# Penalty function

### From Glossary

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 The function, is called a *penalty function* if it satisfies for not feasible and if is feasible. The set depends on the type of penalty function, and there are two classical types, each requiring 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.

More generally, a penalty function could involve many parameters, such as the Lagrangian, or it could be stated in parameter free form. A general model is the generalized penalty-function/surrogate dual, see the supplement on duals. The notion of an exact penalty function leads to a strong dual.