= Slater's condition
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= Slater condition
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For a finite-dimensional <convex optimization> problem with affine equalities $Ax=b$ and convex inequalities $g_i(x)\le0$, <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>.<Convex Optimization by Stephen Boyd and Lieven Vandenberghe>[§5.2.3, equations (5.26)–(5.27), refined condition and dual attainment]{ref}
The relative-domain hypothesis matters. Minimize $-\sqrt x$ over its domain $x\ge0$ subject to $x\le0$. The only feasible point is zero, giving primal value zero. For multiplier $\lambda>0$, the dual function is
$$
\inf_{x\ge0}(-\sqrt x+\lambda x)=-\frac1{4\lambda}.
$$
Its 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.
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