Ellipsoid Algorithms

  • Manfred Padberg
Part of the Algorithms and Combinatorics book series (AC, volume 12)

Abstract

Gaius Julius Caesar (101–44 B.C.) made military use of it when his legions conquered Gaulle, Machiavelli coined it as a political doctrine for Florentine princes to improve his princes’ control over their subjects, and mathematicians employ it e.g. when they use binary search to locate the optimum of a p-unimodal function over a compact convex subset of ℝ1. Divide and conquer, or more precisely divide and reign, is what Niccolo’s doctrine translates to. The compact convex subset of ℝ1 is, of course, some finite interval of the real line and the basic idea of binary search consists in successively halving this interval — like we did in Chapter 7.5.3. Utilizing the p-unimodality (see below) of the function to be optimized, we then discard one half of the original interval forever and continue the search in the other half of the original interval — which is again a compact convex subset of ℝ1. Consequently, the basic idea can be reapplied and we can iterate. Since the length of the left-over interval is half of the length of the previous interval, the 1-dimensional “volume” of the remaining compact convex subset of ℝ1 to be searched shrinks at a geometric rate and we obtain fast convergence of the iterative scheme.

Keywords

Hull Hunt Lution Marquis 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Manfred Padberg
    • 1
  1. 1.Leonard N. Stern School of Business, Statistics and Operations Research DepartmentNew York UniversityNew York CityUSA

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