This question is related to my previous question posted here: Piecewise constraint using big-M notation and this question posted on the math stackexchange: https://math.stackexchange.com/questions/4065112/mip-modelling-of-piecewise-linear-function

Given the answers to the questions posted above, I am a little confused regarding the use of upper and lower bounds (see the variables $U_{1}, U_{2}, U_{3}, L_{1}, L_{2}, L_{3}$ in the math stackexchange post for example).

In that post, I tried setting each $z$ to 1 (one at a time), and sure enough I can see that we obtain the piecewise function the OP wanted. However, what is confusing me is that apparently the $L_{i}$ and $U_{i}$ have to be calculated. It was my (naive) understanding that these could just be "absurdly" large and small numbers. So using the same post, U = 1000 and L = -1000 ($\forall i$) would be fine. But is this not the case? Is there a way to calculate these?

If I can indeed just choose "absurdly" large and small numbers for these, is there any computational drawback? From my basic understanding, my guess would be that the search space would increase and as would the computational time.



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