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A Survey of Quasi-Newton Equations and Quasi-Newton Methods for Optimization

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  • Chengxian Xu
  • Jianzhong Zhang

Abstract

Quasi-Newton equations play a central role in quasi-Newton methods for optimization and various quasi-Newton equations are available. This paper gives a survey on these quasi-Newton equations and studies properties of quasi-Newton methods with updates satisfying different quasi-Newton equations. These include single-step quasi-Newton equations that use only gradient information and that use both gradient and function value information in one step, and multi-step quasi-Newton equations that use the gradient information in last m steps. Main properties of quasi-Newton methods with updates satisfying different quasi-Newton equations are studied. These properties include the finite termination property, invariance, heredity of positive definite updates, consistency of search directions, global convergence and local superlinear convergence properties. Copyright Kluwer Academic Publishers 2001

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  • Chengxian Xu & Jianzhong Zhang, 2001. "A Survey of Quasi-Newton Equations and Quasi-Newton Methods for Optimization," Annals of Operations Research, Springer, vol. 103(1), pages 213-234, March.
    • Handle: RePEc:spr:annopr:v:103:y:2001:i:1:p:213-234:10.1023/a:1012959223138
    DOI: 10.1023/A:1012959223138
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    Cited by:

    1. Yong Li & Gonglin Yuan & Zhou Sheng, 2018. "An active-set algorithm for solving large-scale nonsmooth optimization models with box constraints," PLOS ONE, Public Library of Science, vol. 13(1), pages 1-16, January.
    2. Zhanwen Shi & Guanyu Yang & Yunhai Xiao, 2016. "A limited memory BFGS algorithm for non-convex minimization with applications in matrix largest eigenvalue problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(2), pages 243-264, April.
    3. Mingyuan Cao & Qingdao Huang & Chaoqian Li & Yueting Yang, 2020. "A Subspace Modified Broyden–Fletcher–Goldfarb–Shanno Method for $$\mathcal {B}$$B-eigenvalues of Symmetric Tensors," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 419-432, February.
    4. Mehiddin Al-Baali & Humaid Khalfan, 2012. "A combined class of self-scaling and modified quasi-Newton methods," Computational Optimization and Applications, Springer, vol. 52(2), pages 393-408, June.
    5. Yueting, Yang & Chengxian, Xu, 2007. "A compact limited memory method for large scale unconstrained optimization," European Journal of Operational Research, Elsevier, vol. 180(1), pages 48-56, July.
    6. Ling, Ai-Fan & Xu, Cheng-Xian & Xu, Feng-Min, 2009. "A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems," European Journal of Operational Research, Elsevier, vol. 197(2), pages 519-531, September.
    7. Fahimeh Biglari & Farideh Mahmoodpur, 2016. "Scaling Damped Limited-Memory Updates for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 177-188, July.
    8. Fatemeh Dargahi & Saman Babaie-Kafaki & Zohre Aminifard, 2024. "Eigenvalue Analyses on the Memoryless Davidon–Fletcher–Powell Method Based on a Spectral Secant Equation," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 394-403, January.

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