Finite-budget active-set SQP with a certified KKT-system residual status

Let a finite-dimensional $C^2$ reduced FE control program have a fixed list of $m$ inequality constraints. Its exact objective, constraint, gradient, and Hessian evaluators, and its active-row rank and reduced-curvature checks, must terminate on every query or return failure. For each active set $A\subseteq \{1,\dots ,m\}$, use the active KKT map $F_A$ and exact-Hessian Jacobian $J_A=D{F}_A$ specified by Full-step exact-Hessian SQP after active-set identification, with inactive inequality multipliers set to zero. Given an initial primal vector $x_0$, equality multiplier ${\nu }_0$, and full inequality multiplier vector ${\lambda }_0\in {\mathbb{R}}^m$ with entries outside $A_0$ zero when $R>0$, an ordered finite list of proposed active sets $A_0,\dots ,A_{R-1}$, maximum budget $N\ge 1$, residual tolerance $\tau \ge 0$, a terminating exact-storage routine satisfying the input contract of Finite-precision reduced-QR KKT solve with certified residual status, and its floating-point hypotheses, process at most $\min (R,N)$ candidates. If $R=0$, return inconclusive immediately. At candidate $j$, first set ${\lambda }_{j,i}=0$ for $i\mathrm{\notin }{A}_j$, then let $z_j=(x_j,{\nu }_j,{\lambda }_{j,A_j})$ be the active KKT variable.

1. Evaluate $F_{A_j}(z_j)$ and $K_j=J_{A_j}(z_j)$. Write $G_j=D{c}_{A_j}(x_j)$ for the stacked equality and active-inequality rows, and $H_j={\mathrm{\nabla }}_{xx}^{2}L(z_j)$ for the Lagrangian Hessian block. Verify that $G_j$ has full row rank and $d^T{H}_{j}d>0$ for every nonzero $d\in \ker G_j$, as required by Finite-precision reduced-QR KKT solve with certified residual status. If an evaluation fails or either condition is false or cannot be verified, mark this candidate numerical-failure, set $(x_{j+1},{\nu }_{j+1},{\lambda }_{j+1})=(x_j,{\nu }_j,{\lambda }_j)$, and continue to the next candidate.

2. Otherwise set $b_j=-F_{A_j}(z_j)$ and call Finite-precision reduced-QR KKT solve with certified residual status exactly once on the intended system $K_{j}s=b_j$, supplying the terminating exact-storage routine. Before factorization or residual evaluation, that call must verify that the stored entries equal the intended $K_j,b_j$. An unavailable or unverified admission is a failed QR solve with no residual certificate. On a failed QR solve, record numerical-failure, set $(x_{j+1},{\nu }_{j+1},{\lambda }_{j+1})=(x_j,{\nu }_j,{\lambda }_j)$, and continue.

3. On a successful QR return, write its vector as ${\widehat{s}}_j=({\widehat{p}}_j,\widehat{\delta }{\nu }_j,\widehat{\delta }{\lambda }_{j,A_j})$ and let ${\eta }_j$ be its residual bound. Set $x_{j+1}=x_j+{\widehat{p}}_j$, ${\nu }_{j+1}={\nu }_j+\widehat{\delta }{\nu }_j$, and, for each inequality index $i$, set ${\lambda }_{j+1,i}={\lambda }_{j,i}+\widehat{\delta }{\lambda }_{j,i}$ if $i\in A_j$ and ${\lambda }_{j+1,i}=0$ otherwise. Thus the next candidate has a full, dimensionally fixed multiplier vector; newly active multipliers start at zero. The QR vector is the only step used by this wrapper. If ${\eta }_j\le \tau$, return $(z_j^{+},A_j,\mathtt{\text{active-kkt-residual-certified}},{\eta }_j)$, where $z_j^{+}=(x_{j+1},{\nu }_{j+1},{\lambda }_{j+1,A_j})$; otherwise continue.

For each successful solve, retain the tuple with the smallest computed ${\eta }_j$. If the finite list or budget is exhausted, return budget-exhausted with that tuple, or inconclusive if no solve succeeded. The active-kkt-residual-certified status concerns only the linearized active KKT equation residual $\mathrm{\parallel }{K}_j{\widehat{s}}_j+F_{A_j}(z_j){\mathrm{\parallel }}_2$. A nonlinear next-point KKT residual, box feasibility, multiplier signs, global optimality, and forward accuracy require separate validated checks; none follows from a small linear-system residual alone.

For an exact quadratic reduced objective and affine active constraints, including every equality constraint, $F_A$ is affine and its Jacobian is constant, so the next-point active KKT residual equals the linear solve residual in the algorithm. For a nonlinear active KKT map, a certified step-equation residual is not by itself a certificate for $F_A(z_j^{+})$; the local OPT6 convergence theorem additionally requires identified active set, LICQ, strict complementarity, positive reduced curvature, locally Lipschitz Jacobian, and a sufficiently close start. The exact-Hessian algorithm supplies the map/Jacobian convention; the QR algorithm supplies the rank and curvature input conditions. The exact solve is not run by this finite-precision wrapper.