# Penalty function

The traditional concept is to augment the objective with a function and one positive constant, so that the original mathematical program is replaced by solving a parametric family of the form $LaTeX: \textstyle \max \left \{f(x) - uP(x): x \in X^0\right\}.$ The function, $LaTeX: P,$ is called a penalty function if it satisfies $LaTeX: P(x) > 0$ for $LaTeX: x$ not feasible and $LaTeX: P(x)=0$ if $LaTeX: x$ is feasible. The set $LaTeX: X^0$ depends on the type of penalty function, and there are two classical types, each requiring $LaTeX: P$ to be continuous: interior (or barrier) and exterior. A penalty function is exact if there exists a finite parameter value such that its maximum is a solution to the original mathematical program.