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New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization

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Abstract

In this paper, two new subspace minimization conjugate gradient methods based on p-regularization models are proposed, where a special scaled norm in p-regularization model is analyzed. Different choices of special scaled norm lead to different solutions to the p-regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R-linear convergence of the proposed methods is also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J. Optim. 16(1):170–192, 2005), Dai and Kou (SIAM J. Optim. 23(1):296–320, 2013), Liu and Liu (J. Optim. Theory. Appl. 180(3):879–906, 2019) and Li et al. (Comput. Appl. Math. 38(1):2019), respectively.

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Acknowledgments

We would like to thank the anonymous referees for their useful comments. We also would like to thank Professors Hager,W.W. and Zhang, H.C. for their CG_DESCENT (5.3) code, and thank Professor Dai, Y.H and Dr. Kou, C.X. for their CGOPT code.

Funding

This research is supported by the National Science Foundation of China (No. 11901561), Guangxi Natural Science Foundation (No. 2018GXNSFBA281180) and China Postdoctoral Science Foundation (2019M660833).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710126, China

    Ting Zhao & Hongwei Liu

  2. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Zexian Liu

  3. School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, China

    Zexian Liu

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  1. Ting Zhao
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  2. Hongwei Liu
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Correspondence to Hongwei Liu.

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Zhao, T., Liu, H. & Liu, Z. New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization. Numer Algor 87, 1501–1534 (2021). https://doi.org/10.1007/s11075-020-01017-1

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