Lagrangian augmented function

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. 

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... 

To accommodate the constraint (b)9 a Lagrangian function L is formed by augmenting/with Equation (b), using a Lagrange multiplier (o... 

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

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. 

The problem (12.35) and (12.36) is now transformed into the minimization of the augmented Lagrangian function ... 

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. 

The constraints on the system are efficiently taken into account using the Augmented Lagrangian Method. The involved functional to be minimized is described as follows... 

Converting the NLP into a minimization problem and formulating the augmented Lagrangian function results in the following imconstrained NLP. 

Formulate the maximum distillate problem using the calculus of variations. Solution Since this problem contains equality constraints, we need to use the Euler-Lagrangian formulation. First, all three equality constraints (Equations 5.47 to 5.49) are augmented to the objective function to form a new objective function... 

This method follows a procedure similar to the dual method. However, a quadratic penalization of the type that was discussed earlier is used in order to include the constraints in the dual function (/>. Consequently, the augmented Lagrangian Jfg/ is defined as follows. 

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