Hessian method

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

Aq becomes asymptotically a g/ g, i.e., the steepest descent fomuila 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 fiinction and tmst radius optimizations is that, in the latter, the level shift is applied only if the calculated step exceeds a threshold, while in the fonuer it is imposed smoothly and is automatically reduced to zero as convergence is approached. 

Although it was originally developed for locating transition states, the EF algoritlnn is also efficient for minimization and usually perfonns as well as or better than the standard quasi-Newton algorithm. In this case, a single shift parameter is used, and the method is essentially identical to the augmented Hessian method. 

Unconstrained optimization methods [W. H. Press, et. al.. Numerical Recipes The Art of Scientific Computing, Cambridge University Press, 1986, Chapter 10] can use values of only the objective function, or of first derivatives of the objective function, second derivatives of the objective function, etc. HyperChem uses first derivative information and, in the Block Diagonal Newton-Raphson case, second derivatives for one atom at a time. HyperChem does not use optimizers that compute the full set of second derivatives (the Hessian) because it is impractical to store the Hessian for macromolecules with thousands of atoms. A future release may make explicit-Hessian methods available for smaller molecules but at this release only methods that store the first derivative information, or the second derivatives of a single atom, are used. 

Biegler, L.T., Nocedal, J. and Schmid, C., 1995. A reduced Hessian method for large scale constrained optimization. SIAM Journal of Optimization, 5(2), 314. 

Hessian method, i.e. the step is parameterized as in eq. (14.6). The minimization step is similar to that described in Section 14.3.1 for locating minima, the only difference is for the unique TS mode. 

One method, which avoids the problem with undesired negative eigenvalues of the Hessian, and which introduces an automatic damping of the rotations, is the augmented Hessian method (AM). To describe the properties of this method, let us again consider the Newton-Raphson equation (4 4) ... 

Equation (4 59) can be compared to the unfolded two-step version of the augmented Hessian method, which results in the secular equation ... 

Illustrate the bracketing theorem mentioned in connection with the augmented Hessian method, by solving equation (4 31) for the energy E in the form E = f(E). Plot both the functions E and f(E) and show that the crossing points (the eigenvalues Ej) satisfies the betweenness condition. 

The augmented Hessian method requires an exact Hessian, or an update method on the Hessian itself. The update formula for the Hessian analysis to the inverse Hessian appear in the Appendix. 

The ability of augmented Hessian methods for generating a search toward a first-order ... 

A novel model reduction-based optimization framework for input/output steady state simulators has been presented. It can be considered as an extension of Reduced Hessian methods. Reduced Hessians are efficiently computed through a double-step reduction procedure first onto the dominant system subspace, adaptively computed through subspace iterations, and second onto the subspace of the decision variables. Only low-order Jacobians need to be computed through a few numerical perturbations, using only... 

L.T. Biegler, J. Nocedal and C. Schmidt, 1995, A Reduced Hessian Method for Large-Scale Constrained Optimization, Siam Journal on Optimization 5 314-347... 

In order to minimize the second-order energy approximation (T) for fixed Cl coefficients a step-restricted augmented Hessian method as outlined in Section II.B (Eqs (30)-(33)) is used. While in other MCSCF methods this technique is employed to minimize the exact energy, it is used here to minimize an approximate energy functional. The parameter vector x is made up of the... 

Augmented Hessian method with step-length control results from Ref 70. Start with orbitals of smaller MCSCF. Final energy - 39.0278826738 hartree. 

The later procedure of Knowles and Werner uses instead an augmented Hessian method to define the orbital corrections. An approximate Hamiltonian operator is constructed that accounts for the simultaneous change of the entire vector of orbital rotations. This procedure may be summarized by the following steps ... 

For the current c, construct the density matrices D and d. From the current exact H, D and d, construct B and w and solve for k using an augmented orbital Hessian method. This orbital correction vector k defines the transformation matrix U. 

If the reaction path is not obvious, then the most general techniques require information about the second derivatives. There exist, however, several often successful techniques that do not require this. The MOPAC and AMPAC series of programs utilize, for example, the saddlepoint technique, which attempts to approach the transition state from the reactant and product geometry simultaneously. The ZINDO set of models can utilize a combination of augmented Hessian and analytic geometry techniques. This is a very effective method, but unfortunately the augmented Hessian method does require approximate second derivatives and is somewhat time consuming. 

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