Barrier function

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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)).


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