= \mathcal L(u_\mu,\mu,p)$ for any $p\in V$.

2.  The cost functional ${\hat{\mathcal{J}}}$ is in general non-convex, thus the existence of a unique minimum for ${\hat{\mathcal{J}}}$ (and thus of ${\mathcal{J}}$) can not be guaranteed.

3.  Let $(\bar u, \bar \mu) \in V \times {\mathcal{P}}$ be a local optimal solution to with $\bar u := u_{\bar \mu}$ the solution of the primal equation for the parameter $\bar\mu$. Then the following constraint qualification holds true: For any $f\in V'$ there exists a pair $(u,\mu) \in V\times {\mathbb{R}}^P$ solving $$\begin{aligned}
                    a_{\bar \mu}(u, v) - \partial_\mu r_{\bar \mu}^{\textnormal{pr}}(\bar u)[v] \cdot \mu = f(v) &&\text{for all } v \in V.
            \end{aligned}$$

4.  Theorem [[\[Thm:existence\_optimalsolution\]]{}]{} does not provide any solution method.

One can derive first-order necessary optimality conditions in order to compute candidates for a local optimal solution of . We refer to [@HPUU2009 Cor. 1.3] for a proof of the following result:

\[prop:first\_order\_opt\_cond\] Let $(\bar u, \bar \mu) \in V \times {\mathcal{P}}$ be a local optimal solution to . Moreover, let Assumption [[\[asmpt:differentiability\]]{}]{} hold true. Then there exists a unique Lagrange multiplier $\bar p\in V$ such that the following first-order necessary optimality conditions hold:

\[eq:optimality\_conditions\] $$\begin{aligned}
        r_{\bar \mu}^{\textnormal{pr}}(\bar u)[v] &= 0 &&\text{for all } v \in 