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An Analysis of the Barzilai and Borwein Gradient Method for Unsymmetric Linear Equations

  • Yu-Hong Dai
  • Li-Zhi Liao
  • Duan Li
Part of the Applied Optimization book series (APOP, volume 96)

Abstract

The Barzilai and Borwein gradient method does not ensure descent in the objective function at each iteration, but performs better than the classical steepest descent method in practical computations. Combined with the technique of nonmonotone line search etc., such a method has found successful applications in unconstrained optimization, convex constrained optimization and stochastic optimization. In this paper, we give an analysis of the Barzilai and Borwein gradient method for two unsymmetric linear equations with only two variables. Under mild conditions, we prove that the convergence rate of the Barzilai and Borwein gradient method is Q-superlinear if the coefficient matrix A has the same eigenvalue; if the eigenvalues of A are different, then the convergence rate is R-superlinear.

Key words

Unsymmetric linear equations gradient method convergence Q-superlinear R-superlinear 

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Yu-Hong Dai
    • 1
  • Li-Zhi Liao
    • 2
  • Duan Li
    • 3
  1. 1.Institute of Computational Mathematics and Scientific/Engineering ComputingChinese Academy of SciencesP.R. China
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloon Tong, KowloonHong Kong
  3. 3.Department of Systems Engineering and Engineering ManagementChinese University of Hong KongHong Kong

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