Penalty and Augmented Lagrangian Methods

This chapter introduces two very important concepts in constrained nonlinear optimization. These are penalty and augmented Lagrangian concepts. The idea is that both these concepts replace the original problem by a sequence of sub-problems in which the constraints are expressed by terms added to the objective function. The penalty concept is implemented in two different methods. The quadratic penalty method adds to the objective function a multiple of the square of the violation of each constraint and solves a sequence of unconstrained optimization sub-problems. Simple and enough intuitive, this approach has some important deficiencies. The nonsmooth exact penalty method, on the other hand, solves a single unconstrained optimization problem. In this approach, a popular function is the l 1 penalty function. The problem with this method is that the nonsmoothness may create complications in numerical implementations. Finally, the second concept is the multiplier method or the augmented Lagrangian method, which explicitly uses Lagrange multiplier estimates in order to avoid the ill-conditioning of the quadratic penalty method.