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2012.106: Coverage processes on spheres and condition numbers for linear programming

2012.106: Peter Burgisser, Felipe Cucker and Martin Lotz (2010) Coverage processes on spheres and condition numbers for linear programming. Annals of Probability, 38 (2). pp. 570-604.

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DOI: 10.1214/09-AOP489

Abstract

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,\alpha)$ be the probability that $n$ spherical caps of angular radius $\alpha$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,\alpha)$ in the case $\alpha\in[\pi/2,\pi]$ and an upper bound for $p(n,m,\alpha)$ in the case $\alpha\in [0,\pi/2]$ which tends to $p(n,m,\pi/2)$ when $\alpha\to\pi/2$. In the case $\alpha\in[0,\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius $\alpha$ that are needed to cover $S^m$. Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem $\exists x\in\mathbb{R}^{m+1}Ax\le0,x\ne0$ where $A\in\mathbb{R}^{n\times(m+1)}$ 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.

Item Type:Article
Subjects:MSC 2000 > 52 Convex and discrete geometry
MSC 2000 > 60 Probability theory and stochastic processes
MSC 2000 > 90 Operations research, mathematical programming
MIMS number:2012.106
Deposited By:Dr. Martin Lotz
Deposited On:23 October 2012

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