$f(a) = \begin{cases} a & a \ge 1 \\ 2a-1 & a\lt 1 \end{cases}$

$\text{minimize } f(u_1) + f(u_2)$

$\begin{align*} \text{s.t. } &\text{some constraints involving } u_1, u_2\\ & \text{some constraints involving } f(u_1), f(u_2) \end{align*}$

I know that you can formulate non-smooth functions such as max, min, absolute value as LP. Is it possible to express if statement as LP?

If so, you can provide an example on how to do so

Note: If you can provide a link to a textbook that describe this in depth, it would be appreciated too

  • $\begingroup$ you minimize a concave function which cannot be expressed with linear constraints; you'll have to introduce binary variables $\endgroup$
    – LinAlg
    Jun 30, 2018 at 22:50
  • $\begingroup$ @LinAlg Can you explain how this would work? $\endgroup$
    – samol
    Jul 1, 2018 at 1:57
  • $\begingroup$ see Chapter 7.6 of aimms.com/english/developers/resources/manuals/… $\endgroup$
    – LinAlg
    Jul 1, 2018 at 12:04


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