# Exterior penalty function

A penalty function in which the associated algorithm generates infeasible points, approaching feasibility in the limit towards a solution. An example is to maximize $LaTeX: f(x) - u h(x)^2$ over $LaTeX: x \in X$ and let $LaTeX: u \rightarrow \infty$ in order to approach the solution to
$LaTeX: \max \{f(x): x \in X, \; h(x)=0\}$.
If, during the penalty iterations, $LaTeX: h(x^*)=0$ for some finite $LaTeX: u$, then $LaTeX: x^*$ solves the original mathematical program. Otherwise, the idea is that $LaTeX: x^*(u)$ approaches the feasibility condition, $LaTeX: h(x^*)=0$, as $LaTeX: u$ gets large (though this need not happen without assumptions on $LaTeX: f$ and $LaTeX: h$).