### Abstract

With advances in computer architecture and software, Newton methods are becoming not only feasible for large‐scale nonlinear optimization problems, but also reliable, fast and efficient. Truncated Newton methods, in particular, are emerging as a versatile subclass. In this article we present a truncated Newton algorithm specifically developed for potential energy minimization. The method is globally convergent with local quadratic convergence. Its key ingredients are: (1) approximation of the Newton direction far away from local minima, (2) solution of the Newton equation iteratively by the linear Conjugate Gradient method, and (3) preconditioning of the Newton equation by the analytic second‐derivative components of the “local” chemical interactions: bond length, bond angle and torsional potentials. Relaxation of the required accuracy of the Newton search direction diverts the minimization search away from regions where the function is nonconvex and towards physically interesting regions. The preconditioning strategy significantly accelerates the iterative solution for the Newton search direction, and therefore reduces the computation time for each iteration. With algorithmic variations, the truncated Newton method can be formulated so that storage and computational requirements are comparable to those of the nonlinear Conjugate Gradient method. As the convergence rate of nonlinear Conjugate Gradient methods is linear and performance less predictable, the application of the truncated Newton code to potential energy functions is promising.

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

Pages (from-to) | 1025-1039 |

Number of pages | 15 |

Journal | Journal of Computational Chemistry |

Volume | 8 |

Issue number | 7 |

DOIs | |

State | Published - 1987 |

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### ASJC Scopus subject areas

- Chemistry(all)
- Computational Mathematics

### Cite this

**A powerful truncated Newton method for potential energy minimization.** / Schlick, Tamar; Overton, Michael.

Research output: Contribution to journal › Article

*Journal of Computational Chemistry*, vol. 8, no. 7, pp. 1025-1039. https://doi.org/10.1002/jcc.540080711

}

TY - JOUR

T1 - A powerful truncated Newton method for potential energy minimization

AU - Schlick, Tamar

AU - Overton, Michael

PY - 1987

Y1 - 1987

N2 - With advances in computer architecture and software, Newton methods are becoming not only feasible for large‐scale nonlinear optimization problems, but also reliable, fast and efficient. Truncated Newton methods, in particular, are emerging as a versatile subclass. In this article we present a truncated Newton algorithm specifically developed for potential energy minimization. The method is globally convergent with local quadratic convergence. Its key ingredients are: (1) approximation of the Newton direction far away from local minima, (2) solution of the Newton equation iteratively by the linear Conjugate Gradient method, and (3) preconditioning of the Newton equation by the analytic second‐derivative components of the “local” chemical interactions: bond length, bond angle and torsional potentials. Relaxation of the required accuracy of the Newton search direction diverts the minimization search away from regions where the function is nonconvex and towards physically interesting regions. The preconditioning strategy significantly accelerates the iterative solution for the Newton search direction, and therefore reduces the computation time for each iteration. With algorithmic variations, the truncated Newton method can be formulated so that storage and computational requirements are comparable to those of the nonlinear Conjugate Gradient method. As the convergence rate of nonlinear Conjugate Gradient methods is linear and performance less predictable, the application of the truncated Newton code to potential energy functions is promising.

AB - With advances in computer architecture and software, Newton methods are becoming not only feasible for large‐scale nonlinear optimization problems, but also reliable, fast and efficient. Truncated Newton methods, in particular, are emerging as a versatile subclass. In this article we present a truncated Newton algorithm specifically developed for potential energy minimization. The method is globally convergent with local quadratic convergence. Its key ingredients are: (1) approximation of the Newton direction far away from local minima, (2) solution of the Newton equation iteratively by the linear Conjugate Gradient method, and (3) preconditioning of the Newton equation by the analytic second‐derivative components of the “local” chemical interactions: bond length, bond angle and torsional potentials. Relaxation of the required accuracy of the Newton search direction diverts the minimization search away from regions where the function is nonconvex and towards physically interesting regions. The preconditioning strategy significantly accelerates the iterative solution for the Newton search direction, and therefore reduces the computation time for each iteration. With algorithmic variations, the truncated Newton method can be formulated so that storage and computational requirements are comparable to those of the nonlinear Conjugate Gradient method. As the convergence rate of nonlinear Conjugate Gradient methods is linear and performance less predictable, the application of the truncated Newton code to potential energy functions is promising.

UR - http://www.scopus.com/inward/record.url?scp=84988050306&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988050306&partnerID=8YFLogxK

U2 - 10.1002/jcc.540080711

DO - 10.1002/jcc.540080711

M3 - Article

VL - 8

SP - 1025

EP - 1039

JO - Journal of Computational Chemistry

JF - Journal of Computational Chemistry

SN - 0192-8651

IS - 7

ER -