Is there a name for the following optimization problem? Is it solvable?
\begin{align} \min_{\pmb{u}} & \qquad \frac{1}{2}(\pmb{u}-\pmb{u}^m)^{T}(\pmb{u}-\pmb{u}^m)\\ \textrm{s.t.} & \qquad \pmb{A}(\pmb{u}) \textrm{ is positive definite} \end{align}
Here $\pmb{u}\,\in\mathbb{R}^{n},\pmb{u}^m\,\in\mathbb{R}^n$, $\pmb{u}^m$ is known, and $\pmb{A}(\pmb{u})\in\,\mathbb{R}^{m\times{m}}, m\approx{n}.$ The entries of $\pmb{A}$ depend linearly on the components of $\pmb{u}$ in a known manner. e.g. ${A}_{7,8}=c_1u_3 + c_2u_{12} + c_3u_{4}$ with $c_i$ known.