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Polyhedral sets having a least element

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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 toA T 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.

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This research was supported by the National Science Foundation under Grants GK-18339 and GP-25738, by the Office of Naval Research under Contracts N00014-67-A-0112-0050 (NR-042-264) and N00014-67-A-0112-0011, and by the IBM Corporation. Part of the second author's contribution to this paper was made while he was on sabbatical leave in 1968–9 as a consultant to the IBM Research Center. Reproduction in whole or in part is permitted for any purpose of the United States Government.

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Cottle, R.W., Veinott, A.F. Polyhedral sets having a least element. Mathematical Programming 3, 238–249 (1972). https://doi.org/10.1007/BF01584992

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  • DOI: https://doi.org/10.1007/BF01584992

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