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Convergence of Conjugate Gradient Methods with a Closed-Form Stepsize Formula

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  • C. Labat

    (Institut de Recherche en Communications et Cybernétique de Nantes)

  • J. Idier

    (Institut de Recherche en Communications et Cybernétique de Nantes)

Abstract

Conjugate gradient methods are efficient methods for minimizing differentiable objective functions in large dimension spaces. However, converging line search strategies are usually not easy to choose, nor to implement. Sun and colleagues (Ann. Oper. Res. 103:161–173, 2001; J. Comput. Appl. Math. 146:37–45, 2002) introduced a simple stepsize formula. However, the associated convergence domain happens to be overrestrictive, since it precludes the optimal stepsize in the convex quadratic case. Here, we identify this stepsize formula with one iteration of the Weiszfeld algorithm in the scalar case. More generally, we propose to make use of a finite number of iterates of such an algorithm to compute the stepsize. In this framework, we establish a new convergence domain, that incorporates the optimal stepsize in the convex quadratic case.

Suggested Citation

  • C. Labat & J. Idier, 2008. "Convergence of Conjugate Gradient Methods with a Closed-Form Stepsize Formula," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 43-60, January.
    • Handle: RePEc:spr:joptap:v:136:y:2008:i:1:d:10.1007_s10957-007-9306-x
    DOI: 10.1007/s10957-007-9306-x
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    References listed on IDEAS

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    1. Jie Sun & Jiapu Zhang, 2001. "Global Convergence of Conjugate Gradient Methods without Line Search," Annals of Operations Research, Springer, vol. 103(1), pages 161-173, March.
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