Trust radius method

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. ... 

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)). 

Aq becomes as5miptotically a g/ g, i.e., the steepest descent formula with a step length 1/a. The augmented Hessian method is closely related to eigenvector (mode) following, discussed in section B3.5.5 2. The main difference between rational function and trust radius optimizations is that, in the latter, the level shift is applied only if the calculated step exceeds a threshold, while in the former it is imposed smoothly and is automatically reduced to zero as convergence is approached. 

We find the solution to (35) using the secant method with a trust radius of a/4 at each iteration. The algorithm was terminated once the integral on the right-hand side of (35) was less than 10 in absolute magnitude. The results are presented in Table 1, along with the maximum absolute error as defined by... 

The method therefore reduces to Newton s method in the local region with its rapid rate of convergence. Notice that in each iteration we always first try the Newton step Eq. (42), resorting to the level-shifted step Eq. (41) only if the Newton step is larger than the trust radius. 

The shift parameter can be used to ensure that the optimization proceeds downhill even if the Hessian has negative eigenvalues. In addition, it can be chosen such that the step size is lower or equal to a predefined threshold. Popular methods using a shift parameter are the rational function optimization (RFO) [48] and Trust Radius (TR) methods [49, 50]. A finer control on the step size and direction can be achieved using an approximate line search method, which attempts to fit a polynomial function to the energies and gradients of the best previous points [51]. 

To conclude the discussion of the trust-region method, we describe how the trust radius ft is updated. In iteration n, we generate a vector X , from which we obtain the reference wave function for the next iteration. However, we may also use this vector to predict the energy in the next iteration n -f 1 according to the expression... 

The advantage of the trust-region method is that even if pPl is nearly singular, mS ip) still has a useful minimum within the trust region. As long as F(x) is reduced, x + J = x -I- Ax l is accepted, else the old value is recycled and the trust radius is reduced. If the agreement between / (x + l) — F(xPJ) and j[ i(Axt ) — j[ J(0) is good (poor), the trust radius is increased (decreased) for the next iteration. 

The dogleg method is based upon an analysis of how the trust-region minimum changes as a function of the trust radius A l. When A is very large, we have effectively an unconstrained problem and the trust-region minimum is the fiill Newton step the global minimum of m p). 

In the trust-region or restricted-step method [19], the Newton step (10.8.2) is taken only if it is smaller than or equal to the trust radius h ... 

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