Penalty Function Method

a method of reducing problems of finding a relative extremum of a function.to problems of finding the absolute extremum.

An important application of the penalty function method is to problems of mathematical programming. Suppose we wish to minimize the function Φ(x) on the set X = {x: f_i (x)≥ 0, i = 1, 2, . . ., m} of an n-dimensional Euclidean space. The function Ψ(x, α), which depends on x and on the numerical parameter a > 0, is called the penalty function (or penalty) for violation of the constraints f_i (x)≥ 0, i = 1, 2, . . ., m. This functipn has the following properties: ψ(x,α) = 0 if x ∈ X and ψ(x, α) > 0 if x ∈ X. Let us construct the function M(x, α)= Φ(x) + ψ(x, α) and designate as x(α)any point of its absolute global minimum. Let

The function ψ(x, α) is selected so that ϕ(x(α)) → ϕ* as α → + ∞. The function

often is selected as ψ(x, α). The selection of the specific form of the function ψ(x, α) involves both the problem of the convergence of the penalty function method and the problems that arise in solving the problem of absolute minimization of the function M(x, α).

In a somewhat more general formulation, the penalty function method consists in reducing the problem of minimizing the function ϕ(x) on a set X to the problem of minimizing some parametric function M(x, α) on a set whose structure is simpler, from the standpoint of effectiveness of the application of numerical methods of minimization, than that of X.