Feasible active-set QP solve with blocking constraints and a finite cap

Input a strictly convex quadratic program $q(z)=\frac{1}{2}{z}^{T}Bz+c^{T}z,Az=b,G_{i}z+d_i\le 0(1\le i\le m),$ with $B=B^T\succ 0$, a feasible starting point $z_0$, a working set $W_0$ of active inequality indices whose rows together with the equality rows are independent, and a finite iteration cap $N\ge 1$. Fix ties by least index. For each iteration $k=0,\dots ,N-1$:

1. Solve the equality-constrained direction problem ${\min }_u(B{z}_k+c{)}^{T}u+\frac{1}{2}{u}^{T}Bu\text{subject to}Au=0,G_{i}u=0(i\in W_k)\mathrm{.}$ Its KKT matrix is nonsingular by The SQP KKT matrix is nonsingular under positive reduced curvature. If the linear solve fails, return numerical-failure.
2. If $u=0$, inspect the inequality multipliers ${\mu }_i$ for $i\in W_k$. If every ${\mu }_i\ge 0$, return solution(z_k, multipliers, W_k). Otherwise delete the least-index $j\in W_k$ with ${\mu }_j<0$ and continue at the same feasible point with the reduced working set.
3. If $u\mathrm{\ne }0$, for every $i\mathrm{\notin }{W}_k$ with $G_{i}u>0$ compute ${\alpha }_i=-\frac{G_i{z}_k+d_i}{G_{i}u},{\alpha }_k=\min (1,{\min }_{i\mathrm{\notin }{W}_k,G_{i}u>0}{\alpha }_i),$ using ${\alpha }_k=1$ when there is no such index. If ${\alpha }_k=0$, return degenerate-block. Otherwise set $z_{k+1}=z_k+{\alpha }_{k}u$. If ${\alpha }_k<1$, add the least-index blocker attaining ${\alpha }_k$ to $W_k$; if adding it makes the rows dependent, return dependent-blocker. If ${\alpha }_k=1$, the new point minimizes $q$ on the current working-set face, even if an unselected inequality becomes active at the endpoint. Set all those unselected multipliers to zero, and return if the current inequality multipliers are nonnegative; otherwise delete the least-index negative-multiplier constraint before continuing.

If the cap is reached first, return iteration-budget-exhausted with the last feasible iterate and multipliers. Every returned non-solution status is inconclusive about optimality. The algorithm does not silently add a zero-length blocking step.