Solver-o-matic
Decision Tree for Nonsmooth Optimization Software
Your first choice should
be Quasi-Secant Method (QSM) with
discrete gradients for nonsmooth possibly nonconvex
minimization by A. Bagirov and A. Ganjehlou. You can employ
either analytically calculated or approximated subgradients
in your experiments (this can be done automatically by
selecting one parameter). The Fortran 77 source code of QSM
is available for downloading.
If the function is highly nonlinear you may also try LMBM. Another option is SolvOpt, although theoretically SolvOpt is not supported for nonconvex functions.