I am working on an optimization problem where some of the terms of the objective function to maximize are expressed as a piecewise linear function of variables:

$$ z =  \begin{cases} 
      c^- x, & x \leq 0 \\
      c^+ x, & x    > 0 \\
   \end{cases} $$

as depicted below

[![objective function plot][1]][2]

When I have $c^- \geq c^+$, I can solve the problem by adding a new variable $x'$, and two constraints: 
 - $x' \leq c^- \times x $
 - $x' \leq c^+ \times x $

But what do I do when $c^- < c^+$ ? I don't think there is a way to express the problem as a linear  programming problem in that case, is there ? 

I have heard about [SOS constraints][SOS]. Are there the canonical way to solve this kind of problem ? If my problem contains many such piecewise linear functions, is it reasonable to expect a free solver from being able to solve such a problem with thousands of SOS1 constraints ?

  [1]: https://i.sstatic.net/BO5J9.png
  [2]: https://wbo.ophir.dev/boards/Nd39jocBDSAJ54Vd1HqGxiaeoHt2S4segM12tQqzFx0-
  [SOS]: https://en.wikipedia.org/wiki/Special_ordered_set