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Hello,

I have the following function form as one of my constraints :

f(x) = MIN(0, x)

Because of the MIN, it is non-differentiable.

As I would like to use an optimizer that uses derivative based methods, I need my objective function and constraints to be differentiable.

How could I convert (or approximate) this function into a differentiable one?

Here is an example of data and the function chart, even if it is trivial.

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  • $\begingroup$ Your question is somewhat vague. Are you defining the function $f(x)$ to be $\min(0,x)$, or is $f(x)$ a function that is already defined and you want to impose the constraint that $f(x)=\min(0,x)$? If you're defining $f(x)$ to be $\min(0,x)$, then how else does $f(x)$ appear in your problem? Does it appear in other constraints? Does it appear in the objective function? $\endgroup$
    – Brian Borchers
    Commented May 6, 2011 at 18:41
  • $\begingroup$ The anti-derivative of a suitable bump function does what you want. $\endgroup$
    – Ryan Budney
    Commented May 6, 2011 at 19:22
  • $\begingroup$ What if you add to your objective function the indicator function of $(-\infty,0]$, and then solved your problem with the proximal gradient method or FISTA ? $\endgroup$
    – littleO
    Commented Jul 31, 2011 at 8:08

1 Answer 1

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You can try

$f(\xi) = \frac{\xi}{2}+\frac{\ln 2}{2k} + \frac{\ln(\cosh(k \xi)}{2k},$

whose derivative is

$f'(\xi) = \frac{1}{2} [\tanh(k \xi) + 1].$

This function approaches the maximum as $k\rightarrow \infty$ (change as needed for the minimum). Will this help solve your problem? Perhaps, if the lack of smoothness is not its essential feature - a question related to the domain of application, not strictly a mathematical issue.

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