Slater's condition

For a finite-dimensional convex optimization problem with affine equalities $Ax=b$ and convex inequalities $g_i(x)\le 0$, Slater's condition requires a point in the relative interior of the common effective domain that satisfies the equalities and makes every inequality strict. A refined version only requires strictness of the nonaffine inequalities; affine inequalities may hold with equality. If the primal optimal value is finite, these conditions imply strong duality and dual attainment.^[ref]

The relative-domain hypothesis matters. Minimize $-\sqrt{x}$ over its domain $x\ge 0$ subject to $x\le 0$. The only feasible point is zero, giving primal value zero. For multiplier $\lambda >0$, the dual function isIts supremum is zero but is not attained by a finite multiplier. The feasible point lies outside the relative interior $(0,\infty )$ of the objective domain, so the refined condition does not apply. Feasibility of affine constraints alone suffices only when the relevant domain hypothesis is also met.