of new TR-RB algorithms that differ in their respective reduced gradient information as well as in the enrichment strategies for the construction of the corresponding reduced models. All variants are thoroughly analyzed numerically in Section \[sec:num\_experiments\], where we consider three well defined benchmark problems. We also compare with selected state of the art optimization methods from the literature.

Problem formulation {#sec:problem}
===================

Given $\mu_\mathsf{a},\mu_\mathsf{b}\in\mathbb{R}^P$ with $P \in {\mathbb{N}}$ we consider the compact and convex admissible parameter set $${\mathcal{P}}:= \left\{\mu\in\mathbb{R}^P\,|\,\mu_\mathsf{a} \leq \mu \leq \mu_\mathsf{b} \right\} \subset {\mathbb{R}}^P,$$ where $\leq$ is understood component-wise. Let $V$ be a real-valued Hilbert space with inner product $(\cdot \,,\cdot)$ and induced norm $\|\cdot\|$. We are interested in efficiently approximating PDE-constrained parameter optimization problems with the quadratic continuous cost functional $${\mathcal{J}}: V \times {\mathcal{P}}\to {\mathbb{R}}, \quad  (u,\mu) \mapsto {\mathcal{J}}(u, \mu) = \Theta(\mu) + j_\mu(u) + k_\mu(u, u),$$ where $\Theta: {\mathcal{P}}\to \mathbb{R}$ denotes a parameter functional and, for each $\mu \in {\mathcal{P}}$, $j_\mu \in V'$ is a parameter-dependent continuous linear functional and $k_\mu: V \times V \to {\mathbb{R}}$ a continuous symmetric bilinear form. To be more precise, we consider the following constrained minimization problem: [ $$\tag{P}
    \label{P}$$ ]{}

$$\min_{(u,\mu)\in V\times{\mathcal{P}}} {\mathcal{J}}(u, \mu), 
    \tag{P.a}\label{P.argmin}$$

subject to $(u,\mu)$ satisfying the 