A new condition number for linear programming

Cheung, Dennis; Cucker, Felipe

doi:10.1007/s101070100237

Cited by 52 publications

(56 citation statements)

References 6 publications

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“…Downloaded 05/11/18 to 18.236.120. 13. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php See also below for bounds obtained in [20].…”

Section: Introduction Let a ∈ Rmentioning

confidence: 99%

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

Cheung¹,

Cucker²,

Hauser³

2005

SIAM J. Optim.

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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.

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“…Downloaded 05/11/18 to 18.236.120. 13. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php See also below for bounds obtained in [20].…”

Section: Introduction Let a ∈ Rmentioning

confidence: 99%

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

Cheung¹,

Cucker²,

Hauser³

2005

SIAM J. Optim.

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“…As a consequence, the conic feasibility problem was studied extensively in the LP literature, where ellipsoid and interior-point methods have been established as polynomial-time algorithms under the rational number Turing machine model. In the real number Turing machine model of Blum et al [3] the complexity of the same algorithms is typically of order O log C(A) · q(m, n), (1.1) where q is a polynomial and C(A) is either Renegar's condition number C R (A) [17], the Goffin-Cheung-Cucker condition number C G (A) [5,12,13] or another related concept of geometric measure. See Renegar [17] for an extensive discussion and examples of condition-number based complexity analyses.…”

Section: Introductionmentioning

confidence: 99%

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

Hauser

Müller

2008

Found Comput Math

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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.

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“…The condition measure ρ(A) was introduced by Goffin [13] and later independently studied by Cheung and Cucker [4]. The latter set of authors showed that |ρ(A)| is also a certain distance to ill-posedness in the spirit introduced and developed by Renegar [21,22].…”

Section: Introductionmentioning

confidence: 99%

On the von Neumann and Frank--Wolfe Algorithms with Away Steps

2016

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The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the interior of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm's rate of convergence depends on the radius of the largest ball around the origin contained in the polytope.We show that under the weaker condition that the origin is in the polytope, possibly on its boundary, a variant of the von Neumann algorithm that includes away steps generates a sequence of points in the polytope that converges linearly to zero. The new algorithm's rate of convergence depends on a certain geometric parameter of the polytope that extends the above radius but is always positive. Our linear convergence result and geometric insights also extend to a variant of the Frank-Wolfe algorithm with away steps for minimizing a convex quadratic function over a polytope.

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A new condition number for linear programming

Cited by 52 publications

References 6 publications

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

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

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

On the von Neumann and Frank--Wolfe Algorithms with Away Steps

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