Logarithmic Barrier Function
For logarithmic barrier functions, is defined as when and otherwise (in 1 dimension. See below for a definition in higher dimensions). This essentially relies on the fact that tends to negative infinity as tends to 0.
This introduces a gradient to the function being optimised which favours less extreme values of (in this case values lower than ), while having relatively low impact on the function away from these extremes.
Logarithmic barrier functions may be favoured over less computationally expensive inverse barrier functions depending on the function being optimised.
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