Is the barrier problem for a linear program a convex problem?

Question

By applying the barrier method to the linear programming problem min $c^T x, Ax ≥ b$, we can formulate: $$\underset{x}{\text{minimize}} \hspace{0.5cm} c^T x - \sum_{i=1}^{m} \log(e_i - d_i^T x)$$

But is the sequence of optimization problems solved in the method convex? Why or why not?

Answer 1

Yes it can because of the following reasons:

1. $c^Tx$ is affine (and hence is convex).

2. $-\log x$ is convex for $x>0$

3. If $f(x)$ is convex for $x\in D\subseteq\Bbb R$, then $f(a+b^Ty)$ is convex where $y\in\{ z:a+b^Tz\in D\}$.