Differentiable approximations to the min and max operators

How does this stack up to the softmax[0] function, log(exp(n_1) + exp(n_2) + ... + exp(n_i))? Are they analytically equivalent?

0: https://en.wikipedia.org/wiki/LogSumExp

tl;dr

1. Within the context of optimisation, differentiable approximations of the min and max operators on R^n are very useful.

2. However, in order for these approximations to be useful they must also be numerically stable.