Trust radius

If the computed step size exceeds the trust radius, (, its direction is reoptunized under the condition that Aq = t, i.e., the Lagrangian... 

A different approach comes from the idea, first suggested by Flelgaker et al. [77], of approximating the PES at each point by a harmonic model. Integration within an area where this model is appropriate, termed the trust radius, is then trivial. Normal coordinates, Q, are defined by diagonalization of the mass-weighted Flessian (second-derivative) matrix, so if... 

Anotlrer way of choosing A is to require that the step length be equal to the trust radius R, this is in essence the best step on a hypersphere with radius R. This is known as the Quadratic Approximation (QA) method. ... 

This may again have multiple solutions, but by choosing the lowest A value the minimization step is selected. The maximum step size R may be taken as a fixed value, or allowed to change dynamically during the optimization. If for example the actual energy change between two steps agrees well witlr that predicted from the second-order Taylor expansion, the trust radius for the next step may be increased, and vice versa. 

Trust radius update (maximum step size allowed). 

The A for the minimization modes is determined as for the RFO method, eq. (14.8). The equation for Ays is quadratic, and by choosing the solution which is larger than 8ts it is guaranteed that the step component in this direction is along the gradient, i.e. a maximization. As for the RFO step, there is no guarantee that the total step length will be within the trust radius. 

The QA method uses only one shift parameter, requiring that Ats = —A, and restricts the total step length to the trust radius (compare with eq. (14.9)). 

The exact same formula may be derived using the concept of an image potential (obtained by inverting the sign offrs and ts), and the QA name is often used together with the TRIM acronym, which stands for Trust Radius Image Minimization ... 

The SO model has several stationary points. If the Newton step Eq. (3.9) is shorter than the trust radius sn I < h then the RSO model has a stationary point in the interior. It also has at least two stationary points on the boundary s = h. To see this we introduce the Lagrangian... 

The trust radius h reflects our confidence in the SO model. For highly anharmonic functions the trust region must be set small, for quadratic functions it is infinite. Clearly, during an optimization we must be prepared to modify h based on our experience with the function. We return to the problem of updating the trust radius later. 

Once p has been determined we calculate the step from the modified Newton equations Eq. (3.24). Therefore, the RF and RSO steps Eire calculated in the same way. The only difference is the prescription for determining the level shift. In the RSO approach p reflects the trust radius h, in the RF model p reflects the metric S. By varying h and S freely the same steps are obtained in the two models. 

The trust radius h is obtained by a feedback mechanism. In the first iteration some arbitrary but reasonable value of h is assumed. In the next iteration, h is modified based on a comparison between the predicted reduction in fix) and the actual reduction. If the ratio between actual and predicted reductions... 

The trust radius in the trust region approach is estimated on the basis of the local Hessian s characteristics (positive-definite, positive-semidefinite, indefinite). The basic idea is to choose s nearly in the current negative gradient direction (—gk) when the trust radius is small, and approach the Newton step -Hk x%k as the trust region is increased. (Hk and g denote the Hessian and gradient, respectively, at xk). Note from condition [12] that these two choices correspond to the extremal cases (M = / and M = H) of general descent directions of form p = — M 1g, where M is a positive-definite approximation to the Hessian. 

Another way of choosing A is to require that the step length be equal to the trust radius... 

Snin-other-orbit interaction. 211 Trust radius, 319 Wave package, 389... 

Position vector(s), general or electronic Distance between electrons i and j Trust radius... 

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