Solver-o-matic
Decision Tree for Nonsmooth Optimization Software
So, you need a derivative free method or a method which automatically
calculates the difference approximations. I recommend to use
Quasi-Secant Method (QSM) with discrete gradients.
QSM is a solver for nonsmooth possibly nonconvex
minimization by A. Bagirov and A. Ganjehlou. In QSM one can employ
either analytically calculated or approximated subgradients
(this can be done automatically by selecting one parameter).
The Fortran 77 source code of QSM is available for downloading.
Not satisfied?