Penaltymenetelmää

Penaltymenetelmää, also known as the penalty method, is a class of techniques used in mathematical optimization to handle problems with constraints. The basic idea is to transform a constrained optimisation problem into a series of unconstrained problems by adding to the objective function a penalty term that increases as the constraints are violated. This penalty term typically depends on a parameter that controls the severity of the penalty; as the parameter grows, the solutions of the modified objective approach feasibility for the original constraints.

The method has a long history and was independently introduced in the mid‑20th century by scholars such as Hestenes and Powell. It is commonly divided into two main categories: exact penalty functions, where a finite penalty parameter can enforce feasibility in a single iteration, and inexact or interior‑point penalty functions, which gradually increase the parameter to guide the iterates toward feasibility. Exact penalties are often based on norms such as the L¹ or L∞ norm of the constraint violations, whereas inexact penalties may use quadratic or logarithmic barrier terms.

Penalty methods are valued for their simplicity and for providing a unified framework that can be combined with standard unconstrained optimisation algorithms such as gradient descent or Newton’s method. They are widely applied in fields ranging from operations research and engineering design to economics and machine learning. However, choosing appropriate penalty parameters and dealing with ill‑conditioning as the penalty grows remain significant practical challenges, motivating the development of augmented Lagrangian and other hybrid techniques that have become popular in modern constrained optimisation.