Want to understand how the dual function is derived https://www.stat.cmu.edu/~ryantibs/convexopt-F18/lectures/dual-corres.pdf (Slide number 13)
Minimizing with respect to $\beta$ is clear, however, minimizing with respect to z is not clear.. Let's consider,
$$g(u) = \underset{\beta,z}{min}\frac{1}{2}\left \| y-z \right \|_2^2+\lambda\left \| \beta \right \|_1+u^{\top}(z-X\beta)$$
$$g(u) = \underset{z}{min}\;\underbrace{\frac{1}{2}\left \| y-z \right \|_2^2 + u^{\top}z}_{\Lambda_z}+ \underset{\beta}{min}\; \underbrace{\lambda\left \| \beta \right \|_1-(u^{\top}X)^{\top}\beta }_{\Lambda_\beta}$$
To find the optimal $z^*$
$$\frac{\partial \Lambda_z}{\partial z} = -(y-z^*)+u = 0, \; \Rightarrow z^* = y-u $$
After substituting $z^*$,
$$g(u) = \frac{1}{2}\left \| u \right \|_2^2 + u^{\top}(y-u) + \underset{\beta}{min}\; \underbrace{\lambda\left \| \beta \right \|_1-(u^{\top}X)^{\top}\beta }_{\Lambda_\beta}$$
The results I'm getting don't match the results in the slides. Could someone give me a hint?