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Nonconvex minimization calculations and the conjugate gradient method

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Numerical Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1066))

Abstract

We consider the global convergence of conjugate gradient methods without restarts, assuming exact arithmetic and exact line searches, when the objective function is twice continuously differentiable and has bounded level sets. Most of our attention is given to the Polak-Ribière algorithm, and unfortunately we find examples that show that the calculated gradients can remain bounded away from zero. The examples that have only two variables show also that some variable metric algorithms for unconstrained optimization need not converge. However, a global convergence theorem is proved for the Fletcher-Reeves version of the conjugate gradient method.

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References

  • A. Buckley (1982), "Conjugate gradient methods", in Nonlinear Optimization 1981, ed. M.J.D. Powell, Academic Press (London).

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  • R. Fletcher (1980), Practical Methods of Optimization, Vol. I: Unconstrained Optimization, John Wiley & Sons (Chichester).

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  • R. Fletcher and C.M. Reeves (1964), "Function minimization by conjugate gradients", The Computer Journal, Vol. 7, pp. 149–154.

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  • M.J.D. Powell (1977), "Restart procedures for the conjugate gradient method", Math. Programming, Vol. 12, pp. 241–254.

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  • J.R. Thompson (1977), "Examples of non-convergence of conjugate descent algorithms with exact line searches", Math. Programming, Vol. 12, pp. 356–360.

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  • G. Zoutendijk (1970), "Nonlinear programming, computational methods", in Integer and Nonlinear Programming, ed. J. Abadie, North-Holland Publishing Co. (Amsterdam).

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Authors

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David F. Griffiths

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© 1984 Springer-Verlag

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Powell, M.J.D. (1984). Nonconvex minimization calculations and the conjugate gradient method. In: Griffiths, D.F. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099521

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  • DOI: https://doi.org/10.1007/BFb0099521

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13344-5

  • Online ISBN: 978-3-540-38881-4

  • eBook Packages: Springer Book Archive

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