L1-regularized quadratic function of multiple variables

Definition

A $L^1$-regularized quadratic functions of the variables $x_1,x_2,\dots ,x_n$ is a function of the form:

$(\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}{x}_i{x}_j)+(\sum _{i=1}^n{b}_i{x}_i)+\lambda \sum _{i=1}^n\left|x_i\right|+c$

In vector form, if we denote by $\overrightarrow{x}$ the column vector with coordinates $x_1,x_2,\dots ,x_n$, then we can write the function as:

${\overrightarrow{x}}^{T}A\overrightarrow{x}+{\overrightarrow{b}}^T\overrightarrow{x}+\lambda |\overrightarrow{x}{|}_1+c$

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