Penalty function

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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.

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.


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