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An Algorithm for Determining Irrelevant Constraints and all Vertices in Systems of Linear Inequalities

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  • T. H. Mattheiss

    (Southern Illinois University, Carbondale, Illinois)

Abstract

This paper describes a new method of generating all vertices of a given convex polytope. Additionally, irrelevant constraints are easily identified without the necessity of enumerating any of the vertices of the given convex polytope. The method embeds the given polytope in a one-higher-dimensional space. The projection of the additional vertices formed by the embedding process into the original space lie in the interior of the polytope and have a tree structure for one and two polytopes. For higher dimensions, the embedding process associates a number with each interior point that facilitates the construction of a spanning tree for all of the interior points. The interior points added can be efficiently generated by a variant of the simplex method. The vertices of the original polytope can be generated easily from these internal points by analyzing the appropriate simplex tableaux.

Suggested Citation

  • T. H. Mattheiss, 1973. "An Algorithm for Determining Irrelevant Constraints and all Vertices in Systems of Linear Inequalities," Operations Research, INFORMS, vol. 21(1), pages 247-260, February.
    • Handle: RePEc:inm:oropre:v:21:y:1973:i:1:p:247-260
    DOI: 10.1287/opre.21.1.247
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    Cited by:

    1. Liu, Yanwu & Tu, Yan & Zhang, Zhongzhen, 2021. "The row pivoting method for linear programming," Omega, Elsevier, vol. 100(C).
    2. Alexander Krausz & Ulrich Rieder, 1997. "Markov games with incomplete information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 263-279, June.
    3. Nonås, Sigrid Lise, 2009. "Finding and identifying optimal inventory levels for systems with common components," European Journal of Operational Research, Elsevier, vol. 193(1), pages 98-119, February.
    4. Mo, S.H. & Norton, J.P., 1990. "Fast and robust algorithm to compute exact polytope parameter bounds," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 32(5), pages 481-493.

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