Augmented penalty function

The OA/ER/AP algorithm introduces an augmented penalty function in the lower bound subproblems of the OA/ER approach. 

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 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. 

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... 

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. 

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. 

The penalty approach corresponds to augmenting the energy surface with a biasing potential U, for example a harmonic function centred at position to with a suitable width ku. 

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. 

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