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Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms [article]

Roman Vershynin
2007 arXiv   pre-print
The simplex method in Linear Programming motivates several problems of asymptotic convex geometry.  ...  We discuss some conjectures and known results in two related directions -- computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses  ...  This version would be significant for the analysis of the simplex method, because it allows one to leave the constraints intact and only perturb the objective function.  ... 
arXiv:cs/0703093v1 fatcat:t2gw3yqgh5c7xplnam6dsp3rfe

An asymptotic simplex method for parametric linear programming

J.A. Filar, K.E. Avrachenkov, E. Altman
1999 1999 Information, Decision and Control. Data and Information Fusion Symposium, Signal Processing and Communications Symposium and Decision and Control Symposium. Proceedings (Cat. No.99EX251)  
He proposed simplex-like method, which works over the field of rational functions. Here we develop an alternative asymptotic simplex method based on Laurent series expansions.  ...  Definition 1 The set of basic indices B is said to be asymptotically optimal (or shortly a-optimal) for the perturbed linear program ( l ) , (2), i f it is optimal for the linear program ( l ) , (2) with  ...  To deal with such perturbed LPs, Jeroslow [12, 131 proposed simplexlike method which works directly over the field of rational functions.  ... 
doi:10.1109/idc.1999.754195 fatcat:nxynjay2hbchjgd4h7a7hbpzqq

Page 4759 of Mathematical Reviews Vol. , Issue 89H [page]

1989 Mathematical Reviews  
is a diag- onal matrix containing components of x, the modified algorithm converges to an optimal solution for a feasible LP with bounded primal regardless the degeneracy of its dual.  ...  Current implementations of the simplex method on sequential computers are based on a triangu- lar factorization of the inverse of the current basis.  ... 

A Formalization of Convex Polyhedra Based on the Simplex Method [chapter]

Xavier Allamigeon, Ricardo D. Katz
2017 Lecture Notes in Computer Science  
The cornerstone of our work is a complete implementation of the simplex method, together with the proof of its correctness and termination.  ...  This allows us to define the basic predicates over polyhedra in an effective way (i.e. as programs), and relate them with the corresponding usual logical counterparts.  ...  The authors are very grateful to A. Mahboubi for her help to improve the presentation of this paper, and to G. Gonthier, F. Hivert and P.-Y. Strub for fruitful discussions.  ... 
doi:10.1007/978-3-319-66107-0_3 fatcat:otb6a4tmobdyhoi6vfomk6hwhe

Page 396 of Mathematical Reviews Vol. , Issue 83a [page]

1983 Mathematical Reviews  
Game theory: two-person matrix games; Chapter 14. Further applications; quadratic programming; functional approximation; matrix eigenvalue perturbation analysis; Appendix 1.  ...  Conversion to specified form; basic, feasible and optimum solutions; Chapter 3. The simplex method; Chapter 4. The simplex method continued; Chapter 5.  ... 

A formalization of convex polyhedra based on the simplex method [article]

Xavier Allamigeon, Ricardo D. Katz
2018 arXiv   pre-print
The cornerstone of our work is a complete implementation of the simplex method, together with the proof of its correctness and termination.  ...  This allows us to define the basic predicates over polyhedra in an effective way (i.e., as programs), and relate them with the corresponding usual logical counterparts.  ...  Acknowledgements The authors are very grateful to A. Mahboubi for her help to improve the presentation of this paper, and to G. Gonthier, F. Hivert and P.-Y. Strub for fruitful discussions.  ... 
arXiv:1706.10269v2 fatcat:5zc25sv2mzaghayyu7wacz4smi

Improving a primal–dual simplex-type algorithm using interior point methods

T. Glavelis, N. Ploskas, N. Samaras
2018 Optimization  
Interior point methods and simplex-type algorithms are the most widely-used algorithms for solving linear programming problems. The simplex algorithm has many important applications.  ...  It applies an interior point method for a few iterations leading to significant improvement of the objective function value.  ...  Any known technique for updating the basic inverse matrix (A B ) −1 and the vectors x B , w, and s N can be combined efficiently with the algorithm.  ... 
doi:10.1080/02331934.2018.1523906 fatcat:jldy326e3jayzn2sa5mz2ugo7y

Page 240 of Mathematical Reviews Vol. 41, Issue 1 [page]

1971 Mathematical Reviews  
and second, basis decomposition, where a revised simplex method is used and the basis-inverse is obtained from inverses of smaller matrices.  ...  Of particular interest is the description of methods of finding a first feasible solution and the product form of the inverse matrix method.  ... 

Practical guidelines for solving difficult linear programs

Ed Klotz, Alexandra M. Newman
2013 Surveys in Operations Research and Management Science  
However, a 5 significant number of large linear programs can require hours, or even days, of run time and are 6 not guaranteed to yield an optimal (or near-optimal) solution.  ...  It can be shown that Benders' Decomposition applied to the primal 937 representation of an LP is equivalent to applying Dantzig-Wolfe Decomposition to the dual LP. 938 The details and application of these  ...  to the Phase I primal simplex method.  ... 
doi:10.1016/j.sorms.2012.11.001 fatcat:7iki6aqx6rdvpiydzb6z23o7jm

Exact arithmetic at low cost – A case study in linear programming

Bernd Gärtner
1999 Computational geometry  
The algorithm is an implementation of the simplex method which combines exact (multiple precision) arithmetic with inexact (floating point) arithmetic, where the number of exact arithmetic operations is  ...  that even competes with the inexact state-ofthe-art solver CPLEX 1 for small values of min(n, m) and and is far superior to methods that use exact arithmetic in any operation.  ...  Finally, many discussions with Emo Welzl have contributed to this paper.  ... 
doi:10.1016/s0925-7721(99)00012-7 fatcat:iyq6dsoaenalfhxmbwcgg73bdi

The existence of a strongly polynomial time simplex algorithm for linear programming problems [article]

Zi-zong Yan, Xiang-jun Li, Jinhai Guo
2022 arXiv   pre-print
It is well known how to clarify whether there is a polynomial time simplex algorithm for linear programming (LP) is the most challenging open problem in optimization and discrete geometry.  ...  We show that there is a simplex algorithm whose number of pivoting steps does not exceed the number of variables of a LP problem.  ...  It is believed to be able to use more global information and therefore to avoid myopiness of the simplex method, for example, see Roos [58] , Todd [71] and Tamura et al. [67] .  ... 
arXiv:2006.11466v12 fatcat:h4gv5a3bindwlhea4ibp7s6tpm

The product form for the inverse in the simplex method

George B. Dantzig, Wm. Orchard-Hays
1954 Mathematics of Computation  
In the revised simplex method,1 both the inverse and inverse transpose of a "basic" matrix are needed; more significant, however, is the fact that each iteration replaces one of the columns of the basis  ...  with each iteration.  ...  The Product Form for the Inverse in the Simplex Method Summary: When a matrix is represented as a product of "elementary" matrices, the matrix, its transpose, its inverse and inverse transpose are readily  ... 
doi:10.1090/s0025-5718-1954-0061469-8 fatcat:bwuoga4plzacfd2frgoeiubkvm

Physarum Powered Differentiable Linear Programming Layers and Applications [article]

Zihang Meng, Sathya N. Ravi, Vikas Singh
2021 arXiv   pre-print
Our proposal is easy to implement and can easily serve as layers whenever a learning procedure needs a fast approximate solution to a LP, within a larger network.  ...  Our solver performs comparably with a customized projected gradient descent method on the first task and outperforms the differentiable CVXPY-SCS solver on the second task.  ...  Acknowledgements We would like to thank one of the anonymous AAAI 2021 reviewers who apart from suggestions also provided an alternative implementation that improved the performance of CVXPY-SCS in our  ... 
arXiv:2004.14539v2 fatcat:3nghuh53grdelbxe4f7vdwucau

Lexicographic perturbation for multiparametric linear programming with applications to control

C.N. Jones, E.C. Kerrigan, J.M. Maciejowski
2007 Automatica  
Furthermore, we introduce a new method for computing the optimal solution in an adjacent region, which is very efficient in all cases and reduces to a single simplex pivot for non-degenerate regions.  ...  We show that every optimal solution of the perturbed problem is an optimal solution to the original and that the perturbed solution is continuous, unique and defined over a set of non-overlapping polyhedral  ...  Acknowledgements The authors would like to thank Komei Fukuda for his valuable comments on this work.  ... 
doi:10.1016/j.automatica.2007.03.008 fatcat:h7lnqxlv2jeadjf64zwhacb3ta

Simplex Initialization: A Survey of Techniques and Trends [article]

Mengyu Huang, Yuxing Zhong, Huiwen Yang, Jiazheng Wang, Fan Zhang, Bo Bai, Ling Shi
2021 arXiv   pre-print
The simplex method is one of the most fundamental technologies for solving linear programming (LP) problems and has been widely applied to different practical applications.  ...  One important way to achieve this goal is to find a better initialization method for the simplex.  ...  In some applications, after the original LP problem was solved, a new LP problem, which is derived by making some small modifications to the original one (e.g., perturbing the bound of some variables,  ... 
arXiv:2111.03376v1 fatcat:6cddy34m6bed5hmx4kvnuiap4i
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