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Computation of Minimum-Volume Covering Ellipsoids

Author

Listed:
  • Peng Sun

    (The Fuqua School of Business, Duke University, Box 90120, Durham, North Carolina 27708)

  • Robert M. Freund

    (Sloan School of Management, Massachusetts Institute of Technology, 50 Memorial Drive, Cambridge, Massachusetts 02142)

Abstract

We present a practical algorithm for computing the minimum-volume n -dimensional ellipsoid that must contain m given points a 1 ,…, a m ∈ ℝ n . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances ( m = 30,000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.

Suggested Citation

  • Peng Sun & Robert M. Freund, 2004. "Computation of Minimum-Volume Covering Ellipsoids," Operations Research, INFORMS, vol. 52(5), pages 690-706, October.
    • Handle: RePEc:inm:oropre:v:52:y:2004:i:5:p:690-706
    DOI: 10.1287/opre.1040.0115
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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.
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    Cited by:

    1. Karim Abou-Moustafa & Frank P. Ferrie, 2018. "Local generalized quadratic distance metrics: application to the k-nearest neighbors classifier," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(2), pages 341-363, June.
    2. Wei-jie Cong & Le Wang & Hui Sun, 2020. "Rank-two update algorithm versus Frank–Wolfe algorithm with away steps for the weighted Euclidean one-center problem," Computational Optimization and Applications, Springer, vol. 75(1), pages 237-262, January.
    3. A. Astorino & M. Gaudioso & W. Khalaf, 2014. "Edge detection by spherical separation," Computational Management Science, Springer, vol. 11(4), pages 517-530, October.
    4. Hlubinka, Daniel & Šiman, Miroslav, 2013. "On elliptical quantiles in the quantile regression setup," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 163-171.
    5. Rosa, Samuel & Harman, Radoslav, 2022. "Computing minimum-volume enclosing ellipsoids for large datasets," Computational Statistics & Data Analysis, Elsevier, vol. 171(C).
    6. Wei-jie Cong & Hong-wei Liu & Feng Ye & Shui-sheng Zhou, 2012. "Rank-two update algorithms for the minimum volume enclosing ellipsoid problem," Computational Optimization and Applications, Springer, vol. 51(1), pages 241-257, January.
    7. E. Hendrix & I. García & J. Plaza & A. Plaza, 2013. "On the minimum volume simplex enclosure problem for estimating a linear mixing model," Journal of Global Optimization, Springer, vol. 56(3), pages 957-970, July.
    8. Alberto Seeger & José Vidal -Nuñez, 2017. "Measuring centrality and dispersion in directional datasets: the ellipsoidal cone covering approach," Journal of Global Optimization, Springer, vol. 68(2), pages 279-306, June.
    9. Efsun Kürüm & Gerhard-Wilhelm Weber & Cem Iyigun, 2018. "Early warning on stock market bubbles via methods of optimization, clustering and inverse problems," Annals of Operations Research, Springer, vol. 260(1), pages 293-320, January.

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