Barrier

Barrier Log

The logarithmic barrier $b(x)$ is one of the most classic functions used to regularise other functions. It depends on a parameter $h$ chosen such that $b(h)=1$, moreover for this barrier $b(0)=\infty$ and $b(\infty)=0$. The function $b$ is defined as

\[ b(x) = \begin{cases} \textrm{NaN} & \textrm{for $x\leq 0$} \\[1em] 1-\ln\dfrac{x}{h} & \textrm{for $0

Barrier LogExp

An evolution of the previous barrier is the LogExp barrier, which has the same properties of the Log barrier, that is, it depends on a parameter $h$ chosen such that $b(h)=1$, moreover for this barrier $b(0)=\infty$ and $b(\infty)=0$. The difference is that this one grows faster to infinity for $x\to 0$. The function $b$ is defined as

\[ b(x) \DEF \left(1-\ln\frac{x}{h}\right)e^{3(1-x/h)}.\]

Barrier Log0

This last barrier has a small difference with respect to the previous two logarithmic barriers, it is identically zero for $x\geq h$. Setting the tolerance $h$, it is defined as

\[ b(x) \DEF \left(1-\ln\frac{x}{h}\right)e^{3(1-x/h)}.\]