Is it possible to write the following optimization problem as a quadratic programming problem:
$$ \| y - X\beta\|^2_2 + \lambda \|C\beta\|_1 $$ such that $D\beta = d$ and $D$ has full row rank.
I am thinking whether below is a possibility: $$ \| y - X\beta\|^2_2 + \lambda \|C\beta\|_1 = \| y - X\beta\|^2_2 + \lambda \|z\|_1 \qquad \qquad \text{s.t}\ z = C\beta,\ \text{and}\ D\beta = d $$
Can we replace $\|z\|_1$ as $t^\top1$ such that $-t < z < t$?