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Memory gradient method for the minimization of functions

Part of the Lecture Notes in Mathematics book series (LNM,volume 132)

Abstract

A new accelerated gradient method for finding the minimum of a function f(x) whose variables are unconstrained is investigated. The new algorithm can be stated as follows:

$$\tilde x = x + \delta x,\delta x = - \alpha g(x) + \beta \delta \hat x$$

where δx is the change in the position vector x, (g(x) is the gradient of the function f(x), and α and β are scalars chosen at each step so as to yield the greatest decrease in the function. The symbol \(\delta \hat x\) denotes the change in the position vector for the iteration preceding that under consideration.

For a nonquadratic function, initial convergence of the present method is faster than that of the Fletcher-Reeves method because of the extra degree of freedom available. Three test problems are considered. A comparison is made between the ordinary gradient method, the Fletcher-Reeves method, and the memory gradient method.

This research, supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF- AFOSR-828-67, is a condensed version of the investigation described in Ref. 1.

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References

  1. MIELE, A., and CANTRELL, J.W., Gradient Methods in Mathematical Programming, Part 2, Memory Gradient Method, Rice University, Aero-Astronautics Report No. 56, 1969.

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  2. FLETCHER, R., and REEVES, C.M., Function Minimization by Conjugate Gradients, Computer Journal, Vol. 7, No. 2, 1964.

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  3. BECKMAN, F.S., The Solution of Linear Equations by the Conjugate Gradient Method, Mathematical Methods for Digital COmputers, Edited by A. Ralston and H.S. Wilf, John WIley and Sons, New York, 1960.

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  4. PEARSON, J.D., On Vriable Metric Methods of Minimization, Research Analysis Corporation, Technical Paper No. RAC-TP-302, 1968.

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  5. CANTRELL, J.W., Method of Independent Multipliers for Minimizing Unconstrained Functions, Rice University, M.S. Thesis, 1969.

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  6. CANTRELL, J.W., On the Relation between the Memory Gradient Method and the Fletcher-Reeves Method, Journal of Optimization Theory and Applications, Vol. 4, No. 1, 1969.

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© 1970 Springer-Verlag

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Miele, A., Cantrell, J.W. (1970). Memory gradient method for the minimization of functions. In: Balakrishnan, A.V., Contensou, M., de Veubeke, B.F., Krée, P., Lions, J.L., Moiseev, N.N. (eds) Symposium on Optimization. Lecture Notes in Mathematics, vol 132. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066685

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  • DOI: https://doi.org/10.1007/BFb0066685

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-04921-0

  • Online ISBN: 978-3-540-36275-3

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