Slater's condition for problem with only one feasible solution

Question

Suppose we have the following optimization problem:

minimize x
s.t. x^2 <= 0

where x is a real number.

I think Slater's condition does not hold, as we cannot have x^2 < 0. But since we can rewrite the constraint as a strict equality constraint, x = 0, and Slater's condition holds when a problem only has strict equalities, shouldn't this problem satisfy the Slater's condition?

Thanks!

Answer 1

Slater's condition does not hold for the original formulation. it trivially holds for the reformulated problem, because that has only a linear constraint. Any problem having only linear constraints trivially satisfies Slater's condition because it only imposes a requirement on nonlinear constraints.

As you can see, the satisfaction or not of Slater's condition is not invariant with respect to reformulations preserving the same optimal solution.