  In this paper, we establish the first explicit and non-asymptotic global
convergence analysis of the BFGS method when deployed with an inexact line
search scheme that satisfies the Armijo-Wolfe conditions. We show that BFGS
achieves a global convergence rate of $(1-\frac{1}{\kappa})^k$ for
$\mu$-strongly convex functions with $L$-Lipschitz gradients, where
$\kappa=\frac{L}{\mu}$ denotes the condition number. Furthermore, if the
objective function's Hessian is Lipschitz, BFGS with the Armijo-Wolfe line
search achieves a linear convergence rate only determined by the line search
parameters and independent of the condition number. These results hold for any
initial point $x_0$ and any symmetric positive definite initial Hessian
approximation matrix $B_0$, although the choice of $B_0$ affects the iteration
count required to attain these rates. Specifically, we show that for $B_0 =
LI$, the rate of $O((1-\frac{1}{\kappa})^k)$ appears from the first iteration,
while for $B_0 = \mu I$, it takes $d\log \kappa$ iterations. Conversely, the
condition number-independent linear convergence rate for $B_0 = LI$ occurs
after $O\left(\kappa\left(d +\frac{M
\sqrt{f(x_0)-f(x_*)}}{\mu^{3/2}}\right)\right)$ iterations, whereas for $B_0 =
\mu I$, it holds after $O\left(\frac{M
\sqrt{f(x_0)-f(x_*)}}{\mu^{3/2}}\left(d\log \kappa + \kappa\right)\right)$
iterations. Here, $d$ denotes the dimension of the problem, $M$ is the
Lipschitz parameter of the Hessian, and $x_*$ denotes the optimal solution. We
further leverage these global linear convergence results to characterize the
overall iteration complexity of BFGS when deployed with the Armijo-Wolfe line
search.
