Exact exhaustive active-set solve for the SQP quadratic subproblem

Input an SQP subproblem from The linearized quadratic subproblem in sequential quadratic programming, using exact real arithmetic. Let $q$ and $m$ be the numbers of equality and inequality constraints. Put $A^{\mathrm{e}\mathrm{q}}=Dh(x)$, $b^{\mathrm{e}\mathrm{q}}=-h(x)$, $G=Dg(x)$, $d=g(x)$, and $c=\mathrm{\nabla }f(x)$, so $G_i=D{g}_i(x)$ and $d_i=g_i(x)$ for $1\le i\le m$. First reduce the linearized equality system $A^{\mathrm{e}\mathrm{q}}p=b^{\mathrm{e}\mathrm{q}}$: test consistency, and, if consistent, retain the lexicographically first maximal independent set $E\subseteq \{1,\dots ,q\}$ of equality row indices with their right-hand sides. If it is inconsistent, return QP-infeasible. Within each retained row set, use increasing original indices; an empty row set contributes no rows or right-hand-side entries.

For each subset $J\subseteq \{1,\dots ,m\}$ in increasing bit-mask order:

1. Stack the equality rows $(A^{\mathrm{e}\mathrm{q}}{)}_E$ and the inequality rows $G_J$, in that order. If these rows are dependent, skip $J$.
2. Treat the selected inequalities as equalities $G_{i}p=-d_i$ and solve $\left[\begin{array}{cc}B& A_J^T\\ A_J& 0\end{array}\right]\left(\genfrac{}{}{0px}{}{p}{\eta }\right)=\left(\genfrac{}{}{0px}{}{-c}{b_J}\right),$ where $A_J=\left(\genfrac{}{}{0px}{}{(A^{\mathrm{e}\mathrm{q}}{)}_E}{G_J}\right)$ and $b_J=\left(\genfrac{}{}{0px}{}{(b^{\mathrm{e}\mathrm{q}}{)}_E}{-d_J}\right)$. Thus the equality entries of $b_J$ are $-h_a(x)$ for $a\in E$, followed by $-g_i(x)$ for $i\in J$, in the declared row order. The multiplier vector $\eta$ is stored in this same order. The KKT matrix is nonsingular by The SQP KKT matrix is nonsingular under positive reduced curvature.
3. Discard the candidate if any linearized inequality is violated or any selected inequality multiplier is negative. Otherwise store $(p,\eta ,J)$.

If no candidate is stored, return QP-infeasible. Otherwise return the stored candidate with least quadratic objective, breaking an exact tie by the least subset bit mask, as solution(p, nu, lambda, E, J). Here $\nu$ is the full equality-multiplier vector, initialized to zero and filled from the equality components of $\eta$ at indices $E$; $\lambda$ is the full inequality-multiplier vector, initialized to zero and filled from the selected components at indices $J$. A finite-precision implementation that cannot complete a rank test or solve must return numerical-failure; this definition's exact-arithmetic branch has no such failure. It requires no feasible initial step and specifies no pivoting rule for an iterative working set solver.