  We introduce SPIRAL, a SuPerlinearly convergent Incremental pRoximal
ALgorithm, for solving nonconvex regularized finite sum problems under a
relative smoothness assumption. Each iteration of SPIRAL consists of an inner
and an outer loop. It combines incremental gradient updates with a linesearch
that has the remarkable property of never being triggered asymptotically,
leading to superlinear convergence under mild assumptions at the limit point.
Simulation results with L-BFGS directions on different convex, nonconvex, and
non-Lipschitz differentiable problems show that our algorithm, as well as its
adaptive variant, are competitive to the state of the art.
