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Convergence Analysis of Nonlinear Conjugate Gradient Methods

In: Optimization and Regularization for Computational Inverse Problems and Applications

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  • Yuhong Dai

    (Chinese Academy of Sciences, State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science)

Abstract

Conjugate gradient methods are a class of important methods for unconstrained optimization and vary only with a scalar β k . In this chapter, we analyze general conjugate gradient method using the Wolfe line search and propose a condition on the scalar β k , which is sufficient for the global convergence. An example is constructed, showing that the condition is also necessary in some sense for the global convergence of general conjugate gradient method. To make better use of the condition, we introduce a new property for conjugate gradient methods. It is shown that many conjugate gradient methods have such property, including the FR, PRP, HS, and DY methods and the FR-PRP, and DY-HS hybrid methods. Consequently, convergence results are gained for these methods under mild assumptions. In addition, an analysis is also given to a new conjugate gradient method, which further demonstrates the usefulness of the condition and the new property. Some discussions about the bound in the hybrid conjugate gradient methods are also given.

Suggested Citation

  • Yuhong Dai, 2010. "Convergence Analysis of Nonlinear Conjugate Gradient Methods," Springer Books, in: Yanfei Wang & Changchun Yang & Anatoly G. Yagola (ed.), Optimization and Regularization for Computational Inverse Problems and Applications, chapter 0, pages 157-181, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-13742-6_8
    DOI: 10.1007/978-3-642-13742-6_8
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