$\bar \mu\in {\mathcal{P}}$ satisfies the first-order necessary optimality conditions . If ${\hat{\mathcal{H}}}_\mu(\bar \mu)$ is positive definite on the *critical cone* $\mathcal C(\bar\mu)$ at $\bar\mu\in{\mathcal{P}}$, i.e., if $\nu \cdot({\hat{\mathcal{H}}}_\mu(\bar \mu)\cdot \nu) > 0$ for all $\nu\in\mathcal C(\bar\mu)\setminus\{0\}$, with $$\begin{aligned}
    \mathcal C(\bar\mu):= \big\{\nu\in\mathbb{R}^P\,\big|\, \exists \mu\in{\mathcal{P}},\,c_1>0: \nu = c_1(\mu-\bar \mu),\, \nabla_\mu {\hat{\mathcal{J}}}(\bar\mu)\cdot \nu = 0  \big\},
    \end{aligned}$$ then $\bar \mu$ is a strict local minimum of .

For this result we refer to [@CasTr15; @Nocedal06], for instance.

High dimensional discretization and model order reduction {#sec:mor}
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We first discretize the optimization problem as well as the corresponding optimality conditions using the classical Ritz-Galerkin projection into a possibly high dimensional approximation space $V_h \subset V$, such as conforming Finite Elements. Note that we restrict ourselves to a conforming approximation for simplicity and that we do not further specify the choice of $V_h$, as neither impacts the analysis below. Based on this idea, we then derive different ways for the ROM using the Reduced Basis method with possibly different reduced primal and dual state spaces. Thus, the resulting ROM optimality system will in general not be equivalent to a Ritz-Galerkin projection of the FOM one onto a reduce space $V_{{r}}\subset V_h$. For this reason, we 