  We consider a combined restarting and adaptive backtracking strategy for the
popular Fast Iterative Shrinking-Thresholding Algorithm frequently employed for
accelerating the convergence speed of large-scale structured convex
optimization problems. Several variants of FISTA enjoy a provable linear
convergence rate for the function values $F(x_n)$ of the form $\mathcal{O}(
e^{-K\sqrt{\mu/L}~n})$ under the prior knowledge of problem conditioning, i.e.
of the ratio between the (\L ojasiewicz) parameter $\mu$ determining the growth
of the objective function and the Lipschitz constant $L$ of its smooth
component. These parameters are nonetheless hard to estimate in many practical
cases. Recent works address the problem by estimating either parameter via
suitable adaptive strategies. In our work both parameters can be estimated at
the same time by means of an algorithmic restarting scheme where, at each
restart, a non-monotone estimation of $L$ is performed. For this scheme,
theoretical convergence results are proved, showing that a $\mathcal{O}(
e^{-K\sqrt{\mu/L}n})$ convergence speed can still be achieved along with
quantitative estimates of the conditioning. The resulting Free-FISTA algorithm
is therefore parameter-free. Several numerical results are reported to confirm
the practical interest of its use in many exemplar problems.
