= Solution
Complete the square in the <augmented Lagrangian method> objective:
$$
\boxed{\mathcal L_\mu(x,u)=f(x)+\frac1{2\mu}\|c(x)-\mu u\|_2^2-\frac\mu2\|u\|_2^2.}
$$
For fixed $u,\mu$, the last term is constant in $x$. Thus its minimization is the <quadratic penalty method> applied to the shifted constraint $c(x)=\mu u$.
Its practical advantage is that multiplier updates, typically $u^+=u-c(x)/\mu$, can drive the original constraint residual to zero while $\mu$ stays moderately sized. The plain <quadratic penalty method> generally needs $\mu$ very small to produce the same feasibility accuracy, exposing the conditioning problem in part (ii).
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