Minimization using logarithmic barrier function

Question

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ (i = 1, \cdots, m) \end{align} , where $Q$ is positive definite, using logarithmic penalty method.
Let the interior of the feasible region be $D$, and reformulating, we can get \begin{equation} \underset{x\in D}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x -\mu_k\sum_{i=1}^m\log{(b_i - a_ix)} \end{equation} in the $k$-th iteration.
My question is,
1. Does this penalized problem has a minimum in the feasible area $D$ ?
2. Is the minimum unique?