Augmented Lagrangian Penalty Function

As shown in Section 12.2.1, the quadratic penalty Junction requires high values for the parameters to satisfy the constraints. It implies that the modified objective function has very narrow valleys and the search for the minimum is quite difficult. 

In fact, the i penalty Junction is discontinuous and can create certain difficulties in the algorithms that require function continuity. 

On the other hand, consider the following function applied to a problem with equality constraints only 

It has been demonstrated (Fletcher, 1987) that, for certain finite values of its penalty parameters, the minimization of the function Oalpf exactly solves the original constrained problem. This function is, therefore, an exact penalty function. 

Fletcher has demonstrated that the Maratos effect is avoided if the exact values of Lagrange parameters, Ij, are inserted into the function (12.23). 

---

As above, the SQP method is not iterated without controls, but a merit function of Chapter 12 is usually adopted (for instance, i or the augmented Lagrangian penalty Junction alpf) to deem whether the iterations converge to the solution. 

Again, we make initial guesses of the multipliers, define a Lagrangian augmented with penalty functions that enforce the constraints, find the unconstrained minimum of this function, and use the results to update the multiplier estimates. At iteration k, the multiplier estimates and and the penalty tolerance > 0 define the augmented Lagrangian... 

Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method)... 

The augmented Lagrangian is a smooth exact penalty function. For simplicity, we describe it for problems having only equality constraints, but it is easily extended to problems that include inequalities. The augmented Lagrangian function is... 

The penalty term of an augmented Lagrangian method is designed to add positive curvature so that the Hessian of the augmented function is positive-definite. 

The classical penalty-function methods have now finally become part of history, the early promise of the augmented Lagrangian approach has faded, and there has been a coalescence of the approach used in the projection methods with the exact penalty-function approach. 

Equation (10) holds for any function V vanishing on Fi. The last temi of the augmented Lagragian (for r=0, Lr is a Lagrangian) introduces a penalty of the incompressibility condition and the Uzawa algorithm allows us to satisfy equation (3) as precisely as we wish using moderate values of r. 

---