On the number of solutions to the complementarity problem and spanning properties of complementary cones
dc.contributor.author | Murty, Katta G. | en_US |
dc.date.accessioned | 2006-04-17T16:53:26Z | |
dc.date.available | 2006-04-17T16:53:26Z | |
dc.date.issued | 1972-01 | en_US |
dc.identifier.citation | Murty, Katta G. (1972/01)."On the number of solutions to the complementarity problem and spanning properties of complementary cones." Linear Algebra and its Applications 5(1): 65-108. <http://hdl.handle.net/2027.42/34188> | en_US |
dc.identifier.uri | http://www.sciencedirect.com/science/article/B6V0R-45F5BFR-1C/2/4a760e56e4ab112335fbeca2f9a4fcd5 | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/34188 | |
dc.description.abstract | The relationship between the number of solutions to the complementarity problem, w = Mz + q, w[ges]0, z[ges]0, wTz=0, the right-hand constant vector q and the matrix M are explored. The main results proved in this work are summarized below.The number of solutions to the complementarity problem is finite for all q [epsilon] Rn if and only if all the principal subdeterminants of M are nonzero. The necessary and sufficient condition for this solution to be unique for each q [epsilon] Rn is that all principal subdeterminants of M are strictly positive. When M[ges]0, there is at least one complementary feasible solution for each q [epsilon] Rn if and only if all the diagonal elements of M are strictly positive; and, in this case, the number of these solutions is an odd number whenever q is nondegenerate. If all principal subdeterminants of M are nonzero, then the number of complementary feasible solutions has the same parity (odd or even) for all q [epsilon] Rn which are nondegenerate. Also, if the number of complementary feasible solutions is a constant for each q [epsilon] Rn, then that constant is equal to one and M is a P-matrix.In the cartesian system of coordinates for Rn, an orthant is a convex cone generated by a set of n-column vectors in Rn, {A.1,...,A.n}, where for each j = 1 to n,A.j is either the jth column vector of the unit matrix of order n (denoted by I.j) or its negative - I.j. There are thus 2n orthants in Rn, and they partition the whole space. It is interesting to know what properties these orthants possess if we obtain them after replacing - I.j by some given column vector - M.j for j = 1 to n. Orthants obtained in this manner are called complementary cones, and their spanning properties are studied. | en_US |
dc.format.extent | 2115728 bytes | |
dc.format.extent | 3118 bytes | |
dc.format.mimetype | application/pdf | |
dc.format.mimetype | text/plain | |
dc.language.iso | en_US | |
dc.publisher | Elsevier | en_US |
dc.title | On the number of solutions to the complementarity problem and spanning properties of complementary cones | en_US |
dc.type | Article | en_US |
dc.rights.robots | IndexNoFollow | en_US |
dc.subject.hlbsecondlevel | Mathematics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | The University of Michigan Ann Arbor, Michigan USA | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/34188/1/0000477.pdf | en_US |
dc.identifier.doi | http://dx.doi.org/10.1016/0024-3795(72)90019-5 | en_US |
dc.identifier.source | Linear Algebra and its Applications | en_US |
dc.owningcollname | Interdisciplinary and Peer-Reviewed |
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