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

$$\Delta E= E({\bf q}_k+\Delta {\bf q}_k)-E({\bf q}_k)\cong \frac{\... ...rray}\right) \left(\begin{array}{c} 1 \\ \Delta {\bf q}_k \end{array}\right) }$$ (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). ${\bf B}_k$ is the Hessian (analytic or approximated). The ${\bf S}_k$ matrix is a symmetric matrix that has to be specified but normally is taken as the unit matrix ${\bf I}$. The solution of RFO equation, that is, the displacement vector $\Delta {\bf q}$ that extremalizes $\Delta E$ (i.e. $\nabla_{\bf q}(\Delta E )=0$) is obtained by diagonalization of the Augmented Hessian matrix solving the $(N+1)$-dimensional eigenvalue equation 1.77

$$\left(\begin{array}{cc} 0 & {\bf g}^T_k \\ {\bf g}_k & {\bf B}_k... ...mbda^{(k)}_\theta {\bf v}^{(k)}_\theta \qquad \forall \quad \theta=1,\ldots,N+1$$ (2.77)

and then the displacement vector $\Delta {\bf q}_k$ for the $k_{th}$ step is evaluated as

$$\Delta {\bf q}_k=\frac{1}{ v^{(k)}_{1,\theta}}{\bf v}^{'(k)}_\theta$$ (2.78)

where

$$({\bf v}^{'(k)}_\theta)^T=( v^{(k)}_{2,\theta},\ldots, v^{(k)}_{N+1,\theta})$$ (2.79)

In equation 1.79, if one is interested in locating a minimum then $\theta=1$, and for a transition structure $\theta=2$. As the optimization process converges, $v^{(k)}_{1,\theta}$ tends to 1 and $\lambda^{(k)}_\theta$ to 0.