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.
Similar content being viewed by others
References
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)
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)
Bouville, C.: Bounding ellipsoid for ray-fractal intersection. Comput. Graph. (ACM) 19, 45–52 (1985)
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)
Cong, W.-j., Liu, H.-w.: Modified algorithms for the minimum volume enclosing axis-aligned ellipsoid problem. Discrete Appl. Math. 158, 627–635 (2010)
Dyer, M., Frieze, A., Kannan, R.: Random polynomial-time algorithm for approximating the volume of convex bodies. J. ACM 38, 1–17 (1991)
Glineur, F.: Pattern separation via ellipsoids and conic programming. Master’s thesis, Faculté Polytechnique de Mons, Belgium (1998)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1988)
Khachiyan, L.G.: Rounding of polytopes in the real number model of computation. Math. Oper. Res. 21, 307–320 (1996)
Khachiyan, L.G., Todd, M.: On the complexity of approximating the maximal inscribed ellipsoid for a polytope. Math. Program. 61, 137–159 (1993)
Kumar, P., Yıldırım, E.A.: Minimum volume enclosing ellipsoids and core sets. J. Optim. Theory Appl. 126, 1–21 (2005)
Lenstra, A.K.: Factoring multivariate integral polynomials. Theor. Comput. Sci. 34, 207–213 (1983)
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)
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)
Richtárik, P.: Improved algorithms for convex minimization in relative scale. http://www.optimization-online.org/DB_FILE/2009/02/2226.pdf (2009)
Silverman, B.W., Titterington, D.M.: Minimum covering ellipses. SIAM J. Sci. Stat. Comput. 1, 401–409 (1980)
Silvey, S.D., Titterington, D.M.: A geometric approach to optimal design theory. Biometrika 62, 21–32 (1973)
Sun, P., Freund, R.M.: Computation of minimum volume covering ellipsoids. Oper. Res. 52, 690–706 (2004)
Titterington, D.M.: Optimal design: some geometrical aspects of D-optimality. Biometrika 62, 313–320 (1975)
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)
Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Fundamental Research Funds for the central universities (JY10000970004).
Rights 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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-010-9342-6