  Although the performance of popular optimization algorithms such as
Douglas-Rachford splitting (DRS) and the ADMM is satisfactory in small and
well-scaled problems, ill conditioning and problem size pose a severe obstacle
to their reliable employment. Expanding on recent convergence results for DRS
and ADMM applied to nonconvex problems, we propose two linesearch algorithms to
enhance and robustify these methods by means of quasi-Newton directions. The
proposed algorithms are suited for nonconvex problems, require the same
black-box oracle of DRS and ADMM, and maintain their (subsequential)
convergence properties. Numerical evidence shows that the employment of L-BFGS
in the proposed framework greatly improves convergence of DRS and ADMM, making
them robust to ill conditioning. Under regularity and nondegeneracy assumptions
at the limit point, superlinear convergence is shown when quasi-Newton Broyden
directions are adopted.
