On the elimination of inessential points in the smallest enclosing ball problem

We consider the construction of the smallest ball enclosing a set formed by n points in . We show that any probability measure on , with mean c and variance matrix V, provides a lower bound b on the distance to c of any point on the boundary of , with b having a simple expression in terms of c and V. This inequality permits to remove inessential points from , which do not participate to the definition of , and can be used to accelerate algorithms for the construction of . We show that this inequality is, in some sense, the best possible. A series of numerical examples indicates that, when d is reasonably small (, say) and n is large (up to ), the elimination of inessential points by a suitable two-point measure, followed by a direct (exact) solution by quadratic programming, outperforms iterative methods that compute an approximate solution by solving the dual problem.

[1]  Kenneth L. Clarkson,et al.  Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm , 2008, SODA '08.

[2]  Selin Damla Ahipasaoglu,et al.  Identification and Elimination of Interior Points for the Minimum Enclosing Ball Problem , 2008, SIAM J. Optim..

[3]  D. Titterington.  Estimation of Correlation Coefficients by Ellipsoidal Trimming , 1978 .

[4]  W. J. Studden,et al.  Theory Of Optimal Experiments , 1972 .

[5]  L..  A delimitation of the support of optimal designs for Kiefer ’ s φ p-class of criteria , 2013 .

[6]  L. Pronzato.  Minimax and maximin space-filling designs: some properties and methods for construction , 2017 .

[7]  Luc Pronzato,et al.  Improvements on removing nonoptimal support points in D-optimum design algorithms , 2007, 0706.4394.

[8]  Donald Goldfarb,et al.  A numerically stable dual method for solving strictly convex quadratic programs , 1983, Math. Program..

[9]  Michael J. Todd,et al.  Minimum volume ellipsoids - theory and algorithms , 2016, MOS-SIAM Series on Optimization.

[10]  Changbao Wu,et al.  Some Algorithmic Aspects of the Theory of Optimal Designs , 1978 .

[11]  E. Alper Yildirim,et al.  Two Algorithms for the Minimum Enclosing Ball Problem , 2008, SIAM J. Optim..

[12]  Bernd Gärtner,et al.  Fast and Robust Smallest Enclosing Balls , 1999, ESA.

[13]  L. Pronzato.  DISCUSSION OF THE PAPER > BY H. P. WYNN , 1993 .

[14]  Emo Welzl,et al.  Smallest enclosing disks (balls and ellipsoids) , 1991, New Results and New Trends in Computer Science.

[15]  Kenneth L. Clarkson,et al.  Optimal core-sets for balls , 2008, Comput. Geom..

[16]  C. Atwood.  Sequences Converging to $D$-Optimal Designs of Experiments , 1973 .

[17]  Peng Sun,et al.  Linear convergence of a modified Frank–Wolfe algorithm for computing minimum-volume enclosing ellipsoids , 2008, Optim. Methods Softw..

[18]  Anthony C. Atkinson,et al.  Optimal Experimental Design , 2015 .

[19]  Jesús López-Fidalgo,et al.  Design of experiments for nonlinear models , 2011 .

[20]  A. Vacavant,et al.  Reconstructions of Noisy Digital Contours with Maximal Primitives Based on Multi-Scale/Irregular Geometric Representation and Generalized Linear Programming , 2017 .

[21]  Andrej Pázman,et al.  Foundations of Optimum Experimental Design , 1986 .

[22]  J. Kiefer,et al.  The Equivalence of Two Extremum Problems , 1960, Canadian Journal of Mathematics.

[23]  Byran J. Smucker,et al.  Optimal experimental design , 2018, Nature Methods.

[24]  Yaming Yu.  Monotonic convergence of a general algorithm for computing optimal designs , 2009, 0905.2646.

[25]  D. Hearn,et al.  The Minimum Covering Sphere Problem , 1972 .

[26]  V. L. Turova-Botkina,et al.  An algorithm for finding the Chebyshev center of a convex polyhedron , 1994, System Modelling and Optimization.

[27]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[28]  Luc Pronzato,et al.  A delimitation of the support of optimal designs for Kiefer’s ϕp-class of criteria , 2013, 1303.5046.

[29]  D. Titterington.  Optimal design: Some geometrical aspects of D-optimality , 1975 .

[30]  L. Pronzato,et al.  Design of Experiments in Nonlinear Models: Asymptotic Normality, Optimality Criteria and Small-Sample Properties , 2013 .

[31]  S. Silvey,et al.  A geometric approach to optimal design theory , 1973 .

[32]  Michael J. Todd,et al.  On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids , 2007, Discret. Appl. Math..