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Study on a memory gradient method for the minimization of functions

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Abstract

A new accelerated gradient method for finding the minimum of a functionf(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 vectorx, g(x) is the gradient of the functionf(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. For a test problem, the number of iterations was about 40–50% that of the Fletcher-Reeves method and the computing time about 60–75% that of the Fletcher-Reeves method, using comparable search techniques.

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References

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

  2. Fletcher, R., andReeves, C. M.,Function Minimization by Conjugate Gradients, Computer Journal, Vol. 7, No. 2, 1964.

  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 Variable Metric Methods of Minimization, Research Analysis Corporation, Technical Paper No. RAC-TP-302, 1968.

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

  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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Author notes

  1. J. W. Cantrell (Graduate Student in Aero-Astronautics)

    Present address: LTV Aerospace Corporation, Dallas, Texas

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering and Materials Science, Rice University, Houston, Texas

    A. Miele (Professor of Astronautics and Director of the Aero-Astronautics Group) & J. W. Cantrell (Graduate Student in Aero-Astronautics)

Authors
  1. A. Miele
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  2. J. W. Cantrell
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Additional information

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. Portions of this paper were presented by the senior author at the International Symposium on Optimization Methods, Nice, France, 1969.

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Miele, A., Cantrell, J.W. Study on a memory gradient method for the minimization of functions. J Optim Theory Appl 3, 459–470 (1969). https://doi.org/10.1007/BF00929359

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  • DOI: https://doi.org/10.1007/BF00929359

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