Nonconvex minimization calculations and the conjugate gradient method
- M. J. D. Powell
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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.
- A. Buckley (1982), "Conjugate gradient methods", in Nonlinear Optimization 1981, ed. M.J.D. Powell, Academic Press (London).
- R. Fletcher (1980), Practical Methods of Optimization, Vol. I: Unconstrained Optimization, John Wiley & Sons (Chichester).
- R. Fletcher and C.M. Reeves (1964), "Function minimization by conjugate gradients", The Computer Journal, Vol. 7, pp. 149–154. CrossRef
- M.J.D. Powell (1977), "Restart procedures for the conjugate gradient method", Math. Programming, Vol. 12, pp. 241–254. CrossRef
- J.R. Thompson (1977), "Examples of non-convergence of conjugate descent algorithms with exact line searches", Math. Programming, Vol. 12, pp. 356–360. CrossRef
- G. Zoutendijk (1970), "Nonlinear programming, computational methods", in Integer and Nonlinear Programming, ed. J. Abadie, North-Holland Publishing Co. (Amsterdam).
- Nonconvex minimization calculations and the conjugate gradient method
- Book Title
- Numerical Analysis
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- Proceedings of the 10th Biennial Conference held at Dundee, Scotland, June 28 – July 1, 1983
- pp 122-141
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- Lecture Notes in Mathematics
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- Springer Berlin Heidelberg
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