optimization problem with a positive definiteness constraint

Question

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.

Answer 1

This can be considered to be a Linear Semidefinite Programming Problem (SDP), which is convex, and for which there are many available numerical optimization solvers.

To put it in standard form, which is not necessary with some optimization modeling systems or solvers, the convex quadratic objective function can be moved to the constraints as a Second Order Come Problem (SOCP) constraint (which is a special case of linear SDP constraint), but better left in SOCP form.

If strict positive definiteness is needed, the linear SDP constraint can be formulated as A(u) - small_positive_number*Identity_matrix is positive semidefinite., where small_positive_number is perhaps about 1e-5.