 Solution

Complete the square in the augmented Lagrangian method objective: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).