On tail decay and moment estimates of a condition number for random linear conic systems

Abstract

In this paper we study the distribution tails and the moments of a condition number which arises in the study of homogeneous systems of linear inequalities. We consider the case where this system is defined by a Gaussian random matrix and characterise the exact decay rates of the distribution tails, improve the existing moment estimates, and prove various limit theorems for large scale systems. Our results are of complexity theoretic interest, because interior-point methods and relaxation methods for the solution of systems of linear inequalities have running times that are bounded in terms of the logarithm and the square of the condition number respectively.\ud \ud Felipe Cucker has been substantially funded by a grant from the Research Grants Council of the Hong Kong SAR (project number CityU 1085/02P). Raphael Hauser has been supported by Felipe Cucker's grant from the Research Grants Council of the Hong Kong SAR (project number CityU 1085/02P) and through a grant of the Nuffield Foundation under the "Newly Appointed Lecturers" grant scheme, (project number NAL/00720/G)