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?