EXTREME POINTS OF LEONTIEF SUBSTITUTION SYSTEMS.
Abstract
A Leontief matrix is a matrix A having exactly one positive element in each column and for which there is a nonnegative (column) vector x such that Ax is positive. Let X(b) be the set of nonnegative solutions to Ax = b where A is Leontief and b > or = 0. The following results are established. An element of X(b) is an extreme point of X(b) if and only if it is determined by a Leontief basis matrix. If A is integral, the extreme points of X(b) are integral for all nonnegative integral b if and only if the determinant of each Leontief basis matrix equals one. The class of Leontief matrices for which X(b) is bounded for all b > or = 0 is characterized. The infimum of a concave function over X(b) is concave in b on the nonnegative orthant. The above results are shown to extend easily to matrices with at most one positive element in each column. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1967
- Accession Number
- AD0659045
Entities
People
- Arthur F. Veinott Jr.
Organizations
- Stanford University