Coverage processes on spheres and condition numbers for linear programming

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

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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 2010, the AMS's Mathematics Subject Classification > 52 Convex and discrete geometry
MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
MSC 2010, the AMS's Mathematics Subject Classification > 90 Operations research, mathematical programming
Depositing User: Dr. Martin Lotz
Date Deposited: 23 Oct 2012
Last Modified: 20 Oct 2017 14:13
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1902

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