Condition pp 147-154 | Cite as

The Ellipsoid Method

  • Peter Bürgisser
  • Felipe Cucker
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 349)

Abstract

In this chapter we describe an algorithm, known as the ellipsoid method, that finds a solution to a given system of linear inequalities. Its complexity analysis can also be done in terms of Open image in new window , but in exchange for a loss of simplicity, we obtain bounds linear in Open image in new window (instead of the quadratic dependence in Open image in new window of the perceptron algorithm).

We also introduce in this chapter, in its last section, a new theme: the use of condition numbers in the analysis of algorithms taking integer (as opposed to real) data. We show that if the entries of \(A\in \mathcal {F}_{D}^{\circ}\) are integer numbers, then one can return a solution with a cost—and since all our data are discrete, we mean bit cost—polynomial in n, m and the bit-size of the largest entry in A.

Bibliography

  1. 114.
    M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics: Study and Research Texts. Springer, Berlin, 1988. Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Felipe Cucker
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsCity University of Hong KongHong KongHong Kong SAR

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