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Computing Minimum-Volume Enclosing Axis-Aligned Ellipsoids

Author

Listed:
  • P. Kumar

    (Florida State University)

  • E. A. Yıldırım

    (Bilkent University)

Abstract

Given a set of points $\mathcal{S}=\{x^{1},\ldots,x^{m}\}\subset \mathbb{R}^{n}$ and ε>0, we propose and analyze an algorithm for the problem of computing a (1+ε)-approximation to the minimum-volume axis-aligned ellipsoid enclosing $\mathcal{S}$ . We establish that our algorithm is polynomial for fixed ε. In addition, the algorithm returns a small core set $\mathcal{X}\subseteq \mathcal{S}$ , whose size is independent of the number of points m, with the property that the minimum-volume axis-aligned ellipsoid enclosing $\mathcal{X}$ is a good approximation of the minimum-volume axis-aligned ellipsoid enclosing $\mathcal{S}$ . Our computational results indicate that the algorithm exhibits significantly better performance than the theoretical worst-case complexity estimate.

Suggested Citation

  • P. Kumar & E. A. Yıldırım, 2008. "Computing Minimum-Volume Enclosing Axis-Aligned Ellipsoids," Journal of Optimization Theory and Applications, Springer, vol. 136(2), pages 211-228, February.
    • Handle: RePEc:spr:joptap:v:136:y:2008:i:2:d:10.1007_s10957-007-9295-9
    DOI: 10.1007/s10957-007-9295-9
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    References listed on IDEAS

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    1. Leonid G. Khachiyan, 1996. "Rounding of Polytopes in the Real Number Model of Computation," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 307-320, May.
    2. P. Kumar & E. A. Yildirim, 2005. "Minimum-Volume Enclosing Ellipsoids and Core Sets," Journal of Optimization Theory and Applications, Springer, vol. 126(1), pages 1-21, July.
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