= Solution
With $y=-c/\mu$, <differentiation> of the <quadratic penalty method> objective gives
$$
\boxed{\nabla P_\mu=g-A^Ty,\qquad \nabla^2P_\mu=H(x,y)+\mu^{-1}A^TA.}
$$
The primal <Newton system> is therefore
$$
\boxed{(H+\mu^{-1}A^TA)p=-g+A^Ty.}
$$
For bounded multipliers and regular derivatives, $H$ remains of ordinary size while $\mu^{-1}A^TA$ 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 $\mu\downarrow0$.
Introduce $w=Ap/\mu$. The same step is obtained from the symmetric augmented system
$$
\boxed{\begin{pmatrix}H&A^T\\A&-\mu I\end{pmatrix}\begin{pmatrix}p\\w\end{pmatrix}=\begin{pmatrix}-g+A^Ty\\0\end{pmatrix}.}
$$
Eliminating $w$ 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.
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