Solver-o-matic
Decision Tree for Nonsmooth Optimization Software
It does not really matter. Both the solvers suitable for your problem can handle both convex and nonconvex problems. Your first choice should be Limited Memory Bundle Method (LMBM). LMBM is specifically developed for large-scale nonsmooth optimization by N. Karmitsa. It works best for highly nonlinear functions. The Fortran 77 source code and the mex-driver (for MatLab users) are available.
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