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I am currently trying to solve an MILP in Gurobi. Unfortunately, Gurobi does not support non-linear functions and I would like to do the following. I currently have the following constraint. It calculates the residual power of a machine $j$ at time $t$, where $\alpha$ is a parameter. Currently the relationship is linear:

$$degneration_{jt}=1-\alpha\cdot age_{jt}$$

But I would like to model such a progression $$f(x)=e^{-x}$$ and would like to use a linear approximation for this. How would I have to model something like this, unfortunately I have no idea.

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  • $\begingroup$ Does $f$ appear in the objective function (in which case we need to know whether you are maximizing or minimizing it) or in constraints? $\endgroup$
    – prubin
    Commented Jan 30 at 16:37
  • $\begingroup$ @prubin I am sorry, the OP isn't clear enough. $f$ doesn't appear in either. It's just an example. Currently I have a linear relationship ship between $degneration$ and $age$ but I rather want a non-linear one. I just need a linear approximation for $f$, as I want to implement it in Gurobi $\endgroup$ Commented Jan 30 at 17:03
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    $\begingroup$ Gurobi's new version 11 does support a variety of common nonlinear functions. You can choose between global optimization and automatic piecewise-linear approximation. (Of course, whether this feature is useful for your particular application will depend on the form that $f$ takes.) $\endgroup$
    – 4er
    Commented Jan 30 at 17:35

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One approximation possible is to use the tangents of $f(x)$. Assuming you are minimizing $f(x)$, introduce a new variable $z$ to approximate $f(x)$ over a given set $A$ and use constraints: \begin{align} z &\ge f'(a)(x-a)+f(a)\quad &\forall a \in A \\ &= -e^{-a}(x-a)+e^{-a} \quad &\forall a \in A\\ &=-e^{-a}x+e^{-a}(1+a) \quad &\forall a \in A \end{align}

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