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Mathematical Programming

, Volume 3, Issue 1, pp 238–249 | Cite as

Polyhedral sets having a least element

  • Richard W. Cottle
  • Arthur F. VeinottJr.
Article

Abstract

For a fixedm × n matrixA, we consider the family of polyhedral setsX b ={x|Ax ≥ b}, b ∈ R m , and prove a theorem characterizing, in terms ofA, the circumstances under which every nonemptyX b has a least element. In the special case whereA contains all the rows of ann × n identity matrix, the conditions are equivalent toAT being Leontief. Among the corollaries of our theorem, we show the linear complementarity problem always has a unique solution which is at the same time a least element of the corresponding polyhedron if and only if its matrix is square, Leontief, and has positive diagonals.

Keywords

Unique Solution Identity Matrix Mathematical Method Complementarity Problem Linear Complementarity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Mathematical Programming Society 1972

Authors and Affiliations

  • Richard W. Cottle
    • 1
  • Arthur F. VeinottJr.
    • 1
  1. 1.Stanford UniversityStanfordUSA

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