  We present and analyze an away-step Frank-Wolfe method for the convex
optimization problem ${\min}_{x\in\mathcal{X}} \; f(\mathsf{A} x) +
\langle{c},{x}\rangle$, where $f$ is a $\theta$-logarithmically-homogeneous
self-concordant barrier, $\mathsf{A}$ is a linear operator that may be
non-invertible, $\langle{c},{\cdot}\rangle$ is a linear function and
$\mathcal{X}$ is a nonempty polytope. The applications of primary interest
include D-optimal design, inference of multivariate Hawkes processes, and
TV-regularized Poisson image de-blurring. We establish affine-invariant and
norm-independent global linear convergence rates of our method, in terms of
both the objective gap and the Frank-Wolfe gap. When specialized to the
D-optimal design problem, our results settle a question left open since
Ahipasaoglu, Sun and Todd (2008). We also show that the iterates generated by
our method will land on and remain in a face of $\mathcal{X}$ within a bounded
number of iterations, which can lead to improved local linear convergence rates
(for both the objective gap and the Frank-Wolfe gap). We conduct numerical
experiments on D-optimal design and inference of multivariate Hawkes processes,
and our results not only demonstrate the efficiency and effectiveness of our
method compared to other principled first-order methods, but also corroborate
our theoretical results quite well.
