I want to formally show that the following minimization problem $$ \min_\theta||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ is equivalent to $$ \min_{\beta, \{w_i \}^{n}_{i=1}} \sum_{i=1}^{n}w_i^2 \text{ s.t. } f_i(\theta)\leq w_i \text{ and } w_i \geq 0 \text{ } \forall i $$ where $f_i (\theta):\Theta \subseteq \mathbb{R}^k \rightarrow \mathbb{R}$.
Could you help me?
Let $\theta,w$ solve $$ \min \sum w_i^2 \quad s.t. \quad f_i(\theta)\le w_i, \ w_i \ge 0. $$ Th constraints are equivalent to $w_i \ge \max(f_i(\theta),0)$. Then it follows that $$ w_i = \max(0,f_i(\theta)). $$ If not then $w_i$ could be reduced to make $\sum w_i^2$ smaller without violating the constraints.
Hence both problems are equivalent.