{W}^{\star}$, and $\bar{w},\bar{\xi}\in L^{2}(\Omega)$ such that the convergences

[TABLE]

hold at least along a subsequence. Furthermore, the above limits satisfy the conditions (11a), (11b), (11c), (11d), and (11e).

Proof

We multiply (14a) by $\alpha_{k}$ and subtract the resulting equation from (13b) in order to obtain

[TABLE]

Testing this equation with $\bar{y}_{k}-\psi^{y}(\bar{x}_{k})$ while noticing that the first-order derivative of a convex function is a monotone operator, we have

[TABLE]

This is used to obtain

[TABLE]

for some constant $C>0$ since $\{F^{\prime}_{y}(\bar{x}_{k},\bar{y}_{k},\bar{u}_{k})\}_{k\in\mathbb{N}}$ is bounded. Next, we multiply (14b) by $\alpha_{k}$ and subtract this from (13c) in order to obtain

[TABLE]

Testing this with $\bar{u}_{k}-\psi^{u}(\bar{x}_{k})$ and exploiting the above estimate as well as the definition of the normal cone, we obtain

[TABLE]

for a constant $\hat{C}>0$. Consequently, the sequence $\{\alpha_{k}(\bar{u}_{k}-\psi^{u}(\bar{x}_{k}))\}_{k\in\mathbb{N}}$ is bounded and, therefore, possesses a weakly convergent subsequence (without relabelling) whose weak limit will be denoted by $\bar{w}$. Thus, we have shown (15c). Due to the relation $\bar{y}_{k}-\psi^{y}(\bar{x}_{k})=(\mathtt{A}^{-1}\circ\mathtt{B})(\bar{u}_{k}-\psi^{u}(\bar{x}_{k}))$ and the compactness of $\mathtt{B}$, we obtain the strong convergence $\alpha_{k}(\bar{y}_{k}-\psi^{y}(\bar{x}_{k}))\to\bar{\mu}$ for some $\bar{\mu}\in\mathcal{Y}$ satisfying (11d). Thus, we have (15b).

Since $j$ is assumed to be continuously Fréchet differentiable, $j$ is strictly differentiable. Noting that the strong convergences $\bar{y}_{k}\to\bar{y}$ and $\psi^{y}(\bar{x}_{k})\to\bar{y}$ hold, we have

[TABLE]

Observing that $\{\alpha_{k}(\bar{y}_{k}-\psi^{y}(\bar{x}_{k}))\}_{k\in\mathbb{N}}$ is particularly bounded, we obtain

[TABLE]

Since the convergence $j^{\prime}(\bar{y})(\alpha_{k}(\bar{y}_{k}-\psi^{y}(\bar{x}_{k})))\to j^{\prime}(\bar{y})\bar{\mu}$ is clear from (15b), we infer

[TABLE]

Observing that $j$ is twice continuously Fréchet differentiable, $j^{\prime}$ is strictly differentiable. Thus, we can reprise the above arguments in order to show the convergence

[TABLE]

Next, we combine (16), (19), and the fact that $\mathtt{A}^{\star}$ is continuously invertible in order to obtain (15d) for some $\bar{\rho}\in\mathcal{W}^{\star}$ (along a subsequence) which satisfies (11b). Now, we can infer (15e) for some $\bar{\xi}\in L^{2}(\Omega)$ from (17) in a similar way. Taking the weak limit in (17) yields (11c). Due to Lemmas 4.5 and 4.6, (13a) is equivalent to

[TABLE]

Due to the convergences $F^{\prime}_{x}(\bar{x}_{k},\bar{y}_{k},\bar{u}_{k})\to F^{\prime}_{x}(\bar{x},\bar{y},\bar{u})$ and (18), we infer (15a) for some $\bar{z}\in\mathbb{R}^{n}$ along a subsequence. Particularly, we have (11a). Finally, (11e) follows by definition of the normal cone while observing $z_{k}\to\bar{z}$ and $\bar{x}_{k}\to\bar{x}$. This completes the proof.

In the subsequent lemma, we characterize the multipliers $\bar{w}$ and $\bar{\xi}$ from Lemma 4.11 in more detail.

Lemma 4.12

Let $\bar{w},\bar{\xi}\in L^{2}(\Omega)$ be the multipliers characterized in Lemma 4.11. Then, (11j), (11k), and (12) hold.

Proof

Due to the strong convergence of $\{\bar{u}_{k}\}_{k\in\mathbb{N}}$ and $\{\psi^{u}(\bar{x}_{k})\}_{k\in\mathbb{N}}$ to $\bar{u}$ in $L^{2}(\Omega)$, these convergences hold pointwise almost everywhere on $\Omega$ along a subsequence (without relabelling). From $\lambda_{k}\in\mathcal{N}_{U_{\textup{ad}}}(\bar{u}_{k})$ and $\phi^{\lambda}(\bar{x}_{k})\in\mathcal{N}_{U_{\textup{ad}}}(\psi^{u}(\bar{x}_{k}))$, we have

[TABLE]

by definition of the normal cone. The aforementioned pointwise a.e. convergence yields $\lambda_{k}(\omega)-\alpha_{k}\phi^{\lambda}(\bar{x}_{k})(\omega)\to 0$ for almost every $\omega\in I^{a+}(\bar{u})\cap I^{b-}(\bar{u})$. Since we already have $\lambda_{k}-\alpha_{k}\phi^{\lambda}(\bar{x}_{k})\rightharpoonup\bar{\xi}$ from (15e), (11j) follows.

Next, we show (11k). If $\{\alpha_{k}\}_{k\in\mathbb{N}}$ is bounded, then $\bar{w}=0$ follows from (15c) and (11k) holds trivially. Thus, we assume $\alpha_{k}\to+\infty$. By continuity of $\phi^{p}$ and $\phi^{\lambda}$, see Lemma 4.4, we have $\phi^{p}(\bar{x}_{k})\to\phi^{p}(\bar{x})$ and $\phi^{\lambda}(\bar{x}_{k})\to\phi^{\lambda}(\bar{x})$. Noting that the lower level Lagrange multipliers are uniquely determined while observing that $\psi^{y}(\bar{x}_{k})\to\bar{y}$ and $\psi^{u}(\bar{x}_{k})\to\bar{u}$ hold, we have $\phi^{p}(\bar{x}_{k})\to\bar{p}$ and $\phi^{\lambda}(\bar{x}_{k})\to\bar{\lambda}$. Here, $\bar{p}\in\mathcal{W}^{\star}$ and $\bar{\lambda}\in L^{2}(\Omega)$ satisfy the conditions (11f), (11g), (11h), and (11i). Thus, (15e) yields the strong convergence $\alpha_{k}^{-1}\lambda_{k}\to\bar{\lambda}$. Let $G\subset\Omega$ be measurable and $\chi_{G}\in L^{\infty}(\Omega)$ be its characteristic function which equals $1$ on $G$ and vanishes on $\Omega\setminus G$. By definition of the normal cone, we have

[TABLE]

Taking the limit $k\to\infty$, we thus obtain $\left\langle\bar{\lambda},\chi_{G}\bar{w}\right\rangle_{L^{2}(\Omega)}=0$. Since $G\subset\Omega$ was chosen arbitrarily, (11k) follows.

Finally, we are going to prove (12). Therefore, we fix an arbitrary measurable set $G\subset\Omega$. We first observe that due to (15c) and (15d), we have

[TABLE]

Now, we can exploit (11c), (17), the weak sequential lower semicontinuity of the map $L^{2}(\Omega)\ni u\mapsto\left\langle\sigma u,\chi_{G}u\right\rangle_{L^{2}(\Omega)}\in\mathbb{R}$, and the definition of the normal cone in order to obtain

[TABLE]

Noting that $G\subset\Omega$ has been chosen arbitrarily, (12) follows.

Above, we have shown that the particular global minimizer $(\bar{x},\bar{y},\bar{u})$ which results from the relaxation approach suggested in Section 4.2 is C-stationary. In order to carry over this analysis to arbitrary local minimizers of (IOC), we exploit a localization argument.

Theorem 4.13

Let $(\bar{x},\bar{y},\bar{u})\in\mathbb{R}^{n}\times\mathcal{Y}\times L^{2}(\Omega)$ be a local minimizer of (IOC). Then, it is C-stationary.

Proof

Invoking Lemma 4.2, there is some $\varepsilon>0$ such that $\bar{x}$ is the unique globally optimal solution of

[TABLE]

where $\mathbb{B}^{\varepsilon}(\bar{x})$ denotes the closed $\varepsilon$-ball around $\bar{x}$. Thus, $(\bar{x},\bar{y},\bar{u})$ is the unique global minimizer of the bilevel programming problem

[TABLE]

Combining Theorem 4.9 as well as Lemmas 4.11 and 4.12, $(\bar{x},\bar{y},\bar{u})$ is a C-stationary point of (20). Noting that the derivative of the functional $\mathbb{R}^{n}\ni x\mapsto\tfrac{1}{2}\left|x-\bar{x}\right|_{2}^{2}\in\mathbb{R}$ vanishes at $\bar{x}$ while $\mathcal{N}_{X_{\textup{ad}}\cap\mathbb{B}^{\varepsilon}(\bar{x})}(\bar{x})=\mathcal{N}_{X_{\textup{ad}}}(\bar{x})$ holds since $\bar{x}$ is an interior point of $\mathbb{B}^{\varepsilon}(\bar{x})$, the C-stationarity conditions of (20) and (IOC) coincide at $(\bar{x},\bar{y},\bar{u})$. This completes the proof.

Remark 4.14

The counterexample from (HarderWachsmuth2019, , Section 3.2) shows that the local minimizers of (IOC) are not S-stationary in general. However, it remains an open question whether the multipliers which solve the C-stationarity system associated with a local minimizer of (IOC) additionally satisfy

[TABLE]

In line with the terminology of finite-dimensional complementarity programming, the resulting stationarity condition may be referred to as the system of (pointwise) Mordukhovich-stationarity. We would like to briefly note that this system cannot be obtained by computing the limiting normal cone to the set $\operatorname{gph}\mathcal{N}_{U_{\textup{ad}}}$ since the latter turns out to be uncomfortably large, see MehlitzWachsmuth2018 . More precisely, this strategy results in the W-stationarity system of (IOC) from Definition 4.10. Additionally, one cannot rely on the limiting variational calculus in $L^{2}(\Omega)$ due to an inherent lack of so-called sequential normal compactness, see Mehlitz2018 . Taking into account the outstanding success of variational analysis in the finite-dimensional setting, these observations are quite unexpected.