I have to show that infimum of the following function $\inf_{y_i}(\lambda_i \|y_i\|_2 + \nu_i^T y_i) = \begin{cases}0 \text{ if } \|\nu_i\|_2 \leq \lambda_i\\ -\infty \text{ otherwise} \end{cases} $, where $y_i \in \mathbb{R}^{n_i}, \lambda_i \in \mathbb{R}$ and $\nu \in \mathbb{R}^{n_i}$.
I know that $\lambda_i \|y_i\|_2 + \nu_i^T y_i$ would be some sort of equation of second degree. So me idea was that in order for this function to have a minimum/infimum that is not $-\infty$, the coefficient of the expression $\lambda_i \|y_i\|_2$ would have to be positive, but then I didn't see how to relate this ideas would help me get an expression of the form $\|\nu_i\|_2\leq \lambda_i$. I also thought about taking the derivative of this expression, but ended up not knowing how to relate this to $\|\nu_i\|_2\leq \lambda_i$
I really don't see how this infimum is obtained. Can anyone help me how this is obtained?
Thanks!
Hint: $\lambda\|y\|_2+v^Ty=\lambda\|y\|_2+\|v\|_2\|y\|_2cos(\theta)=\|y\|_2(\lambda+\|v\|_2cos(\theta))$, where $\theta$ is the angle between $v$ and $y$.