I'm trying to formulate and determine the variables, objective, and constraints for the minimization problem $\min_\vec{x}f(\vec{x})$ for the following functions $f \in$ ($q,r,s,t$) as linear program (LP) or quadratic program (QP) or neither (N):
Where $\vec{x} \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, $\vec{y} \in \mathbb{R}^{m}$ and $|\vec{v}|_p$ is the $p$-norm.
$$\displaystyle \\ q(\vec{x}) = |A\vec{x}-\vec{y}|_∞ + |\vec{x}|_1 \\ r(\vec{x}) = |A\vec{x}-\vec{y}|^2_2 + |\vec{x}|_1 \\ s(\vec{x}) = |A\vec{x}-\vec{y}|^2_2 - |\vec{x}|_1 \\ t(\vec{x}) = |A\vec{x}-\vec{y}|^2_2 + |\vec{x}|_1^2 \\ $$
My best guess right now is $q$ is LP, $r$ is QP, $s$ is N, and $t$ is QP.