Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems

Abstract

Abstract. In this paper we study the distribution of C (A) and log C (A), where C (A) is a condition number for the linear conic system Ax ≤ 0, x = 0, with A ∈ R n×m . For Gaussian matrices A we develop both upper and lower bounds on the decay rates of the distribution tails of C (A), showing that P [C (A) ≥ t] ∼ c/t for large t, where c is a factor that depends only on the problem dimensions (m, n). Using these bounds, we derive moment estimates for C (A) and log C (A) and prove various limit theorems for the cases where m and/or n are large. Combined with condition number based complexity analyses, our results yield tail information on the distribution of running times for interior-point or relaxation methods designed to solve the feasibility problem Ax ≤ 0, x = 0.

Conditioning of Random Conic Systems Under a General Family of Input Distributions

“…Having the application of the above theory to random input ma- Note that in this case Theorem 3.1 below shows that P[C G (A) > t] =Õ(t −1 ). As mentioned earlier, special cases of this result were already established in [6,8,9].…”

“…Hence, the new terminology of smoothed analysis has been introduced for studies of this kind, a framework that has also been applied to the complexity analysis of the simplex and perceptron algorithms (see Spielman-Teng [21] and Blum-Dunagan [2], respectively), as well as in other contexts. We also remark that since C G (A) is independent of row scaling, the framework of study of [9] becomes the same as [6,8] when σ 2 → ∞.…”

“…In the mentioned papers [8,9] and [6], the distributions of the random matrix A have smoothness parameter 1, so that the tails of C(A) decay likeÕ(t −1 ). In fact, all of these distributions fall under the framework of Example 1 of Sect.…”

“…For instance, in the above mentioned papers [6,8,9], it is shown that under the probability models considered, the expected running time of interior point and ellipsoid methods is bounded by a polynomial in the problem dimensions (n, m). Such results are referred to as strong polynomiality on average.…”

Smoothed analysis of condition numbers and complexity implications for linear programming

“…These algorithms include the ellipsoid algorithm [Kha79,FV00], von Neumann's algorithm [EF00], and the recent perceptron algorithm with rescaling [DV04]. [CC02], and Cheung, Cucker, and Hauser [CCH03] studied the distribution of condition numbers of random linear programs drawn from various distributions, and their bounds also imply that the average-case complexity of the interior-point method is O(n 3 log n). If one specializes our results to perturbations of the all-zero matrixĀ and the all-zero vectorsb andc, then one obtains a similar average-case analysis of the distribution of condition numbers under the Gaussian distribution.…”