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"These days, even the most pure and abstract mathematics is in danger to be applied."
- Anonymous

Decision Tree for Nonsmooth Optimization Software

So, you need a derivative free method or a method which automatically calculates the difference approximations. Your first choice should be Discrete Gradient Method (DGM) for derivative free optimization by A. Bagirov, B. Karasozen and M. Sezer. To apply DGM, you only need to compute at every point the value of the objective function. The subgradient will be approximated. The Fortran 77 source code of DGM is available for downloading.

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