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Stationarity of the equality-constrained quadratic subproblem gives $B_k{s}_k+g_k-A_k^T{y}_{k+1}=0$. Together with feasibility, this isMultiplying the bottom block row by $-1$ gives a symmetric indefinite saddle-point matrix, with right side $(-g_k,c_k{)}^T$.

One solution strategy uses the Schur complement. From $s_k=B_k^{-1}(A_k^T{y}_{k+1}-g_k)$ obtainThis is an $m\times m$ positive-definite system when $A_k$ has full row rank; factor $B_k$ and solve with it rather than forming its inverse.

A second strategy uses a nullspace basis $Z$ for $\ker A_k$. Choose one particular $s_p$ with $A_k{s}_p=-c_k$, write $s_k=s_p+Zz$, and solve the reduced systemThe reduced Hessian is positive definite. These range-space and nullspace reductions exploit different relative sizes of $m$ and $n-m$.

With $y=-c/\mu$, differentiation of the quadratic penalty method objective givesThe primal Newton system is thereforeFor bounded multipliers and regular derivatives, $H$ remains of ordinary size while ${\mu }^{-1}{A}^{T}A$ becomes large in directions normal to the constraints. If the feasible tangent space is nontrivial and retains ordinary-size curvature, this produces a large condition number as $\mu \downarrow 0$.

Introduce $w=Ap/\mu$. The same step is obtained from the symmetric augmented systemEliminating $w$ recovers the primal system exactly. This matrix contains no explicit ${\mu }^{-1}$ entries and tends to a saddle-point matrix. Under full row rank of the limiting constraint Jacobian matrix and nonsingularity of the limiting reduced Hessian, that limiting matrix is nonsingular, so this formulation avoids the artificial penalty-induced separation of scales.

These conditioning statements require regularity beyond bounded $y$ alone. A rank-deficient constraint Jacobian matrix or singular reduced Hessian can still make the augmented system ill-conditioned. If $m=n$ and $A$ is nonsingular, all directions may scale together in the primal system, so a large condition number is not automatic there either.

Stationarity of the equality-constrained quadratic model gives $g_k+H_k{s}_k-A_k^T{y}_{k+1}=0$, while feasibility gives $A_k{s}_k=-c_k$. A symmetric system linking the step and new multiplier isLet $Z$ have columns forming a basis of $\ker A_k$. The saddle-point matrix is nonsingular exactly when $A_k$ has full row rank and $Z^T{H}_{k}Z$ is nonsingular. Full row rank is necessary because a nonzero vector in $\ker A_k^T$ would give a null vector of the block matrix. Under full row rank, solving a homogeneous system forces $s=Zq$ and then $Z^T{H}_{k}Zq=0$; this proves sufficiency and the converse. Positive definiteness of the reduced Hessian is sufficient for a constrained minimum but is stronger than the nonsingularity condition asked here.

A null-space method first finds $s_p$ with $A_k{s}_p=-c_k$, then solvesand recovers $y_{k+1}$ from $A_k^T{y}_{k+1}=g_k+H_k{s}_k$. It needs the stated full row rank and nonsingular reduced matrix; Cholesky decomposition is available when the latter is positive definite.

A range-space method additionally assumes $H_k$ is invertible. It solves the Schur complement equationsIf $H_k$ is positive definite and $A_k$ has full row rank, the Schur matrix is positive definite and both solves can use Cholesky decomposition. An indefinite but invertible $H_k$ instead requires a suitable indefinite factorization; the Schur matrix must remain nonsingular.

For equality-constrained optimization, restrict the Hessian matrix $H$ of the optimization Lagrangian to feasible tangent directions. If the columns of $Z$ form a basis of the nullspace of the constraint Jacobian matrix $A$, the resulting quadratic form has matrix $Z^{T}HZ$. Positive definiteness gives positive quadratic curvature in every nonzero tangent direction. Nonsingularity of this reduced matrix, together with full row rank of $A$, characterizes nonsingularity of the corresponding equality-constrained saddle-point matrix.