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Alternative methods for representing the inverse of linear programming basis matrices
Methods for representing the inverse of Linear Programming (LP) basis matrices are closely related to techniques for solving a system of sparse unsymmetric linear equations by direct methods. It is now well accepted that for these problems the static process of reordering the matrix in the lower block triangular (LBT) form constitutes the initial step. We introduce a combined static and dynamic factorisation of a basis matrix and derive its inverse which we call the partial elimination form of the inverse (PEFI). This factorization takes advantage of the LBT structure and produces a sparser representation of the inverse than the elimination form of the inverse (EFI). In this we make use of the original columns (of the constraint matrix) which are in the basis. To represent the factored inverse it is, however, necessary to introduce special data structures which are used in the forward and the backward transformations (the two major algorithmic steps) of the simplex method. These correspond to solving a system of equations and solving a system of equations with the transposed matrix respectively. In this paper we compare the nonzero build up of PEFI with that of EFI. We have also investigated alternative methods for updating the basis inverse in the PEFI representation. The results of our experimental investigation are presented in this pape
Mechanism Design via Dantzig-Wolfe Decomposition
In random allocation rules, typically first an optimal fractional point is
calculated via solving a linear program. The calculated point represents a
fractional assignment of objects or more generally packages of objects to
agents. In order to implement an expected assignment, the mechanism designer
must decompose the fractional point into integer solutions, each satisfying
underlying constraints. The resulting convex combination can then be viewed as
a probability distribution over feasible assignments out of which a random
assignment can be sampled. This approach has been successfully employed in
combinatorial optimization as well as mechanism design with or without money.
In this paper, we show that both finding the optimal fractional point as well
as its decomposition into integer solutions can be done at once. We propose an
appropriate linear program which provides the desired solution. We show that
the linear program can be solved via Dantzig-Wolfe decomposition. Dantzig-Wolfe
decomposition is a direct implementation of the revised simplex method which is
well known to be highly efficient in practice. We also show how to use the
Benders decomposition as an alternative method to solve the problem. The
proposed method can also find a decomposition into integer solutions when the
fractional point is readily present perhaps as an outcome of other algorithms
rather than linear programming. The resulting convex decomposition in this case
is tight in terms of the number of integer points according to the
Carath{\'e}odory's theorem
Bounding Stability Constants for Affinely Parameter-Dependent Operators
In this article we introduce new possibilities of bounding the stability
constants that play a vital role in the reduced basis method. By bounding
stability constants over a neighborhood we make it possible to guarantee
stability at more than a finite number of points and to do that in the offline
stage. We additionally show that Lyapunov stability of dynamical systems can be
handled in the same framework.Comment: Accepted version (C. R. Math.), 6 pages, 3 figure
Numerical Analysis
Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud
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In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud
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With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright
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