is approximated by the *discrete reduced functional* $$\begin{aligned}
{\hat{\mathcal{J}}}_h(\mu) := {\mathcal{J}}(u_{h, \mu}, \mu) = \mathcal L(u_{h,\mu},\mu,p_h) && \text{for all } p_h\in V_h,
\label{eq:Jhat_h}\end{aligned}$$ where $u_{h, \mu} \in V_h$ is the solution of and we formulate the discrete optimization problem $$\begin{aligned}
\min_{\mu \in {\mathcal{P}}} {\hat{\mathcal{J}}}_h(\mu).
\tag{$\hat{\textnormal{P}}_h$}\label{Phat_h}\end{aligned}$$ Further, $\bar\mu_h$ denotes a locally optimal solution to satisfying the first- and second-order optimality conditions.

Since $u_{h,\mu}$ and $p_{h,\mu}$ belong to the same space $V_h$, Propositions [[\[prop:first\_order\_opt\_cond\]-[\[prop:grad\_Jhat\]]{},[\[prop:hessian\_Jhat\]]{}-\[prop:second\_order\]]{}]{} from Section [[\[sec:problem\_fom\]]{}]{} hold for the FOM as well, with all quantities replaced by their discrete counterparts.

As usual in the context of RB methods, we eliminate the issue of “truth” by assuming that the high dimensional space $V_h$ is accurate enough to approximate the true solution.

\[asmpt:truth\] We assume that the primal discretization error $\|u_\mu - u_{h, \mu}\|$, the dual error $\|p_\mu - p_{h, \mu}\|$, the primal sensitivity errors $\|d_{\mu_i} u_\mu - d_{\mu_i} u_{h, \mu}\|$ and the dual sensitivity errors $\|d_{\mu_i} p_\mu - d_{\mu_i} p_{h, \mu}\|$ are negligible for all $\mu \in {\mathcal{P}}$, $1 \leq i \leq P$.

To define suitable ROMs, in what follows, we assume that we have computed problem adapted RB spaces $V_{{r}}^{\textnormal{pr}}, V_{{r}}^{\textnormal{du}}\subset V_h$, the construction of which is detailed in Section \[sec:construct\_RB\]. 