Karistusfunktsioon

Karistusfunktsioon, often translated as penalty function, is a concept used in optimization and control theory. It refers to a function that is added to an objective function to penalize constraint violations. The goal of a penalty function is to transform a constrained optimization problem into an unconstrained one by incorporating the constraints directly into the objective. When a constraint is violated, the penalty function adds a positive value to the objective function, making the solution less desirable. The magnitude of this penalty typically increases with the degree of constraint violation.

The basic idea is to minimize a modified objective function, P(x, r) = f(x) + r * phi(x), where f(x) is the original objective function, x represents the decision variables, r is a positive penalty parameter, and phi(x) is the penalty function that measures the constraint violation. As the penalty parameter r increases, the optimizer is increasingly deterred from violating the constraints. For many penalty function methods, a sequence of unconstrained minimization problems is solved with increasing values of r. The solutions to these unconstrained problems converge to the solution of the original constrained problem as r approaches infinity.

Common examples of penalty functions include the squared violation penalty, where phi(x) is the sum of the squares of the violated constraints, and the absolute value penalty, where phi(x) is the sum of the absolute values of the violated constraints. The choice of penalty function and the strategy for increasing the penalty parameter can significantly affect the efficiency and convergence of the optimization algorithm. Karistusfunktsioon plays a crucial role in numerical methods for solving constrained optimization problems.