  In this paper we propose an adaptively extrapolated proximal gradient method,
which is based on the accelerated proximal gradient method (also known as
FISTA), however we locally optimize the extrapolation parameter by carrying out
an exact (or inexact) line search. It turns out that in some situations, the
proposed algorithm is equivalent to a class of SR1 (identity minus rank 1)
proximal quasi-Newton methods. Convergence is proved in a general non-convex
setting, and hence, as a byproduct, we also obtain new convergence guarantees
for proximal quasi-Newton methods. The efficiency of the new method is shown in
numerical experiments on a sparsity regularized non-linear inverse problem.
