Exact-Hessian SQP converges quadratically on the identified active KKT system

Let $A$ stack equality rows and active inequality rows at $x_{\ast }$. For $d\in \ker A$, $d$ is linearly feasible and has zero derivative on every active inequality. Conversely, if $d$ is in the critical cone, stationarity gives $\mathrm{\nabla }f(x_{\ast }{)}^{T}d=-{\sum }_{i\in I}{\lambda }_{\ast ,i}D{g}_i(x_{\ast })d$; each $D{g}_i(x_{\ast })d\le 0$, so equality to zero and strict positivity of every ${\lambda }_{\ast ,i}$ force each such derivative to vanish. Thus the critical cone is exactly $\ker A$. The stated second-order condition gives positive curvature on $\ker A$, and LICQ gives independent rows.

Define $F(x,\mu )=({\mathrm{\nabla }}_{x}L(x,\mu ),c_A(x))$. Its derivative at the root is the active KKT block matrix, nonsingular by [F2] and step 1.1. By [A1], this derivative is locally Lipschitz on a small ball. The exact step of [F3] satisfies $DF(x_k,{\mu }_k)s_k+F(x_k,{\mu }_k)=0$. Apply [F4] to the combined primal-dual variable: for a sufficiently close starting pair, all full steps are defined, remain local, and converge quadratically.