A review of the LU update in the simplex algorithm

Elble, Joseph M.; Sahinidis, Nikolaos V.

doi:10.1504/ijmor.2012.048901

Cited by 10 publications

(7 citation statements)

References 30 publications

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“…The standard choices are the relatively simple product form (PF) update [18] or the efficient Forrest-Tomlin (FT) update [5]. A comprehensive report on simplex update techniques is given by Elble and Sahinidis [3] and novel techniques, some motivated by the design and development of pami, are described by Huangfu and Hall [14]. For the purpose of this report, the features of all relevant update methods are summarised as follows.…”

Section: Simplex Update Techniquesmentioning

confidence: 99%

Parallelizing the dual revised simplex method

Huangfu¹,

Hall

2017

Math. Prog. Comp.

127

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This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems. One approach, called PAMI, extends a relatively unknown pivoting strategy called suboptimization and exploits parallelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible. Computational results show that the performance of PAMI is superior to that of the leading open-source simplex solver, and that SIP complements PAMI in achieving speedup when PAMI results in slowdown. One of the authors has implemented the techniques underlying PAMI within the FICO Xpress simplex solver and this paper presents computational results demonstrating their value. In developing the first parallel revised simplex solver of general utility, this work represents a significant achievement in computational optimization.

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Section: Simplex Update Techniquesmentioning

confidence: 99%

Parallelizing the dual revised simplex method

Huangfu¹,

Hall

2017

Math. Prog. Comp.

127

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“…The matrix T = I + (a q − a p )ê T p is in the general form (14) so, using (15), the inverse of the updated basis matrix is given by…”

Section: Alternate Product Form Updatementioning

confidence: 99%

“…Thus hyper-sparsity may be exploited fully with the PF or FT updates. However, in the case of the APF and MPF updates, the form of the general elementary matrix (14) prohibits the extension of these techniques within the phase of FTRAN and BTRAN associated with UPDATE .…”

Section: New Techniques and Hyper-sparsitymentioning

confidence: 99%

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Novel update techniques for the revised simplex method

Huangfu

Hall

2014

Comput Optim Appl

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This paper introduces three novel techniques for updating the invertible representation of the basis matrix when solving practical sparse linear programming problems using a high performance implementation of the dual revised simplex method, being of particular value when suboptimization is used. Two are variants of the product form update and the other permits multiple Forrest-Tomlin updates to be performed. Computational results show that one of the product form variants is significantly more efficient than the traditional approach, with its performance approaching that of the Forrest-Tomlin update for some problems. The other is less efficient, but valuable in the context of the dual revised simplex method with suboptimization. Results show that the multiple Forrest-Tomlin updates are performed with no loss of serial efficiency.

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“…This leads to a sequence of small dense matrices B and B T that need to be solved. Since these matrices appear in a sequence where they differ only by rank-one updates, we can easily update their QR factorization [8,4].…”

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confidence: 99%

Krylov-Simplex method that minimizes the residual in $\ell_1$-norm or $\ell_\infty$-norm

Vanroose,

Cornelis

2021

Preprint

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The paper presents two variants of a Krylov-Simplex iterative method that combines Krylov and simplex iterations to minimize the residual r = b−Ax. The first method minimizes r ∞, i.e. maximum of the absolute residuals. The second minimizes r 1 , and finds the solution with the least absolute residuals. Both methods search for an optimal solution x k in a Krylov subspace which results in a small linear programming problem. A specialized simplex algorithm solves this projected problem and finds the optimal linear combination of Krylov basis vectors to approximate the solution. The resulting simplex algorithm requires the solution of a series of small dense linear systems that only differ by rank-one updates. The QR factorization of these matrices is updated each iteration. We demonstrate the effectiveness of the methods with numerical experiments.

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A review of the LU update in the simplex algorithm

Cited by 10 publications

References 30 publications

Parallelizing the dual revised simplex method

Parallelizing the dual revised simplex method

Novel update techniques for the revised simplex method

Krylov-Simplex method that minimizes the residual in $\ell_1$-norm or $\ell_\infty$-norm

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