penaltymethoden

Penalty methods are a class of optimization algorithms used to solve constrained optimization problems. The core idea is to transform a constrained optimization problem into a sequence of unconstrained optimization problems. This is achieved by augmenting the objective function with a penalty term that penalizes constraint violations.

For a minimization problem with constraints g(x) <= 0 and h(x) = 0, a typical penalty function might be of the form P(x, r) = f(x) + r Phi(x), where f(x) is the original objective function, r is a positive penalty parameter, and Phi(x) is a penalty term that increases as the constraints are violated. For example, Phi(x) could be the sum of squared positive parts of g(x) and squared h(x).

The algorithm proceeds by iteratively solving the unconstrained problem P(x, r) for increasing values of r. As r approaches infinity, the minimizer of the penalized problem converges to the minimizer of the original constrained problem, provided the penalty function is chosen appropriately. The penalty parameter r is typically increased geometrically or linearly in each iteration.

Penalty methods are conceptually simple and can be applied to a wide range of problems. However, they can suffer from numerical difficulties, particularly when the penalty parameter r becomes very large, leading to ill-conditioned Hessians. This can make the unconstrained optimization steps computationally expensive or unstable. Variants of penalty methods, such as augmented Lagrangian methods, have been developed to mitigate these issues.