TY - JOUR
T1 - On the number of solutions to the linear comple-mentarity problem
AU - Tamir, Arie
PY - 1976/12
Y1 - 1976/12
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=34250385738&partnerID=8YFLogxK
U2 - 10.1007/BF01580680
DO - 10.1007/BF01580680
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:34250385738
VL - 10
SP - 347
EP - 353
JO - Mathematical Programming
JF - Mathematical Programming
SN - 0025-5610
IS - 1
ER -