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

Abstract

We consider the conic feasibility problem associated with the linear homogeneous system Ax ≤ 0, x = 0. The complexity of iterative algorithms for solving this problem depends on a condition number C(A). When studying the typical behavior of algorithms under stochastic input, one is therefore naturally led to investigate the fatness of the tails of the distribution of C(A). Introducing the very general class of uniformly absolutely continuous probability models for the random matrix A, we show that the distribution tails of C(A) decrease at algebraic rates, both for the Goffin-Cheung-Cucker number C G and the Renegar number C R . The exponent that drives the decay arises naturally in the theory of uniform absolute continuity, which we also develop in this paper. In the case of C G , we also discuss lower bounds on 336 Found Comput Math (2009) 9: 335-358 the tail probabilities and show that there exist absolutely continuous input models for which the tail decay is subalgebraic.

Coverage processes on spheres and condition numbers for linear programming

“…A bound for E(ln C (A)) of the form O(min{n, m ln n}) was shown in [4]. This bound was improved [7] to max{ln m, ln ln n} + O(1) assuming that n is moderately larger than m. Still, in [5], the asymptotic behavior of both C (A) and ln C (A) was exhaustively studied, and these results were extended in [14] to matrices A ∈ (S m ) n drawn from distributions more general than the uniform. Independently of this stream of results, in [8], a smoothed analysis for a related condition number is performed from which it follows that E(ln C (A)) = O(ln n).…”

“…Differentiating te = Ap we get e dt = dAp + A dp. By multiplying both sides with A −1 and using formula (14) we obtain p(dt/t) − dp = A −1 dAp.…”

Robust smoothed analysis of a condition number for linear programming

“…An advantage of the uniform model is that, for conic condition numbers, results on smoothed analysis easily extend to more general families of probability distributions. To obtain such robustness results we rely on a general boosting technique developed in Hauser and Müller [18] and apply it similarly as in Cucker et al [11] The framework is the following (cf. §2.4).…”

“…A bound for E(ln C (A)) of the form O(min{n, m ln n}) was shown in [9]. This bound was improved in [13] to max{ln m, ln ln n} + O(1) assuming that n is moderately larger than m. Still, in [10], the asymptotic behavior of both C (A) and ln C (A) was exhaustively studied and these results were extended in [18] to matrices A ∈ (S m ) n drawn from distributions more general than the uniform. Finally, in [5], the exact distribution of C (A) conditioned to A being feasible was found and asymptotically sharp tail bounds for the infeasible case were given.…”

A Condition Number for Multifold Conic Systems

“…Such condition number is close in spirit to C(A) but is invariant under row scaling and is defined in terms of a best conditioned solution to (6). This condition number can also be used in the analysis of algorithms (e.g., the analysis in [11] carries over to C (A)) and has also been studied as a random variable [8,12,20].…”