On/off variables in MILPs with infinite bounds

Question

I have an LP defined by

$$A x = b$$ $$0 \leq x \leq U$$

and would like to extend it to an MILP through introduction of binary on/off variables $z$ such that

$$z_i = 0 \implies x_i = 0.$$

This seems to be typically done through introduction of constraints

$$x_i \leq z_i U_i.$$

However, what do we do if $x_i$ is unbounded, so $U_i = \infty$? I suppose we could introduce arbitrarily large $U_i$, but this feels inelegant, and may lead to scaling issues for the solver.

Edit: if finite, large $U_i$ are required, what factors must be considered in choosing their value?

Answer 1

Would this work?

$$(1-z_i)x_i = 0$$

Edit: Oh wait, this is not linear. But on second thought I don't think you can do it without introducing some kind of complementarity condition.