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Method of ellipsoids, its generalizations and applications

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Cybernetics Aims and scope

Conclusion

After the burst of enthusiasm caused by the Khachiyan result [5], the methods of ellipsoids turned out to be the center of attention of many optimizers. After some time, only their theoretical value began to be achnowledged, raising doubts about their practical efficiency. And this is valid to an extent if we speak of the MCGM method and its negligible modifications.

In this paper the authors have attempted to show that ellipsoid methods are just a very particular case of the families of gradient type algorithms using a space stretching operation. Other representatives of this family, as r-algorithms, say, are an effective practical means for solving many complex mathematical programming problems reducing to nondifferentiable optimization. The theory of a whole class of algorithms with space stretching is still far from completion. It seems to us a sufficiently realistic aim to construct, such an algorithm, which would be no less practically efficient than the r-algorithm and would have as good a foundation as the method of ellipsoids.

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Authors
  1. V. I. Gershovich
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  2. N. Z. Shor
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Additional information

Translated from Kibernetika, No. 5, pp. 61–69, September–October, 1982.

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Gershovich, V.I., Shor, N.Z. Method of ellipsoids, its generalizations and applications. Cybern Syst Anal 18, 606–617 (1982). https://doi.org/10.1007/BF01068741

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