Annals of Probability

Coverage processes on spheres and condition numbers for linear programming

Peter Bürgisser, Felipe Cucker, and Martin Lotz

Full-text: Open access

Abstract

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let p(n, m, α) be the probability that n spherical caps of angular radius α in Sm do not cover the whole sphere Sm. We give an exact formula for p(n, m, α) in the case α∈[π/2, π] and an upper bound for p(n, m, α) in the case α∈[0, π/2] which tends to p(n, m, π/2) when απ/2. In the case α∈[0, π/2] this yields upper bounds for the expected number of spherical caps of radius α that are needed to cover Sm.

Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem ∃x∈ℝm+1Ax≤0, x≠0 where A∈ℝn×(m+1) is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to A being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all n>m, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.

Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 570-604.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143527

Digital Object Identifier
doi:10.1214/09-AOP489

Mathematical Reviews number (MathSciNet)
MR2642886

Zentralblatt MATH identifier
1205.60027

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 90C05: Linear programming

Keywords
Condition numbers covering processes geometric probability integral geometry linear programming

Citation

Bürgisser, Peter; Cucker, Felipe; Lotz, Martin. Coverage processes on spheres and condition numbers for linear programming. Ann. Probab. 38 (2010), no. 2, 570--604. doi:10.1214/09-AOP489. https://projecteuclid.org/euclid.aop/1268143527


Export citation

References

  • [1] Agmon, S. (1954). The relaxation method for linear inequalities. Canad. J. Math. 6 382–392.
  • [2] Bürgisser, P. and Amelunxen, D. (2008). Uniform smoothed analysis of a condition number for linear programming. Accepted for Math. Program. A. Available at arXiv:0803.0925.
  • [3] Cheung, D. and Cucker, F. (2001). A new condition number for linear programming. Math. Program. 91 163–174.
  • [4] Cheung, D. and Cucker, F. (2002). Probabilistic analysis of condition numbers for linear programming. J. Optim. Theory Appl. 114 55–67.
  • [5] Cheung, D., Cucker, F. and Hauser, R. (2005). Tail decay and moment estimates of a condition number for random linear conic systems. SIAM J. Optim. 15 1237–1261.
  • [6] Cucker, F. and Peña, J. (2002). A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine. SIAM J. Optim. 12 522–554.
  • [7] Cucker, F. and Wschebor, M. (2002). On the expected condition number of linear programming problems. Numer. Math. 94 419–478.
  • [8] Dunagan, J., Spielman, D. A. and Teng, S.-H. (2009). Smoothed analysis of condition numbers and complexity implications for linear programming. Math. Programming. To appear. Available at http://arxiv.org/abs/cs/0302011v2.
  • [9] Dvoretzky, A. (1956). On covering a circle by randomly placed arcs. Proc. Natl. Acad. Sci. USA 42 199–203.
  • [10] Gilbert, E. N. (1965). The probability of covering a sphere with N circular caps. Biometrika 52 323–330.
  • [11] Goffin, J.-L. (1980). The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5 388–414.
  • [12] Hall, P. (1985). On the coverage of k-dimensional space by k-dimensional spheres. Ann. Probab. 13 991–1002.
  • [13] Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
  • [14] Hauser, R. and Müller, T. (2009). Conditioning of random conic systems under a general family of input distributions. Found. Comput. Math. 9 335–358.
  • [15] Janson, S. (1986). Random coverings in several dimensions. Acta Math. 156 83–118.
  • [16] Kahane, J.-P. (1959). Sur le recouvrement d’un cercle par des arcs disposés au hasard. C. R. Math. Acad. Sci. Paris 248 184–186.
  • [17] Miles, R. E. (1968). Random caps on a sphere. Ann. Math. Statist. 39 1371.
  • [18] Miles, R. E. (1969). The asymptotic values of certain coverage probabilities. Biometrika 56 661–680.
  • [19] Miles, R. E. (1971). Isotropic random simplices. Adv. in Appl. Probab. 3 353–382.
  • [20] Moran, P. A. P. and Fazekas de St. Groth, S. (1962). Random circles on a sphere. Biometrika 49 389–396.
  • [21] Motzkin, T. S. and Schoenberg, I. J. (1954). The relaxation method for linear inequalities. Canad. J. Math. 6 393–404.
  • [22] Reitzner, M. (2002). Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc. 354 2243–2278.
  • [23] Rosenblatt, F. (1962). Principles of Neurodynamics. Perceptrons and the Theory of Brain Mechanisms. Spartan Books, Washington, DC.
  • [24] Santaló, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.
  • [25] Siegel, A. F. (1979). Asymptotic coverage distributions on the circle. Ann. Probab. 7 651–661.
  • [26] Siegel, A. F. and Holst, L. (1982). Covering the circle with random arcs of random sizes. J. Appl. Probab. 19 373–381.
  • [27] Solomon, H. (1978). Geometric Probability. SIAM, Philadelphia, PA.
  • [28] Stevens, W. L. (1939). Solution to a geometrical problem in probability. Ann. Eugenics 9 315–320.
  • [29] Wendel, J. G. (1962). A problem in geometric probability. Math. Scand. 11 109–111.
  • [30] Whitworth, W. A. (1965). DCC Exercises in Choice and Chance. Dover, New York.
  • [31] Zähle, M. (1990). A kinematic formula and moment measures of random sets. Math. Nachr. 149 325–340.