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Stability of Huang's update for the conjugate gradient method

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Abstract

The conditions under which Huang's conjugate gradient method generates descent directions are given and discussed. Bounds for the condition number of the inverse Hessian matrix are estimated for the case of a symmetric update.

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References

  1. Huang, H. Y.,Unified Approach to Quadratically Convergent Algorithms for Function Minimization, Journal of Optimization Theory and Applications, Vol. 5, No. 6, 1970.

  2. Dixon, L. C. W.,Variable Metric Algorithms: Necessary and Sufficient Conditions for Identical Behaviour on Non-Quadratic Functions, The Hatfield Polytechnic, Numerical Optimization Centre, Technical Report No. 26, 1971.

  3. Fletcher, R.,A New Approach to Variable Metric Algorithms, Computer Journal, Vol. 13, No. 3, 1970.

  4. Davidon, W. C.,Variable Metric Method for Minimization, Atomic Energy Commission, Research and Development Report No. ANL-5990, 1959.

  5. Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 2, 1963.

  6. Broyden, C. G.,Quasi-Newton Methods and Their Application to Function Minimization, Mathematics of Computation, Vol. 21, No. 99, 1967.

  7. Murtagh, B. A., andSargent, R. H. W.,Computational Experience with Quadratically Convergent Minimization Methods, Computer Journal, Vol. 13, No. 2, 1970.

  8. Abbot, J.,Factors Affecting the Stability of Methods of the Davidon-Fletcher-Powell Type, Essex University, Computing Centre, MS Thesis, 1971.

  9. Shanno, D. F.,Conditioning of Quasi-Newton Methods for Function Minimization, Mathematics of Computation, Vol. 24, No. 111, 1970.

  10. Biggs, M. C.,A New Variable Metric Technique Taking Account of Nonquadratic Behaviour of the Objective Function, The Hatfield Polytechnic, Numerical Optimization Centre, Technical Report No. 17, 1971.

  11. Greenstadt, J.,Product Form for Variable Metric Corrections, IBM Working Paper, Palo Alto, California, 1971.

  12. Broyden, C. G.,The Convergence of a Class of Double-Rank Minimization Algorithms, Journal of the Institute of Mathematics and Its Applications, Vol. 6, No. 3, 1970.

  13. Shanno, D. F., andKettler, P. C.,Optimal Conditioning of Quasi-Newton Methods, Mathematics of Computation, Vol. 24, No. 111, 1970.

  14. Greenstadt, J.,Variations on Variable Metric Methods, Mathematics of Computation, Vol. 24, No. 109, 1970.

  15. Spedicato, E.,Conditioning in Huang's Two-Parameter Class for Conjugate Gradients, CISE Report No. N-154, Milano, Italy, 1972.

  16. Gourlay, A. R., Brodlie, K. W., andGreenstadt, J.,Product Form Corrections for Variable Metric Methods of Minimization, I, Report No. IBM UKSC-0010, Peterlee, England, 1972.

  17. Gourlay, A. R., Brodlie, K. W., andGreenstadt, J.,Product Form Corrections for Variable Metric Methods of Minimization, II, Report No. IBM UKSC-0033, Peterlee, England, 1972.

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Authors and Affiliations

  1. Computing Center, CISE, Segrate, Milano, Italy

    E. Spedicato (Research Associate)

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  1. E. Spedicato
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Communicated by A. Miele

The author is much indebted to Professor C. G. Broyden of the Essex University Computing Center, for valuable advice and criticism; he is also grateful to Drs. J. Greenstadt and A. R. Gourlay for having sent copies of their unpublished papers.

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Spedicato, E. Stability of Huang's update for the conjugate gradient method. J Optim Theory Appl 11, 469–479 (1973). https://doi.org/10.1007/BF00935660

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  • DOI: https://doi.org/10.1007/BF00935660

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