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.
Similar content being viewed by others
References
P. Bod, “Megjegyzes G. Wintgen egy tetelehez,”MTA III.Osztaly Kozlemenyei 16 (1966) 275–279.
P. Collatz, “Aufgaben monotoner Art,”Arch. Math. 3 (1952) 366–376.
G.B. Dantzig, “Optimal solution of a dynamic Leontief model with substitution,”Econometrica 23 (1955) 295–302.
P. Du Val, “The unloading problem for plane curves,”American Journal of Mathematics 62 (1940) 307–311.
M. Fiedler and V. Pták, “On matrices with nonpositive off-diagonal elements and positive principal minors,”Czechoslovak Mathematical Journal 12 (1962) 382–400.
O. Mangasarian, “Characterizations of real matrices of monotone kind,”SIAM Review 10 (1968) 439–441.
R. Saigal, “On a generalization of Leontief systems,” Univ. of California, Berkeley, 1970.
H. Samelson, R.M. Thrall and O. Wesler, “A partition theorem for Euclideann-space,”Proceedings of the American Mathematical Society 9 (1958) 805–807.
A.F. Veinott, Jr., “Extreme points of Leontief substitution systems,”Linear Algebra and its Applications 1 (1968) 181–194.
G. Wintgen, “Indifferente Optimierungsprobleme,” Beitrag zur Internationalen Tagung. Mathematik und Kybernetik in der Ökonomie, Berlin, 1964. Konferenzprotokoll. Teil II (Akademie Verlag, Berlin) 3–6.
Author information
Authors and Affiliations
Additional information
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.
Rights and permissions
About this article
Cite this article
Cottle, R.W., Veinott, A.F. Polyhedral sets having a least element. Mathematical Programming 3, 238–249 (1972). https://doi.org/10.1007/BF01584992
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584992