### Abstract

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 in approximating 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 or 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.

Original language | English (US) |
---|---|

Pages (from-to) | 132-154 |

Number of pages | 23 |

Journal | SIAM Journal on Optimization |

Volume | 10 |

Issue number | 1 |

State | Published - 1999 |

### Fingerprint

### Keywords

- Descent direction
- Indefinite preconditioner
- Modified Cholesky factorization
- Molecular potential minimization
- Truncated-Newton method
- Unconventional modified Cholesky factorization

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Optimization*,

*10*(1), 132-154.

**Efficient implementation of the truncated-newton algorithm for large-scale chemistry applications.** / Xie, Dexuan; Schlick, Tamar.

Research output: Contribution to journal › Article

*SIAM Journal on Optimization*, vol. 10, no. 1, pp. 132-154.

}

TY - JOUR

T1 - Efficient implementation of the truncated-newton algorithm for large-scale chemistry applications

AU - Xie, Dexuan

AU - Schlick, Tamar

PY - 1999

Y1 - 1999

N2 - 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 in approximating 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 or 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.

AB - 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 in approximating 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 or 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.

KW - Descent direction

KW - Indefinite preconditioner

KW - Modified Cholesky factorization

KW - Molecular potential minimization

KW - Truncated-Newton method

KW - Unconventional modified Cholesky factorization

UR - http://www.scopus.com/inward/record.url?scp=0033269652&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033269652&partnerID=8YFLogxK

M3 - Article

VL - 10

SP - 132

EP - 154

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 1

ER -