Uniform Smoothed Analysis of a Condition Number for Linear Programming

Bürgisser, Cucker, and Lotz [arxiv:math.NA/0610270] proved a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Applications to linear and polynomial equation solving were given. We show that a suitable modification of the general theorem in that paper, adapted to a spherical convex setting, allows to analyze condition numbers of convex optimization. More specifically, we perform a smoothed analysis of the condition number of the linear programming feasibility problem. Some of our techniques heavily rely on ideas developed by Dunagan, Spielman, and Teng [arXiv:cs.DS/0302011]. AMS subject classifications: 90C05, 90C31, 52A22, 60D05