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

\[\begin{split}b(x) = \begin{cases} \textrm{NaN} & x\leq 0 \\[1em] 1-\ln\dfrac{x}{h} & 0<x<h \\[1em] \dfrac{12} {3+\dfrac{x}{h}\left(8+\dfrac{x^3}{h^3}\right)} & x\geq h. \end{cases}\end{split}\]

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

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

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

\[b(x) := \left(1-\ln\dfrac{x}{h}\right)\mathrm{e}^{3(1-x/h)}.\]