Theorem 4.2, p. 7 — AGP-regularity is the weakest strict constraint qualification associated with AGP

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

Here AGP-regularity is the outer semicontinuity at $(x^*,0)$ of $(x,\varepsilon)\mapsto N_{\Omega(x,-\infty)}(x+\varepsilon)$ (Definition 4.1), and AGP($-\infty$) asks for $x^k\to x^*$ with $P_{\Omega(x^k,-\infty)}(x^k-\nabla f(x^k))-x^k\to0$ (4.1).

The forward implication says AGP-regularity is a strict constraint qualification for AGP; the backward implication says every property of the constraints under which AGP implies KKT for all objectives implies AGP-regularity, so it is the weakest one.

Formalization Note The paper's "AGP" is AGP($\gamma$) for $\gamma\in[-\infty,0)$, which Martínez and Svaiter showed to be independent of $\gamma$; this paper does not prove that. The goal pins $\gamma=-\infty$. The forward direction for every $\gamma\in[-\infty,0)$ is the separate theorem agp_regular_agp_implies_kkt, and the backward direction for $\gamma=-\infty$ is the strongest form, since its hypothesis is only about $\gamma=-\infty$. KKT is in multiplier form; the objective is quantified inside the equivalence, for fixed constraints and fixed $x^*$.