New stepsizes for the gradient method

Abstract

Gradient methods are famous for their simplicity and low complexity, which attract more and more attention for large scale optimization problems. A good stepsize plays an important role to construct an efficient gradient method. This paper proposes a new framework to generate stepsizes for gradient methods applied to convex quadratic function minimization problems. By adopting different criterions, we propose four new gradient methods. For 2-dimensional unconstrained problems with convex quadratic objective functions, we prove that the new methods either terminate in finite iterations or converge R-superlinearly; for n-dimensional problems, we prove that all the new methods converge R-linearly. Numerical experiments show that the new methods enjoy lower complexity and outperform the existing gradient methods.

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Notes

  1. 1.

    For example, in this test problem, the per iteration computational cost of \({\text {ABB}}_{\min }\) is \(n^2+3n+\frac{5}{2}\), while that of Alg. 3 is \(\frac{1}{5}n^2+\frac{2}{5}n+\frac{3}{10}\).

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Acknowledgements

The authors would like to thank Prof. Ya-xiang Yuan from Chinese Academy of Sciences, the editor and the two anonymous reviewers for their valuable suggestions and comments.

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Correspondence to Cong Sun.

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This work is partially supported by NSFC Grants 11771056, 91630202 and 11871115, and the Young Elite Scientists Sponsorship Program by CAST 2017QNRC001.

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Sun, C., Liu, JP. New stepsizes for the gradient method. Optim Lett 14, 1943–1955 (2020). https://doi.org/10.1007/s11590-019-01512-y

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Keywords

  • Gradient method
  • Steepest descent
  • R-linear convergence rate
  • Finite termination