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What would be the numerical method of choice to find minima in a non-smooth, non-convex, locally Lipschitz function $f: \mathbb{R}^n\rightarrow \mathbb{R}$.

The function $f$ is mostly smooth but contains three-dimensional cusps of the following form:
$$ g: \mathbb{R}^3\rightarrow \mathbb{R}\\ g(\boldsymbol{x})= -\exp(-{\lVert \boldsymbol x \rVert}_2) $$ with $\lVert \cdot \rVert_2$ being the Euclidean Norm.

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    $\begingroup$ I just tried to minimize the function $g$ that you present using SQP in Python and it converged after 8 iterations. $\endgroup$
    – nicoguaro ♦
    Commented Feb 7, 2017 at 17:12

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I think I found the right article myself:

Lewis, A.S. & Overton, M.L. Math. Program. (2013) 141: 135. doi:10.1007/s10107-012-0514-2

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    $\begingroup$ Yes, that is exactly the right approach: non-smooth Newton methods. $\endgroup$
    – Wolfgang Bangerth
    Commented Feb 8, 2017 at 0:42

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