Skip to main content
SpringerLink
Log in

Rank-two update algorithms for the minimum volume enclosing ellipsoid problem

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We consider the problem of computing a (1+ε)-approximation to the minimum volume enclosing ellipsoid (MVEE) of a given set of m points in R n. Based on the idea of sequential minimal optimization (SMO) method, we develop a rank-two update algorithm. This algorithm computes an approximate solution to the dual optimization formulation of the MVEE problem, which updates only two weights of the dual variable at each iteration. We establish that this algorithm computes a (1+ε)-approximation to MVEE in O(mn 3/ε) operations and returns a core set of size O(n 2/ε) for ε∈(0,1). In addition, we give an extension of this rank-two update algorithm. Computational experiments show the proposed algorithms are very efficient for solving large-scale problem with a high accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahipasaoǧlu, S.D., Sun, P., Todd, M.J.: Linear convergence of a modified Frank-Wolfe algorithm for computing minimum volume enclosing ellipsoids. Optim. Methods Softw. 23, 5–19 (2008)

    Article  MathSciNet  Google Scholar 

  2. Barequet, G., Har-Peled, S.: Efficiently approximating the minimum volume bounding box of a point set in three dimensions. J. Algorithms 38, 91–109 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bouville, C.: Bounding ellipsoid for ray-fractal intersection. Comput. Graph. (ACM) 19, 45–52 (1985)

    Article  Google Scholar 

  4. Chen, P.-H., Fan, R.-E., Lin, C.-J.: A study on SMO-type decomposition methods for support vector machines. IEEE Trans. Neural Netw. 17, 893–908 (2006)

    Article  Google Scholar 

  5. Cong, W.-j., Liu, H.-w.: Modified algorithms for the minimum volume enclosing axis-aligned ellipsoid problem. Discrete Appl. Math. 158, 627–635 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dyer, M., Frieze, A., Kannan, R.: Random polynomial-time algorithm for approximating the volume of convex bodies. J. ACM 38, 1–17 (1991)

    MATH  MathSciNet  Google Scholar 

  7. Glineur, F.: Pattern separation via ellipsoids and conic programming. Master’s thesis, Faculté Polytechnique de Mons, Belgium (1998)

  8. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1988)

    MATH  Google Scholar 

  9. Khachiyan, L.G.: Rounding of polytopes in the real number model of computation. Math. Oper. Res. 21, 307–320 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Khachiyan, L.G., Todd, M.: On the complexity of approximating the maximal inscribed ellipsoid for a polytope. Math. Program. 61, 137–159 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kumar, P., Yıldırım, E.A.: Minimum volume enclosing ellipsoids and core sets. J. Optim. Theory Appl. 126, 1–21 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lenstra, A.K.: Factoring multivariate integral polynomials. Theor. Comput. Sci. 34, 207–213 (1983)

    Article  MathSciNet  Google Scholar 

  13. Nesterov, Yu.E.: Rounding of convex sets and efficient gradient methods for linear programming problems. Discussion Paper 2004-4, CORE, Catholic University of Louvain, Louvain-la-Neuve, Belgium (2004)

  14. Platt, J.: Fast training of support vector machines using sequential minimal optimization. In: Schölkopf, B., Burges, C.J.C., Smola, A.J. (eds.) Kernel Methods-Support Vector Learning, pp. 185–208. MIT Press, Cambridge (1999)

    Google Scholar 

  15. Richtárik, P.: Improved algorithms for convex minimization in relative scale. http://www.optimization-online.org/DB_FILE/2009/02/2226.pdf (2009)

  16. Silverman, B.W., Titterington, D.M.: Minimum covering ellipses. SIAM J. Sci. Stat. Comput. 1, 401–409 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Silvey, S.D., Titterington, D.M.: A geometric approach to optimal design theory. Biometrika 62, 21–32 (1973)

    Article  MathSciNet  Google Scholar 

  18. Sun, P., Freund, R.M.: Computation of minimum volume covering ellipsoids. Oper. Res. 52, 690–706 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Titterington, D.M.: Optimal design: some geometrical aspects of D-optimality. Biometrika 62, 313–320 (1975)

    MATH  MathSciNet  Google Scholar 

  20. Todd, M.J., Yıldırım, E.A.: On Khachiyan’s algorithm for the computation of minimum-volume enclosing ellipsoids. Discrete Appl. Math. 155, 1731–1744 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998)

    MATH  Google Scholar 

  22. Welzl, E.: Smallest enclosing disks (balls and ellipsoids). In: Proceedings of New Results and New Trends in Computer Science. LNCS, vol. 555, pp. 359–370. Springer, Berlin (1991)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Xidian University, Xi’an, 710071, P.R. China

    Wei-jie Cong, Hong-wei Liu, Feng Ye & Shui-sheng Zhou

Authors
  1. Wei-jie Cong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong-wei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shui-sheng Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-jie Cong.

Additional information

This work was supported by the Fundamental Research Funds for the central universities (JY10000970004).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cong, Wj., Liu, Hw., Ye, F. et al. Rank-two update algorithms for the minimum volume enclosing ellipsoid problem. Comput Optim Appl 51, 241–257 (2012). https://doi.org/10.1007/s10589-010-9342-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-010-9342-6

Keywords

Search

Navigation