= Solution
For the convention above, after approximately minimizing $\Phi(\cdot,\lambda^{[k]},\mu_k)$, update
$$
\boxed{\lambda_i^{[k+1]}=
\max\left\{0,\lambda_i^{[k]}-\frac{g_i(x^{[k+1]})}{\mu_k}\right\}.}
$$
The exact shifted-residual identity is
$$
\min\{g_i(x^{[k+1]}),\mu_k\lambda_i^{[k]}\}
=\mu_k(\lambda_i^{[k]}-\lambda_i^{[k+1]}).
$$
If the constraint is violated, $g_i(x^{[k+1]})\leq0$, its left side is $\min\{g_i,0\}$, and replacing the new multiplier by its limiting value gives the printed local estimate
$$
\boxed{\min\{g_i(x^{[k+1]}),0\}
\approx\mu_k(\lambda_i^{[k]}-\lambda_i^*).}
$$
For arbitrary iterates, the precise unshifted statement includes the negative part:
$$
\boxed{\min\{g_i(x^{[k+1]}),0\}
=\mu_k\min\{\lambda_i^{[k]}-\lambda_i^{[k+1]},0\}.}
$$
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 <condition number>[ill-conditioning] of the inner minimization. Inactive constraints eventually have zero multiplier and zero violation.
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