A revised simplex method with integer Q-matrices

Azulay, David-Olivier; Pique, Jean-François

doi:10.1145/502800.502804

Cited by 4 publications

(4 citation statements)

References 5 publications

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“…Pivoting is done by row operations on the system followed by a division by the old determinant, which always produces integers (see Avis 2000, Sect. 7 or Azulay andPique 2001). In that way, the dictionary entries are kept from growing indefinitely.…”

Section: S-intpivmentioning

confidence: 99%

Enumeration of Nash equilibria for two-player games

et al. 2009

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This paper describes algorithms for finding all Nash equilibria of a twoplayer game in strategic form. We present two algorithms that extend earlier work. Our presentation is self-contained, and explains the two methods in a unified framework using faces of best-response polyhedra. The first method is based on the known vertex enumeration program lrs, for "lexicographic reverse search". It enumerates the vertices of only one best-response polytope, which determine a complementary face in the other polytope. The second method is a modification of the known EEE algorithm, for "enumeration of extreme equilibria". We also describe a second, as yet not implemented, variant that is space efficient. We discuss details of implementations of the lrs-based and the EEE algorithm, and report on computational experiments that compare the two algorithms, which show that both have their strengths and weaknesses.

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Section: S-intpivmentioning

confidence: 99%

Enumeration of Nash equilibria for two-player games

et al. 2009

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“…The performance of the REF framework for solving dense SLEs is measured in relation to exact rational arithmetic LU factorization, which is a primary basic solution validation subroutine used in exact LP solvers (see Section 2.1). In addition, we devise and test an efficient adaptation of the Q-matrix revised simplex method (Azulay and Pique 2001), which is an alternative IPGE-based approach. The metrics of interest are threefold: (1) runtimes of the construction phases for each tool (i.e., calculating exact LU factorizations or adapted Q-matrices), (2) runtimes of the respective solution phases (i.e., performing exact forward and backward triangular solves or multiplying the adapted Q-matrix with the SLE RHS), and (3) storage requirements.…”

Section: Featured Computational Testsmentioning

confidence: 99%

“…Each Q-matrix corresponds to the adjunct matrix of a basis and, like the REF factorizations, it is constructed via IPGE. Azulay and Pique (2001) expanded this tool by developing the Q-matrix revised simplex method, which was designed to operate as a full-fledged exact LP solver. In the limited accompanying set of experiments, their method decreased the respective exact rational arithmetic revised simplex runtimes by at most one order of magnitude-achieving a maximum speedup of 12 on 1 of 24 NETLIB instances.…”

Section: Storage the Second Experiments Performs A New Set Of Runs Tomentioning

confidence: 99%

Solution of Dense Linear Systems via Roundoff-Error-Free Factorization Algorithms

Escobedo

Moreno-Centeno

Lourenco

2018

ACM Trans. Math. Softw.

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Exact solving of systems of linear equations (SLEs) is a fundamental subroutine within number theory, formal verification of mathematical proofs, and exact-precision mathematical programming. Moreover, efficient exact SLE solution methods could be valuable for a growing body of science and engineering applications where current fixed-precision standards have been deemed inadequate. This article contains key derivations relating, and computational tests comparing, two exact direct solution frameworks: roundoff-error-free (REF) LU factorization and rational arithmetic LU factorization. Specifically, both approaches solve the linear system Ax = b by factoring the matrix A into the product of a lower triangular (L) and upper triangular (U) matrix, A = LU . Most significantly, the featured findings reveal that the integer-preserving REF factorization framework solves dense SLEs one order of magnitude faster than the exact rational arithmetic approach while requiring half the memory. Since rational LU is utilized for basic solution validation in exact linear and mixed-integer programming, these results offer preliminary evidence of the potential of the REF factorization framework to be utilized within this specific context. Additionally, this article develops and analyzes an efficient streamlined version of Edmonds’s Q-matrix approach that can be implemented as another basic solution validation approach. Further experiments demonstrate that the REF factorization framework also outperforms this alternative integer-preserving approach in terms of memory requirements and computational effort. General purpose codes to solve dense SLEs exactly via any of the aforementioned methods have been made available to the research and academic communities.

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“…When dealing with black-box optimization problems, traditional mathematical programming methods such as the simplex algorithm [Dantzig and Thapa 1997] [Weglarz et al 1977] [Azulay and Pique 2001], and gradient-based methods such as the Quasi-Newton method [Zhu et al 1997] and the conjugate gradient method [Hager and Zhang 2006] are no longer applicable, since the internal information about the problem, such as the coefficients and derivative, is unavailable, or only partially available. In such a case, derivative-free algorithms are promising methods for solving black-box optimization problems as they take only the input and output of a function into account.…”

Section: Introductionmentioning

confidence: 99%

A Competitive Divide-and-Conquer Algorithm for Unconstrained Large-Scale Black-Box Optimization

Mei

Omidvar

et al. 2016

ACM Trans. Math. Softw.

199

107

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Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.

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A revised simplex method with integer Q-matrices

Cited by 4 publications

References 5 publications

Enumeration of Nash equilibria for two-player games

Enumeration of Nash equilibria for two-player games

Solution of Dense Linear Systems via Roundoff-Error-Free Factorization Algorithms

A Competitive Divide-and-Conquer Algorithm for Unconstrained Large-Scale Black-Box Optimization

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