Linearize minimum and maximum constraint with variable and constant

Question

Let's say I want to linearize the restrictions: $ \min(0, y) \leq x \leq \max(0, y) $

Then I can define $y_{\max}$ and $y_{\min}$ such that:

$$ y_{\max} \geq 0 \\ y_{\max} \geq y \\ y_{\min} \leq 0 \\ y_{\min} \leq y \\ y_{\max} + y_{\min} = y $$

If I now reformulate the original restriction as $ y_{\min} \leq x \leq y_{\max} $, have I now successfully linearized the constraint without needing to add binary decision variables? E.g. can I use this restriction in a simple LP?

Answer 1

Consider $(x,y,y_\min,y_\max)=(1,0,-1,1)$, which satisfies your proposed linearization but violates the original constraints.

Furthermore, notice that $(x,y)=(1,0)$ is halfway between two points $(2,2)$ and $(0,-2)$ that do satisfy the original constraints, so the feasible region is not convex, which implies that no LP formulation is possible.