Theorem 4.5, p. 11 — SAKKT-regularity is the weakest strict constraint qualification associated with SAKKT

Consider problem (1.1) with continuously differentiable constraint functions $h_1,\dots,h_m$, $g_1,\dots,g_p$ on $\mathbb R^n$, and let $x^*$ be a feasible point. Then

Here SAKKT-regularity (Definition 4.3) is the outer semicontinuity $\limsup_{x\to x^*}N_{\Omega(x,0)}(x)\subset N_{\Omega(x^*,0)}(x^*)$, where $\Omega(x,0)$ is the linearization of the equality constraints and of the inequality constraints that are active or violated at $x$; SAKKT asks for $x^k\to x^*$ and multipliers $\lambda^k$, $\mu^k\ge0$ with $\mu_j^k=0$ whenever $g_j(x^k)<0$ and with the KKT residual tending to zero.

The forward implication says that SAKKT-regularity is a strict constraint qualification for SAKKT; the backward implication says that every constraint property which makes SAKKT imply KKT for all objectives implies SAKKT-regularity. Together they make SAKKT-regularity the weakest such property.

Formalization Note "Smooth objective" is read as $\mathrm C^1$, which is all either direction uses. The objective is quantified inside the equivalence, for fixed constraints and a fixed feasible $x^*$. KKT is the multiplier form with sign and complementarity conditions.