Slater's condition for problem with only one feasible solution 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!

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.