I wish to find a positive semidefinite matrix $Q \in \mathbb{R}^{n \times n}$ such that $$A \,\mbox{vec} (Q) = b$$ where matrix $A \in \mathbb{R}^{m \times n^{2}}$ and vector $b \in \mathbb{R}^m$ are given.
I know there are many solvers that do this, but how would one use the barrier-method to solve this problem? Say, using the log-determinant as the barrier function. I'm not sure how barrier method works for semidefinite programs.