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In A differentiable approximation to the minimum function, a differentiable approximation of the minimum function is given, but it seems it only works for positive reals.

Is there an easy-to-implement approximation to the minimum function $f: \mathbf{R^N} \rightarrow \mathbf{R}$ that behaves correctly over all $\mathbf{R^N}$, even when two elements of the input vector are equal?

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If you have a smooth approximation $f_k$ which is ok for positive numbers, for $x:=(x_1,\dots,x_N)\in\mathbb{R}^N$ you may translate everything, for instance $f_k(x_1+\|x\|_2^2+1,\dots,x_N+\|x\|_2^2+1)-\|x\|_2^2-1.$ –  Pietro Majer Sep 13 '12 at 9:51
    
Thank you, this indeed solves my problem. –  Antonio El Khoury Sep 13 '12 at 11:50

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