On the number of solutions to the linear comple-mentarity problem

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Abstract

Given an n × n matrix A and an n-dimensional vector q let N(A, q) be the cardinality of the set of solutions to the linear complementarity problem defined by A and q. It is shown that if A is nondegenerate then N(A, q) + N(A, -q) ≤ 2n, which in turn implies N(A, q) ≤ 2n - 1 if A is also a Q-matrix. It is then demonstrated that minq≠0N(A, q) ≤ 2n-1 - 1, which concludes that the complementary cones cannot span Rn more than 2n-1 - 1 times around. For any n, an example of an n × n nondegenerate Q-matrix spanning all Rn, but a subset of empty interior, 2[n/3] times around is given.

Original languageEnglish
Pages (from-to)347-353
Number of pages7
JournalMathematical Programming
Volume10
Issue number1
DOIs
StatePublished - Dec 1976
Externally publishedYes

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