$\inf\limits_{y_i} (\lambda_i \lVert y_i \rVert_2 + \nu_i^T y_i) = 0$ if $\lVert \nu_i \rVert_2 \leq \lambda_i$

Question

I am trying to solve the following problem:

$\inf\limits_{y_i} (\lambda_i \lVert y_i \rVert_2 + \nu_i^T y_i) = 0$ if $\lVert \nu_i \rVert_2 \leq \lambda_i$ and $-\infty$ otherwise.

I know that $\lambda_i \lVert y_i \rVert_2 + \nu_i^T y_i \stackrel{\text{Cauchy-Schwartz}}{\leq} \lambda_i \lVert y_i \rVert_2 + \lVert \nu \rVert_2 \lVert y_i \rVert_2$ but this is a upper bound, and I would like a lower bound.

In dimension 1, this would be simply $\lambda |x| + \nu x$ so we would like $\nu \leq \lambda$ but I can not generalize.

Thank you

Answer 1

Suppose $\|v_i\|_2 \leq \lambda_i$. Then, $-\lambda_i \lVert y_i \rVert_2 - \nu_i^T y_i\le -\lambda_i \lVert y_i \rVert_2+\lVert y_i \rVert_2\lVert v_i \rVert_2\le 0$ so $\lambda_i \lVert y_i \rVert_2 + \nu_i^T y_i\ge 0$. The minimum value is attained when $y_i=0$.

Now suppose Suppose $\|v_i\|_2 > \lambda_i$.

$\inf\limits_{y_i} (\lambda_i \lVert y_i \rVert_2 + \nu_i^T y_i)\leq \lambda_i \lVert nv_i \rVert_2 + \nu_i^T (-nv_i)=-n(\|v_i\|^{2}-\lambda_i\|v_i\|_2)\to -\infty$ as $ n \to \infty$.