Is it possible to approximate the $max\{x,y\}$ by a differentiable function? $f(x,y)=max \{x,y\} ;\ x,y>0$


Yes it is. One possibility is the following: Note that $\def\abs#1{\left|#1\right|}$ $$ \max\{x,y\} = \frac 12 \bigl( x+ y + \abs{x-y}\bigr), $$ take a differentiable approximation of $\abs\cdot$, for example $\def\abe{\mathop{\rm abs}\nolimits_\epsilon}$$\abe \colon \mathbb R \to \mathbb R$ for $\epsilon > 0$ given by $$ \abe(x) := \sqrt{x^2 + \epsilon}, \quad x \in \mathbb R $$ and define $\max_\epsilon \colon \mathbb R^2 \to \mathbb R$ by $$ \max\nolimits_\epsilon(x,y) := \frac 12 \bigl( x+y+\abe(x-y)\bigr). $$ Another possibility is to take a smooth mollifier $\phi_\epsilon$ and let $\max'_\epsilon :=\mathord\max * \phi_\epsilon$.

  • $\begingroup$ (+1), good advice. One smoothing I stumbled across recently is $|x| \approx x\operatorname{erf}(ax)$ with $a \gg 1$. Taking $a=5$ works well enough. $\endgroup$ – Antonio Vargas Oct 8 '13 at 22:37

Another possibility is given by: \begin{equation} \max (x,y) \approx \ln(e^x+e^y) \end{equation} This approximation doesn't work well for similar values of $x$ and $y$. We can however, remedy this by introducing a scaling parameter $N$: \begin{equation} \max (x,y) \approx \frac1N\ln(e^{N x}+e^{N y}) \end{equation} for large values of $N$.

A general definition is given by:

\begin{equation} \max\limits_{x \in S} \approx \frac1N\ln(\sum\limits_{x\in S} e^{N x}) \end{equation}

Note that in practice $e^{Nx}$ will give unworkably large answers for large $N$.

  • $\begingroup$ How could deal with overflow and underflow in case of large numbers multiplied by large scaling parameter N? $\endgroup$ – Feras Oct 22 '17 at 17:49
  • $\begingroup$ I don't know. That's why I said it is unworkable for large N. $\endgroup$ – Angelorf Oct 23 '17 at 22:34
  • $\begingroup$ I'm testing another function $max(x,y) = pow((pow(x,n) + pow(y,n), 1/n)$ it is doesn't have overflow but in one case when x or y is zero so I'm adding this small number 1e-5 to avoid it. Do you have any idea ? $\endgroup$ – Feras Oct 24 '17 at 16:46
  • $\begingroup$ That doesn't work for negative numbers. Why avoid zero? $0^x = 0$ (except $0^0$, but $n \neq 0$ anyway) $\endgroup$ – Angelorf Oct 25 '17 at 19:42
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    $\begingroup$ Overflow issues can be (mostly) avoided using the identity ln(sum(exp(Nn)))/N = ln(sum(exp(N(n-m))))/N + m, where m is the actual maximum. $\endgroup$ – Ian Fellows May 18 '18 at 17:03

Maximum norm is maximum of absolutes and you can approximate it with a p-norm with large(ish) p. If you have negative numbers as well, transform them with log(exp(x)+1) first which, in turn, can be approximated with x for larger values.


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