L1-regularized quadratic function

Definition

A $\ell ^{1}$-regularized quadratic function of one variable is a function of the form:

$f(x):=ax^{2}+bx+c+\lambda |x|$

where $a,b,c,\lambda \in \mathbb {R} ,a\neq 0,\lambda >0$.

The function has a piecewise quadratic function definition:

$f(x):=\left\lbrace {\begin{array}{rl}ax^{2}+(b+\lambda )x+c,&x\geq 0\\ax^{2}+(b-\lambda )x+c,&x<0\\\end{array}}\right.$

Differentiation

First derivative

The expression for the first derivative is obtained from the piecewise definition of the function, using the differentiation rule for piecewise definition by interval. We obtain the expression:

$f'(x)=\left\lbrace {\begin{array}{rl}2ax+(b+\lambda ),&x>0\\2ax+(b-\lambda ),&x<0\\\end{array}}\right.$

The first derivative is undefined at $x=0$. At this point, the left-hand derivative is $b-\lambda$ and the right-hand derivative is $b+\lambda$.

Second derivative

The second derivative is defined as:

$f''(x)=2a,\qquad x\neq 0$

The second derivative is undefined at zero.