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An adaptive sizing BFGS method for unconstrained optimization

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Abstract

In this paper, an adaptive sizing BFGS method for unconstrained optimization is proposed, whose scaling factor is automatically chosen by the signature of some terms of the approximate model’s curvature. The scaling factor is always chosen less than or equal to one as required by the convergence property of the self-scaling BFGS method of Al-Baali (Comput Optim Appl 9:191–203, 1998), while the choosing strategy can ensure the sufficient positive definiteness of the updating matrices. Under mild conditions, the global convergence properties of Al-Baali on convex functions are proved. Numerical results on some test problems show the proposed method is competitive with its counterparts provided that the scaling factor is not too small.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (71071075), the Natural Science Fundation of the Jiangsu Higher Education Institutions of China (12KJB110006) and the funding of Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents. The authors would like to thank one anonymous referee for careful reading of the earlier draft and valuable comments which greatly improve the quality of the paper, and Professor Dr. Mehiddin Al-Baali at Sultan Qaboos University of Oman for sending some related references to us. We are also grateful to Dr. Kathryn Lockwood at University of Florida for improving our English writing, Professor Dr. William Hager at University of Florida for providing the computer to revise the paper.

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Authors and Affiliations

  1. College of Sciences, Nanjing Tech University, Nanjing, 210009, Jiangsu, People’s Republic of China

    Hao Liu, Jianfeng Shao & Baoxian Chang

  2. College of Sciences, China University of Mining and Technology, Xuzhou, 221008, Jiangsu, People’s Republic of China

    Haijun Wang

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  1. Hao Liu
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  2. Jianfeng Shao
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  4. Baoxian Chang
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Correspondence to Haijun Wang.

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Liu, H., Shao, J., Wang, H. et al. An adaptive sizing BFGS method for unconstrained optimization. Calcolo 52, 233–244 (2015). https://doi.org/10.1007/s10092-014-0115-y

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