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Volume 17, Issue 6
Convergence Properties of a Modified BFGS Algorithm for Minimization with Armijo-Goldstein Steplengths

Nai-Yang Deng & Zheng-Feng Li

J. Comp. Math., 17 (1999), pp. 645-652.

Published online: 1999-12

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  • Abstract

The line search strategy is crucial for an efficient unconstrained optimization algorithm. One of the reason why the Wolfe line searches is recommended lies in that it ensures positive definiteness of BFGS updates. When gradient information has to be obtained costly, the Armijo-Goldstein line searches may be preferred. To maintain positive definiteness of BFGS updates based on the Armijo-Goldstein line searches, a slightly modified form of BFGS update is proposed by I.D. Coope and C.J. Price (Journal of Computational Mathematics, 13 (1995), 156-160), while its convergence properties is open up to now. This paper shows that the modified BFGS algorithm is globally and superlinearly convergent based on the Armijo-Goldstein line searches.

  • Keywords

BFGS methods, Convergence, Superlinear convergence.

  • AMS Subject Headings

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COPYRIGHT: © Global Science Press

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@Article{JCM-17-645, author = {Deng , Nai-Yang and Li , Zheng-Feng}, title = {Convergence Properties of a Modified BFGS Algorithm for Minimization with Armijo-Goldstein Steplengths}, journal = {Journal of Computational Mathematics}, year = {1999}, volume = {17}, number = {6}, pages = {645--652}, abstract = {

The line search strategy is crucial for an efficient unconstrained optimization algorithm. One of the reason why the Wolfe line searches is recommended lies in that it ensures positive definiteness of BFGS updates. When gradient information has to be obtained costly, the Armijo-Goldstein line searches may be preferred. To maintain positive definiteness of BFGS updates based on the Armijo-Goldstein line searches, a slightly modified form of BFGS update is proposed by I.D. Coope and C.J. Price (Journal of Computational Mathematics, 13 (1995), 156-160), while its convergence properties is open up to now. This paper shows that the modified BFGS algorithm is globally and superlinearly convergent based on the Armijo-Goldstein line searches.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9135.html} }
TY - JOUR T1 - Convergence Properties of a Modified BFGS Algorithm for Minimization with Armijo-Goldstein Steplengths AU - Deng , Nai-Yang AU - Li , Zheng-Feng JO - Journal of Computational Mathematics VL - 6 SP - 645 EP - 652 PY - 1999 DA - 1999/12 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9135.html KW - BFGS methods, Convergence, Superlinear convergence. AB -

The line search strategy is crucial for an efficient unconstrained optimization algorithm. One of the reason why the Wolfe line searches is recommended lies in that it ensures positive definiteness of BFGS updates. When gradient information has to be obtained costly, the Armijo-Goldstein line searches may be preferred. To maintain positive definiteness of BFGS updates based on the Armijo-Goldstein line searches, a slightly modified form of BFGS update is proposed by I.D. Coope and C.J. Price (Journal of Computational Mathematics, 13 (1995), 156-160), while its convergence properties is open up to now. This paper shows that the modified BFGS algorithm is globally and superlinearly convergent based on the Armijo-Goldstein line searches.

Nai-Yang Deng & Zheng-Feng Li. (1970). Convergence Properties of a Modified BFGS Algorithm for Minimization with Armijo-Goldstein Steplengths. Journal of Computational Mathematics. 17 (6). 645-652. doi:
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