Mathematical Programming

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

Polyhedral sets having a least element

  • Richard W. Cottle
  • Arthur F. VeinottJr.


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.


Unique Solution Identity Matrix Mathematical Method Complementarity Problem Linear Complementarity Problem 
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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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