I was reading on second order cone programs https://www.di.ens.fr/~aspremon/PDF/MVA/Duality.pdf page 33 and have trouble trying to derive its dual. While I am able to formulate the lagrangian easily, I do not understand how the following steps are carried out.
For the steps below, I perfectly understand that the minimum over x and t is only bounded if the expression in the green and blue boxes (excluding x and t) are zero. But what about the minimum over y ? Why is the infimum over y = 0 if the L2 norm of $v_i$ is less than $\lambda_i$ ?