Penalty Function Method

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: fi (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 fi (x)≥ 0, i = 1, 2, . . ., m. This functipn has the following properties: ψ(x,α) = 0 if xX and ψ(x, α) > 0 if xX. 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.

REFERENCES

Moiseev, N. N. Elementy leorii optimal’nykh sistem. Moscow, 1975.
Fiacco, A., and G. McCormick. Nelineinoe programmirovanie. Moscow, 1972. (Translated from English.)
Céa, J. Optimizatsiia. Moscow, 1973. (Translated from French.)

V. G. KARMANOV

References in periodicals archive ?
When we use penalty function method in literature [9] to inverse optimization problem (7), the optimization problem (7) is turned into the following form by the method in literature [9]:
0466 in solving the same problem through comparison of neural network method and penalty function method in literature [9].
A Penalty Function Method based on Bilevel Programming for Solving Inverse Optimal Value Problems.
Application of genetic algorithm and penalty function method in machine optimal design, Journal of China Jiliang University 15(4): 290-293.
Penalty function method is a technique which is used to solve the constrained optimization problems.
Most of the authors studied exact penalty function method under the notions of convexity (see, for example, [10, 16]).
1] exact exponential penalty function method for solving a class of differentiable optimization problems with (p, r) - [rho] - ([eta], [theta])-invex functions.
In constrained optimization problem, penalty function method has been adopted to transform problem into non-constrained ones.
3) Process the constraint by the penalty function method to.
This paper adopts the penalty function method to deal with the constraints.
They used a penalty function method and divided the constitutive equation into shear stress and normal stress components to decrease memory capacity.
Therefore we use the decoupled method (13, 14) (separate velocity, pressure fields and stress field), the streamwise integration method (15-18) (integrate constitutive equation along streamline in order to decrease the computational memory in stress field), the penalty function method (8) (pressure variable reduced in coalition equations) and the skyline method (contract region of matrix memory needed analysis).