| This dissertation considers an important class of large scale unconstrained minimization problems of the form min x&isinN&subD imize f&parl0x&parr0 1 where f : D&rarrR is a twice continuously differentiable nonlinear function defined on a domain D of Rn with n being large, N&subD is a neighborhood of a local minimum point x* of f such that f(x*) &le f(x) for all x &isin N , and the Hessian matrix H(x) is dense but can be approximated by a sparse incomplete Hessian matrix, M(x).A numerical minimization algorithm based on a modified Newton method and a line search strategy, called the incomplete Hessian Newton method (IHN), was proposed by Professor Dexuan Xie to solve (1) and was analyzed under the assumptions that M(x) is symmetric and positive definite and that the incomplete Hessian Newton equation is solved exactly for generating the search direction. Under those assumptions, IHN is a well defined descent search method. In practice, however, M( x) may be indefinite when x is far away from x*. As a result, IHN may not be a well defined descent search method globally. To develop a practical and global IHN-type scheme, IHN was modified as truncated IHN (T-IHN), which solves the incomplete Hessian Newton equation inexactly using a modified preconditioned conjugate gradient solver, resulting in a well defined descent search method even with an indefinite incomplete Hessian matrix or an indefinite preconditioner.This dissertation gives a general convergence analysis of T-IHN. It shows that T-IHN is globally convergent even with an indefinite incomplete Hessian matrix or an indefinite preconditioner, which may happen in practice. It also proves that when the T-IHN iterates are close enough to a minimum point, T-IHN has a linear rate of convergence, and an admissible line search step length of one satisfying the Wolfe conditions. A user friendly Matlab package of T-IHN is provided in this dissertation. Moreover, as an important application, a particular T-IHN algorithm is constructed for the minimization of the force field of a biomolecule, and numerically tested for a protein model problem based on a widely used molecular simulation package, CHARMM. Numerical results confirm the theoretical results, and demonstrate that T-IHN can have a better performance (in terms of computer CPU time) than most CHARMM minimizers. |
