\begin{align*} \delta_r^*({\nu}) &= \max_{{\beta \in R^M}}\bigg\{{\beta}'{\nu}\mid \frac{1}{P}\|{z}-{Y}{\beta}\|_2^2+\lambda\|{\beta}\|_1\leq\hat{\rho}+r\bigg\} \\ &= \max_{{\beta\in R^M},g\geq0,\,h\geq0}\bigg\{{\beta}'{\nu}\mid \|{z}-{Y}{\beta}\|_2^2\leq g,\,\|{\beta}\|_1\leq h,\, \frac{1}{P}g+\lambda h\leq\hat{\rho}+r\bigg\}. \end{align*} It follows from strong duality that $$ \begin{array}{rll} \delta_r^*({\nu}) = \min & (\hat{\rho}+r) s + {u-v}+{z}' \\ \mathrm{s.t.} & {Y}' {w} - {t} = {\nu}, \\ & \bigg \| \begin{array}{c} 2v\\ {w} \end{array} \bigg \|_2 \leq 2u, \\ & \| {t} \|_\infty \leq \lambda s, \\ & Pu + Pv \leq s ,\\ & u,s \in \mathbb{R}_+,\,v\in \mathbb{R},\,{w}\in \mathbb{R}^P,\,{t}\in \mathbb{R}^M. \end{array} $$
How is the dual of this optimization problem solved