Random linear programs with many variables and few constraints

Blair, Charles E.

doi:10.1007/bf01582163

Cited by 13 publications

(2 citation statements)

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“…Since the regression quantile problem is a parametric program, we may have hoped to apply known results on the efficiency of appropriate simplex algorithms in random situations. Indeed, there are results on the probabilistic behavior of the number of pivots in parametric linear programming problems (see Smale (1983), Shamir (1984), Borgwardt (1987) and Blair (1986)). However, most of these results require rather artificial uniform distributions on spheres, which are completely unreasonable in statistical applications.…”

Section: Bo { R P" E Po(yi-xi)=min}mentioning

confidence: 98%

Asymptotic Behavior of the Number of Regression Quantile Breakpoints

Portnoy¹

1991

SIAM J. Sci. and Stat. Comput.

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In the general regression model yi=xfl+ei, for i= 1,..., n and tiER p, the "regression quantile"/(0) estimates the coefficients of thelinear regression function parallel to x/3 and roughly lying above a fraction 0 of the data. As introduced by Koenker and Bassett [Econometrica, 46 (1978), pp. 33-50], these regression quantiles are analogous to order statistics and provide a natural and appealing approach to the analysis of the general linear model. Computation of/(0) can be expressed as a parametric linear programming problem with J, distinct extremal solutions as 0 goes from zero to one. That is, there will be J, breakpoints {0j}, for j-1,... ,J, such that/(0j) is obtained from/(0_) by a single simplex pivot. Each/(0) is characterized by a specific subset of p observations. Although no previous result restricts J to be less than the upper bound (,)= O(n P), practical experience suggests that J grows roughly linearly with n. Here it is shown that, in fact, Jn O(n log n) in probability, where the distributional assumptions are those typical of multiple regression situations. The result is based on a probabilistic rather than combinatoric approach which should have general application to the probabilistic behavior of the number of pivots in (parametric) linear programming problems. The conditions are roughly that the constraint coefficients form random independent vectors, and that the number of variables is fixed while the number of constraints tends to infinity.

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Section: Bo { R P" E Po(yi-xi)=min}mentioning

confidence: 98%

Asymptotic Behavior of the Number of Regression Quantile Breakpoints

Portnoy¹

1991

SIAM J. Sci. and Stat. Comput.

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“…Blair [ 7 ] proves that the expected number of undominated columns in a problem of order m x n , under an even more general model, is less than c(m)(ln n ) m ( m +~) l n ( m + l ) + m . In general, estimations of numbers of vertices, numbers of undominated columns, or numbers of nonredundant constraints, lead to exponential estimates on the number of steps.…”

Section: Introductionmentioning

confidence: 99%

Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm

Megiddo

1986

Mathematical Programming

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In this paper we analyze the average number of steps performed by the self-dual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem of n variables with m constraints. Smale established that for every number of constraints m, there is a constant c ( m ) such that the number of pivot steps of the self-dual algorithm, p(m, n ) , is less than c(m)(ln n)"""'+". We improve upon this estimate by showing that p(m, n ) is bounded by a function of m only. The symmetry of the function in m and n implies that p(m, n ) is in fact bounded by a function of the smaller of m and n.

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Probabilistic Analysis of Algorithms

Frieze

Reed

1998

Algorithms and Combinatorics

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Random linear programs with many variables and few constraints

Cited by 13 publications

References 9 publications

Asymptotic Behavior of the Number of Regression Quantile Breakpoints

Asymptotic Behavior of the Number of Regression Quantile Breakpoints

Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm

Probabilistic Analysis of Algorithms

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