Lagrangian penalty function

Bound-Constrained Formulation for Lagrangian Penalty Function... 

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... 

Murray, W., and Wright, M., Projected Lagrangian methods based on trajectories of barrier and penalty functions, SOL Report 78-23, Stanford University, Stanford, California (1978). 

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. 

Very recently Sargent 88 has shown that, although the Lagrangian function is not an exact penalty function, it can indeed be used in a descent test to force global convergence. Moreover the test is satisfied without step-reduction in the final stages, so the convergence is superlinear. Thus the penalty parameter and its associated problems are eliminated. 

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... 

The main idea of the algorithm of Caroe and Schultz [11] is to decompose a 2S-MILP into its scenarios by Lagrangian relaxation of the non-anticipativity constraints. In a Lagrangian relaxation, constraints are removed and included in the objective function with a penalty term. 

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. 

Step 1. For a given set of Lagrange multipliers and penalty parameter minimize the Lagrangian function L R) to obtain an improved estimate of the factorized 2-RDM at the energy minimum. 

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. 

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