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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.

For constraints $h=0$, $g\ge 0$, minimize the displayed merit function for progressively smaller $\mu >0$, where $g_i^{-}=\min (g_i,0)$. Increasing the quadratic penalty discourages constraint violation. In a violated inequality, differentiation gives the multiplier estimate ${\lambda }_i=-g_i/\mu$; an inactive inequality has estimate zero. Thus obtaining small violations by this method alone typically requires a very small $\mu$, which may cause a large condition number in the inner minimization.