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Towards a practical parallelisation of the simplex method

J. A. J. Hall
2008 Computational Management Science  
This paper reviews previous attempts to parallelise the simplex method in relation to efficient serial simplex techniques and the nature of practical LP problems.  ...  For the major challenge of solving general large sparse LP problems, there has been no parallelisation of the simplex method that offers significantly improved performance over a good serial implementation  ...  As a consequence, although work on parallelising the simplex method using dense matrix algebra has generally exploited data parallelism, it was a commonly held view that when using sparse matrix algebra  ... 
doi:10.1007/s10287-008-0080-5 fatcat:7rdsk7ic7bca7dy5e6yvybuioq

Minimum volume simplicial enclosure for spectral unmixing of remotely sensed hyperspectral data

Eligius M.T. Hendrix, Inmaculada Garcia, Javier Plaza, Antonio Plaza
2010 2010 IEEE International Geoscience and Remote Sensing Symposium  
In order to interpret the results in Table 1 , let us first focus on the scene with opaque trees and sparse population (CASI01 01).  ...  Canopies of both opaque and translucent trees were designed with sparse and dense populations.  ... 
doi:10.1109/igarss.2010.5649694 dblp:conf/igarss/HendrixGPP10 fatcat:45jkjjml55d7ha3novwhdseez4

GPU Acceleration of the Matrix-Free Interior Point Method [chapter]

Edmund Smith, Jacek Gondzio, Julian Hall
2012 Lecture Notes in Computer Science  
The matrix-free method is one such approach and is so named since the iterative solution procedure requires only the results of operations Ax and A T y, where A is the matrix of constraint coefficients  ...  Since the computational cost of these operations may well dominate the total solution time for the problem, it is important that the techniques used to perform them are efficient.  ...  Its focus on a small core of sparse operations makes highly optimized implementations using state of the art hardware possible without excessive difficulty.  ... 
doi:10.1007/978-3-642-31464-3_69 fatcat:3klpdqf55vawpigj7wdcag7fla

Parallelizing the dual revised simplex method

Q. Huangfu, J. A. J. Hall
2017 Mathematical Programming Computation  
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.  ...  This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems.  ...  For many sparse LP problems the matrix B −1 is dense, so solutions of linear systems involving B or B T can be expected to be dense even when, as is typically the case in the revised simplex method, the  ... 
doi:10.1007/s12532-017-0130-5 fatcat:7l3elmcu2jg4ffeswgq4oqlywq

Parallelizing the dual revised simplex method [article]

Q. Huangfu, J. A. J. Hall
2015 arXiv   pre-print
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.  ...  This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems.  ...  For many sparse LP problems the matrix B −1 is dense, so solutions of linear systems involving B or B T can be expected to be dense even when, as is typically the case in the revised simplex method, the  ... 
arXiv:1503.01889v1 fatcat:dp3migoocjd35ou7bmg7yd3v3y

Implementing interior point linear programming methods in the Optimization Subroutine Library

J. J. H. Forrest, J. A. Tomlin
1992 IBM Systems Journal  
This class of methods uses quite different computational kernels than the traditional simplex method.  ...  In particular, interior methods can benefit greatly from use of vector architectures on the IBM 3090'" series computers and "super-scalar" processing on the RlSC System/6000" series. s . = x .  ...  Acknowledgments We are indebted to Jenny Edwards for converting the ESSLl6000 dense Cholesky routines to function under OSL, to Fred Gustavson for helpful discussions, and to Irvin Lustig for generating  ... 
doi:10.1147/sj.311.0026 fatcat:kpcxuhvq6fd45ojhsx4vcxh6lq

Hyper-Sparsity in the Revised Simplex Method and How to Exploit it

J. A. J. Hall, K. I. M. McKinnon
2005 Computational optimization and applications  
Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product.  ...  Analysis of the commonly-used techniques for implementing each step of the revised simplex method shows them to be inefficient when hyper-sparsity is present.  ...  The authors would like to thank John Reid who brought the Gilbert-Peierls algorithm to their attention and made valuable comments on an earlier version of this paper.  ... 
doi:10.1007/s10589-005-4802-0 fatcat:vsma6uo4vzb4xflgrsq5kwzwqq

Leveraging Sparsity to Speed Up Polynomial Feature Expansions of CSR Matrices Using K-Simplex Numbers [article]

Andrew Nystrom, John Hughes
2018 arXiv   pre-print
The algorithm performs a K-degree expansion by using a bijective function involving K-simplex numbers of column indices in the original matrix to column indices in the expanded matrix.  ...  This work derives the required function for the cases of K=2 and K=3 and shows its use in the K=2 algorithm.  ...  Note that the point at which the sparse and dense algorithms intersect when varying d is to the right of 0.5, which is when a matrix technically becomes sparse.  ... 
arXiv:1803.06418v3 fatcat:arpgwegxevdu3pnhajrddrwsie

Optimization solvers: the missing link for a fully open-source energy system modelling ecosystem

Maximilian Parzen, Julian Hall, Jesse Jenkins, Tom Brown
2022 Zenodo  
The proposal promotes making the world's fastest open-source solver HiGHS better and describes a plan on how to achieve this.  ...  However, the developments require funds which we aim to collect with this proposal. Funds are collected over the Linux Foundation platform and can be tax-deductible.  ...  One particular property of A that must be handled carefully is the existence of dense columns.  ... 
doi:10.5281/zenodo.6534004 fatcat:cojv53lfrncenfcmk73nrwjrj4

Fast and efficient linear programming and linear least-squares computations

V. Pan, J. Reif
1986 Computers and Mathematics with Applications  
Pan gratefully acknowledges the support given by NSF Grants MCS 8203232 and DCR 8507573. J. Reif was supported by the Office of Naval Research, Contract No. N00014-80-C-0647.  ...  On the other hand, all the estimates have been extended to the case of an arbitrary integer input matrix & in Refs [19, 20] by using some different techniques, in particular, using variable diagonals  ...  This stabilization can be combined with the customary techniques of threshold pivoting, used in sparse matrix computations at the stage of determining the elimination ordering [15] (see also Remark 3  ... 
doi:10.1016/0898-1221(86)90006-4 fatcat:lhm2zra3njf4nh6m6cepifq4zq

A basis-deficiency-allowing primal phase-I algorithm using the most-obtuse-angle column rule

Wei Li, P. Guerrero-García, A. Santos-Palomo
2006 Computers and Mathematics with Applications  
Our computational experiments with the smallest test problems from the standard NETLIB set show that a dense projected-gradient implementation largely outperforms that of the variation of the primal simplex  ...  of the primal simplex method included in the commercial code TOMLAB LPSOLVE V3.0. (~)  ...  If the number of basic columns equals the number of rows of the coefficient matrix, it is a normal basis; else, it is a deficient basis. Clearly, traditional simplex variants use normal bases only.  ... 
doi:10.1016/j.camwa.2005.11.033 fatcat:zow6jdurxrfrxdfnhymiz5y4nu

Sparse Matrix Methods in Optimization

Philip E. Gill, Walter Murray, Michael A. Saunders, Margaret H. Wright
1984 SIAM Journal on Scientific and Statistical Computing  
product-form update; use of the Schur complement).  ...  Since significant advances continue to be made with single-system solvers, we give special attention to methods that allow such solvers to be used repeatedly on a sequence of modified systems (e.g., the  ...  sparse matrix techniques.  ... 
doi:10.1137/0905041 fatcat:yf4a5l2t5vabzi2p3ktl73hnqi

The impact of high-performance computing in the solution of linear systems: trends and problems

Iain S. Duff
2000 Journal of Computational and Applied Mathematics  
We will concentrate on direct methods of solution and consider both the case when the coe cient matrix is dense and when it is sparse.  ...  We review the in uence of the advent of high-performance computing on the solution of linear equations.  ...  Acknowledgements I would like to thank my colleagues Patrick Amestoy, Jacko Koster, Xiaoye Li, John Reid, and Jennifer Scott for some helpful remarks on a draft of this paper.  ... 
doi:10.1016/s0377-0427(00)00401-5 fatcat:v6vcoq2a7ncjtexywdeiwjmjte

Efficient parallel linear programming

Victor Pan, John Reif
1986 Operations Research Letters  
~ce of the Karmarkar and the simplex algorithms for dense and sparse linear programs are examined.  ...  Linear programming and least squares computations ate accelerated using authors' parallel algorithms for solving linear systems. The implications on the performa,.  ...  of threshold pivoting, used in sparse matrix comp-'a¢ions at the stage of determining (be elimination ordering (Pissanetsky [21] ; see also [20] on the stabilization based on variable diagonal techniques  ... 
doi:10.1016/0167-6377(86)90085-4 fatcat:wcs2luk5dzhwhk5rva6zphmgkm

Efficient Implementation of the Simplex Method on a CPU-GPU System

Mohamed Esseghir Lalami, Vincent Boyer, Didier El-Baz
2011 2011 IEEE International Symposium on Parallel and Distributed Processing Workshops and Phd Forum  
Double precision implementation is used in order to improve the quality of solutions. Computational tests have been carried out on randomly generated instances for non-sparse LP problems.  ...  The Simplex algorithm is a well known method to solve linear programming (LP) problems. In this paper, we propose a parallel implementation of the Simplex on a CPU-GPU systems via CUDA.  ...  Computing the entering and leaving variables: Finding the entering or leaving variables results in finding a minimum within a set of values. This can be done on GPU via reduction techniques.  ... 
doi:10.1109/ipdps.2011.362 dblp:conf/ipps/LalamiBB11 fatcat:lccgbtguvnguhkghhrrpyonkey
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