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### Rational Function Optimization

While standard Newton-Raphson is based on the optimization on a quadratic model, by replacing this quadratic model by a rational function approximation we obtain the RFO method [130,131].2.6

 (2.76)

The numerator in equation 1.76 is the quadratic model of equation 1.74. The matrix in this numerator is the so called Augmented Hessian (AH). is the Hessian (analytic or approximated). The matrix is a symmetric matrix that has to be specified but normally is taken as the unit matrix . The solution of RFO equation, that is, the displacement vector that extremalizes (i.e. ) is obtained by diagonalization of the Augmented Hessian matrix solving the -dimensional eigenvalue equation 1.77

 (2.77)

and then the displacement vector for the step is evaluated as

 (2.78)

where

 (2.79)

In equation 1.79, if one is interested in locating a minimum then , and for a transition structure . As the optimization process converges, tends to 1 and to 0.

Next: Direct Inversion of Iterative Up: Second derivative methods Previous: Newton Raphson and quasi-Newton   Contents
Xavier Prat Resina 2004-09-09