How to impose that both real variables can't be simultaneously strictly positive linearly?

Question

I have two variables: $\alpha$ and $\beta$ that are reals (NOT binary) and I want to express the fact that they can't be strictly positive at the same time (i.e. $\alpha=0$ or $\beta=0$) with a linear constraint but I don't manage to do it.

Answer 1

You want a linear function $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f(\alpha, \beta) > 0$ or $\ge 0$ (i.e. a linear constraint) if and only if $(\alpha, \beta)$ is in the complement of the open first quadrant; call that set $Q$.

This is impossible for $>$ because the constraint defines an open set in that case, and $Q$ is closed. So take $f(\alpha, \beta) \ge 0$. (We could consider $\le$, but by symmetry it's the same; just switch $f$ with $-f$.)

Now you still have a problem because $f^{-1}[0,\infty)$ for $f$ linear consists either of the empty set (if $f < 0$ is constant), the portion of the plane on one side of the line $f^{-1}(0)$ (if $f$ is non-constant), or the full plane (if $f \ge 0$ is constant).

So it can't be done.