| Zusammenfassung |
We perform a smoothed analysis of the GCC-condition number $C(A)$ of the linear programming feasibility problem $\exists x\in\mathbb R^m+1 Ax < 0$. Suppose that $\barA$ is any matrix with rows $\bara_i$ of euclidean norm $1$ and, independently for all $i$, let $a_i$ be a random perturbation of $\bara_i$ following the uniform distribution in the spherical disk in $S^m$ of angular radius $\arcsin\sigma$ and centered at $\bara_i$. We prove that $E(łn C(A)) = O(mn / \sigma)$. A similar result was shown for Renegar's condition number and Gaussian perturbations by Dunagan, Spielman, and Teng [arXiv:cs.DS/0302011]. Our result is robust in the sense that it easily extends to radially symmetric probability distributions supported on a spherical disk of radius $\arcsin\sigma$, whose density may even have a singularity at the center of the perturbation. Our proofs combine ideas from a recent paper of Bürgisser, Cucker, and Lotz (Math. Comp. 77, No. 263, 2008) with techniques of Dunagan et al. |
