# Barrier function

$LaTeX: P$ is a barrier function on the strict interior, say $LaTeX: I$, of the set $LaTeX: S$ if $LaTeX: P$ is continuous, and $LaTeX: P(x) \rightarrow \infty$ as $LaTeX: x$ approaches any point in $LaTeX: S\backslash I$. Barrier functions arises in penalty function methods for certain classes that include convex programs of the form:

$LaTeX: \min \{ f(x) : g(x) \ge 0 \},$

where $LaTeX: I = \{ x : g(x) > 0 \}$. Two popular choices are

$LaTeX: \textstyle P(x) = \sum_i 1 / g_i(x),$

and

$LaTeX: \textstyle P(x) = -\sum_i \ln (g_i(x)).$