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      On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids

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      Author
      Todd, M. J.
      Yıldırım, E. A.
      Date
      2007
      Source Title
      Discrete Applied Mathematics
      Print ISSN
      0166-218X
      Electronic ISSN
      1872-6771
      Publisher
      Elsevier
      Volume
      155
      Issue
      13
      Pages
      1731 - 1744
      Language
      English
      Type
      Article
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      Abstract
      Given A {colon equals} { a1, ..., am } ⊂ Rd whose affine hull is Rd, we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum-volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yi{dotless}ldi{dotless}ri{dotless}m, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yi{dotless}ldi{dotless}ri{dotless}m or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.
      Keywords
      Löwner ellipsoid
      Core sets
      Rounding of polytopes
      Ellipsoid method
      Approximation algorithms
      Permalink
      http://hdl.handle.net/11693/23405
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/j.dam.2007.02.013
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