 Solution

For the convention above, after approximately minimizing $\Phi (\cdot ,{\lambda }^{[k]},{\mu }_k)$, updateThe exact shifted-residual identity isIf the constraint is violated, $g_i(x^{[k+1]})\le 0$, its left side is $\min \{g_i,0\}$, and replacing the new multiplier by its limiting value gives the printed local estimateFor arbitrary iterates, the precise unshifted statement includes the negative part:Thus the printed estimate describes approach from the violated side; it is not a universal signed equality on the feasible side. This qualification follows directly from the projection in the update.

Its numerical significance is that the constraint violation is controlled by ${\mu }_k$ times the multiplier error, rather than by ${\mu }_k$ alone. Once multipliers are accurate, high feasibility accuracy can be obtained with a moderate positive penalty parameter. This avoids forcing $1/{\mu }_k$ to become enormous and reduces the ill-conditioning of the inner minimization. Inactive constraints eventually have zero multiplier and zero violation.