An Efficient Gradient Method with Approximately Optimal Stepsize Based on Tensor Model for Unconstrained Optimization

  • Zexian Liu
  • Hongwei LiuEmail author


A new type of stepsize, which was recently introduced by Liu et al. (Optimization 67(3):427–440, 2018), is called approximately optimal stepsize and is very efficient for gradient method. Interestingly, all gradient methods can be regarded as gradient methods with approximately optimal stepsizes. In this paper, we present an efficient gradient method with approximately optimal stepsize based on tensor model for unconstrained optimization. In the proposed method, if the objective function is not close to a minimizer and a quadratic function on a line segment between the current and latest iterates, then a tensor model is exploited to generate approximately optimal stepsize for gradient method. Otherwise, quadratic approximation models are constructed to generate approximately optimal stepsizes for gradient method. The global convergence of the proposed method is established under weak conditions. Numerical results indicate that the proposed method is very promising.


Gradient method Approximately optimal stepsize Quadratic model Tensor model Global convergence 

Mathematics Subject Classification

90C06 65K 



We would like to thank Professors Hager and Zhang, H. C. for their C code of CG_DESCENT, and thank Professor Dai, Y. H. for his help in the numerical experiments. This research is supported by National Science Foundation of China (No.11461021), Shangxi Science Foundation (No. 2017JM1014), Guangxi Science Foundation (Nos. 2018GXNSFBA281180, 2017GXNSFBA198031), Project of Guangxi Education Department Grant (2017KY0648), Scientific Research Project of Hezhou University (Nos. 2014YBZK06, 2016HZXYSX03).


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China
  2. 2.School of Mathematics and Computer ScienceHezhou UniversityHezhouPeople’s Republic of China

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