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A new subspace minimization conjugate gradient method based on modified secant equation for unconstrained optimization

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Abstract

In this paper, a new subspace minimization conjugate gradient method based on modified secant equation is proposed and analyzed. For a classical subspace minimization conjugate gradient method, the search direction is derived by minimizing an approximate quadratic model of objective function in a two-dimensional subspace. Generally, the approximate Hessian matrix in the above quadratic model is required to satisfy the standard secant equation, while we consider an approximate Hessian matrix which satisfies the modified secant equation. We give some rules such that if these rules are satisfied, we choose the standard secant equation, otherwise we choose the modified one. We can prove that the proposed directions satisfy the sufficient descent property under some extra conditions. We also present a modified nonmonotone Wolfe line search and establish the global convergence of the proposed method for general nonlinear functions under mild assumptions. Numerical comparisons are given with famous CG_DESCENT (5.3) (Hager and Zhang in SIAM J Optim 16(1):170–192, 2005) and SMCG_BB (Liu and Liu in J Optim Theory Appl 180(3):879–906, 2019), and show that the proposed algorithm is very promising.

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Acknowledgements

We would like to thank Pofessors Hager,W.W. and Zhang, H.C. for their CG_DESCENT code. This research is supported by National Science Foundation of China (No. 11901561), GuangXi 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

    Xinliu Diao & 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, 100080, China

    Zexian Liu

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

    Zexian Liu

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  1. Xinliu Diao
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  2. Hongwei Liu
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  3. Zexian Liu
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Correspondence to Hongwei Liu.

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Communicated by Paulo J. S. Silva.

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Diao, X., Liu, H. & Liu, Z. A new subspace minimization conjugate gradient method based on modified secant equation for unconstrained optimization. Comp. Appl. Math. 39, 251 (2020). https://doi.org/10.1007/s40314-020-01301-9

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