Augmented Lagrangian method

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

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. 

Considering such recent relevance of SDP in quantum chemistry, this chapter discusses some practical aspects of this variational calculation of the 2-RDM formulated as an SDP problem. We first present the definition of an SDP problem, and then the primal and dual SDP formulations of the variational calculation of the 2-RDM as SDP problems (Section II), an efficient algorithm to solve the SDP problems the primal-dual interior-point method (Section III), a brief section about alternative and also efficient augmented Lagrangian methods (Section IV), and some computational aspects when solving the SDP problems (Section V). 

Software based on the augmented Lagrangian method (Section IV) is also available PENSDP by Kocvara and Stingl [24] (unique commercial code) and SDPLR by Burer, Monteiro, and Choi [26-28]. 

Farhat et al. (1990) used augmented Lagrangian method to solve the optimisation problem presented above. See the original reference for further details. 

The constraints are first applied to all components of the displacements by the Lagrange multiplier method then the modifications needed for the augmented Lagrangian method are added. Lastly, the equations of motion for the Lagrange multiplier method are obtained. A detailed derivation of such equations of motion can be found in Ref. 171, together with the explicit recipe for the Verlet algorithm used to integrate such equations. 

The coupling between the two regions is then completed by imposing the same constraints as in the ODD method and, as before, the equations of motion can be obtained using the Lagrange multiplier method and the augmented Lagrangian method. 

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... 

Here we consider the augmented Lagrangian method, which converts the constrained problem into a sequence of imconstrained minimizations. We first treat equality constraints, and then extend the method to include inequality constraints. 

To find Xmin, we again use the augmented Lagrangian method, writing each inequality constraint hj x) > 0 in an equivalent form similar to an equality constraint by introducing a slack variable Sj,... 

The augmented Lagrangian method is not the only approach to solving constrained optimization problems, yet a complete discussion of this subject is beyond the scope of this text. We briefly consider a popular, and efficient, class of methods, as it is used by fmincon, sequential quadratic programming (SQP). We wUl find it useful to introduce a common notation for the equality and inequality constraints using slack variables. 

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