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Efficient Implementation of the Truncated-Newton Algorithm for Large-Scale
Chemistry Applications.

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To efficiently implement the truncated-Newton (TN) optimization method
for large scale, highly-nonlinear functions in chemistry, an unconventional
modified Cholesky (UMC) factorization is proposed to avoid large modifications
to a problem-derived preconditioner, used in the inner loop used to approximate
the TN search vector at each step. The main motivation is to reduce the
computational time of the overall method: large changes in standard modified
Cholesky factorizations are found to increase the number of total iterations,
as well as computational time, significantly. Since the UMC may generate
an indefinite, rather than a positive definite, effective preconditioner,
we prove that directions of descent still result. Hence, convergence to
a local minimum can be shown as in classic TN methods for our UMC-based
algorithm. Our incorporation of the UMC also requires changes in the TN
inner loop regarding the negative-curvature test (which we replace by a
descent direction test) and the choice of exit directions. Numerical experiments
demonstrate that the unconventional use of an indefinite preconditioner
works much better than the minimizer without preconditioning, as well as
other minimizers available in the molecular mechanics package CHARMM. Good
performance of the resulting TN method for large potential energy problems
is also shown with respect to the limited-memory BFGS method, tested both
with and without preconditioning.

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