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Variable metric relaxation methods, part II: The ellipsoid method

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Abstract

The deepest, or least shallow, cut ellipsoid method is a polynomial (time and space) method which finds an ellipsoid, representable by polynomial space integers, such that the maximal ellipsoidal distance relaxation method using this fixed ellipsoid is polynomial: this is equivalent to finding a linear transforming such that the maximal distance relaxation method of Agmon, Motzkin and Schoenberg in this transformed space is polynomial. If perfect arithmetic is used, then the sequence of ellipsoids generated by the method converges to a set of ellipsoids, which share some of the properties of the classical Hessian at an optimum point of a function; and thus the ellipsoid method is quite analogous to a variable metric quasi-Newton method.

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Authors and Affiliations

  1. Faculty of Management, McGill University, 1001 Sherbrooke St. W., H3A 1G5, Montreal, Quebec, Canada

    Jean-Louis Goffin

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  1. Jean-Louis Goffin
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This research was supported in part by the F.C.A.C. of Quebec, and the N.S.E.R.C. of Canada under Grant A 4152.

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Goffin, JL. Variable metric relaxation methods, part II: The ellipsoid method. Mathematical Programming 30, 147–162 (1984). https://doi.org/10.1007/BF02591882

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  • DOI: https://doi.org/10.1007/BF02591882

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