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Optimization

  • Stan Wagon

Abstract

The top image is the machine-precision trajectory of 0.1 under the quadratic map 4x (1 − x). It suffers from roundoff error and the terms beyond the 60th are not the same as the values in the true trajectory of \(\frac{1}{{10}}\) shown just below it. But it turns out that the noisy trajectory can be shadowed, meaning that there is a value near \(\frac{1}{{10}}\)— it turns out to be 0.09999999999999998884314845320503 — whose true trajectory under f matches the noisy trajectory very closely (to within 10−15). Finding this value is a true needle-in-a-gigantic-haystack problem, but sophisticated optimization algorithms are capable of getting it in under one second, and with only a few lines of code. The third image shows the absolute difference between the noisy trajectory and true trajectories in the vicinity of the shadow value; this gives an indication of the difficulty of finding this value, since the spike that defines it is very narrow.

Keywords

Differential Evolution Integer Linear Program Linear Programming Problem Stable Marriage Roundoff Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Stan Wagon
    • 1
  1. 1.Department of Mathematics and Computer ScienceMacalester CollegeSt. PaulUSA

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