Slater's condition

A convex problem has a point that strictly satisfies every inequality constraint.

For a convex problem, Slater's condition holds when there is a point $\tilde x$ with $A\tilde x=b$ and $f_i(\tilde x)<0$ for every inequality constraint (strict, not just $\le0$ ). When it holds, the KKT conditions become not just sufficient but necessary for optimality — see slater-condition in the theorem registry.