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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. 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.
    2. Liu, Yanwu & Tu, Yan & Zhang, Zhongzhen, 2021. "The row pivoting method for linear programming," Omega, Elsevier, vol. 100(C).
    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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