Projected Hessian

Gurwitz, C. B., and Overton, M., SQP methods based on approximating a projected Hessian matrix, SIAM J. Sci. Stat. Comput. 10(4) 631 (1989). 

Nocedal, Jorge, and Overton, Michael L., Projected Hessian updating algorithms for non-linearly constrained optimization, SIAM J. Numer. Anal. 22(5), 821 (1985). 

J. Nocedal and M. L. Overton, 1985, Projected Hessian updating algorithms for nonlinearly constrained optimization, SIAM Journal of Numerical Analysis, 22 821-850. 

This projected Hessian matrix of cr(x) is symbolized by M. Forming the derivatives in matrix (10), and sorting out the corresponding terms, the eigenvalues of the following matrix [19] have to be calculated... 

In order to use variational transition state theory or reaction path Hamiltonian methods (see Transition State Theory) to calculate rate constants, one needs vibrational frequencies perpendicular to the reaction path. These are obtained by projecting out motion along the tangent, D. The projector, P, and the projected Hessian, H, are given by... 

The projected frequencies can then be calculated from the projected Hessian in the usual fashion. If mass-weighted Cartesian coordinates are u.sed, rigid body translation and rotation are also projected out. In addition to the projected frequencies, one also needs the coupling matrix elements, B,... 

In another promising method, based on the effective Hamiltonian theory used in quantum chemistry [19], the protein is divided into blocks that comprise one or more residues. The Hessian is then projected into the subspace defined by the rigid-body motions of these blocks. The resulting low frequency modes are then perturbed by the higher... 

The parameters a are obtained by inserting (4 27) into (4 24) and project onto each of the The process normally converges in less than ten iterations, although, as in all iterative schemes of this kind, convergence can be slowed down considerably in near singular situations (small eigenvalues to the Hessian... 

The difference between the gradients at the two points thus gives a finite difference approximation to the projection of the exact Hessian H on the direction (p - p n+1)), at the point p n+1 This information about the Hessian can be used to improve the convergence of the calculation. Suppose that we have computed a step on the energy surface as Ap(n+2 We can always formally write this as ... 

---