  The quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) method has proven to
be very reliable and efficient for the minimization of smooth objective
functions since its inception in the 1960s. Recently, it was observed
empirically that it also works remarkably well for non-smooth problems when
combined with the Armijo-Wolfe line search, but only very limited theoretical
convergence theory could be established so far. In this paper, we expand these
results by considering convex piecewise linear functions with two pieces that
are not bounded below. We prove that the algorithm always terminates in a
finite number of iterations, eventually generating an unbounded direction. In
other words, in contrast to the gradient method, the BFGS algorithm does not
converge to a non-stationary point.
