On the matrix cuts of Lovasz and Schrijver and their use in integer programming

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dc.contributor.advisorCook, William J.en_US
dc.creatorDash, Sanjeeben_US
dc.date.accessioned2009-06-04T08:37:48Zen_US
dc.date.available2009-06-04T08:37:48Zen_US
dc.date.issued2001en_US
dc.description.abstractAn important approach to solving many discrete optimization problems is to associate the discrete set (over which we wish to optimize) with the 0-1 vectors in a given polyhedron and to derive linear inequalities valid for these 0-1 vectors from a linear inequality system defining the polyhedron. Lovasz and Schrijver (1991) described a family of operators, called the matrix-cut operators, which generate strong valid inequalities, called matrix cuts, for the 0-1 vectors in a polyhedron. This family includes the commutative, semidefinite and division operators; each operator can be applied iteratively to obtain, in n iterations for polyhedra in n-space, the convex hull of 0-1 vectors. We study the complexity of matrix-cut based methods for solving 0-1 integer linear programs. We first prove bounds on the (rank) number of iterations required to obtain the integer hull. We show that the upper bound of n, mentioned above, can be attained in the case of the semidefinite operator, answering a question of Goemans. We also determine the semidefinite rank of the standard linear relaxation of the traveling salesman polytope up to a constant factor. We study the use of the semidefinite operator in solving numerical instances and present results on some combinatorial examples and also on a few instances from the MIPLIB test set. Finally, we examine the lengths of cutting-plane proofs based on matrix cuts. We answer a question of Pudlak on such proofs, and prove an exponential lower bound on the length of cutting-plane proofs based on one class of matrix cuts.en_US
dc.format.extent112 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH.SCI. 2001 DASHen_US
dc.identifier.citationDash, Sanjeeb. "On the matrix cuts of Lovasz and Schrijver and their use in integer programming." (2001) Diss., Rice University. <a href="https://hdl.handle.net/1911/17952">https://hdl.handle.net/1911/17952</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/17952en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectMathematicsen_US
dc.subjectOperations researchen_US
dc.titleOn the matrix cuts of Lovasz and Schrijver and their use in integer programmingen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematical Sciencesen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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