I want to write a quadratic program to the problem
$$\text {minimize}_{z\in\mathbb{R}^d} \|z\|_1^2 + F(z)$$
where $\|\cdot\|_1$ is the $l_1$ norm and $F$ is some function. How can I do that?
Attempt:
Define $t_j:=|z_j|$. (We can constraint $t_j\geq \pm z_j$. Hence, $$ ||z||_1^2=(\sum_{j\leq d}|z_j|)^2 \\=(\sum_{j\leq d}t_j)^2=\sum_{k,l\leq d}t_kt_l=t\cdot t^T $$ But I'm not sure how it could help.