L1-regularized quadratic function of multiple variables

Definition

A $L^{1}$-regularized quadratic function of multiple variables of the variables $x_{1},x_{2},\dots ,x_{n}$ is a function of the form:

$\left(\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}x_{i}x_{j}\right)+\left(\sum _{i=1}^{n}b_{i}x_{i}\right)+\lambda \sum _{i=1}^{n}|x_{i}|+c$

In vector form, if we denote by ${\vec {x}}$ the column vector with coordinates $x_{1},x_{2},\dots ,x_{n}$, then we can write the function as:

${\vec {x}}^{T}A{\vec {x}}+{\vec {b}}^{T}{\vec {x}}+\lambda \|{\vec {x}}\|_{1}+c$

where $A$ is the $n\times n$ matrix with entries $a_{ij}$ and ${\vec {b}}$ is the column vector with entries $b_{i}$.