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dc.contributor.authorTodd, M. J.en_US
dc.contributor.authorYıldırım, E. A.en_US
dc.date.accessioned2016-02-08T10:13:30Z
dc.date.available2016-02-08T10:13:30Z
dc.date.issued2007en_US
dc.identifier.issn0166-218X
dc.identifier.urihttp://hdl.handle.net/11693/23405
dc.description.abstractGiven A {colon equals} { a1, ..., am } ⊂ Rd whose affine hull is Rd, we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum-volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yi{dotless}ldi{dotless}ri{dotless}m, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yi{dotless}ldi{dotless}ri{dotless}m or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.en_US
dc.language.isoEnglishen_US
dc.source.titleDiscrete Applied Mathematicsen_US
dc.relation.isversionofhttp://dx.doi.org/10.1016/j.dam.2007.02.013en_US
dc.subjectLöwner ellipsoiden_US
dc.subjectCore setsen_US
dc.subjectRounding of polytopesen_US
dc.subjectEllipsoid methoden_US
dc.subjectApproximation algorithmsen_US
dc.titleOn Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoidsen_US
dc.typeArticleen_US
dc.departmentDepartment of Industrial Engineeringen_US
dc.citation.spage1731en_US
dc.citation.epage1744en_US
dc.citation.volumeNumber155en_US
dc.citation.issueNumber13en_US
dc.identifier.doi10.1016/j.dam.2007.02.013en_US
dc.publisherElsevieren_US
dc.identifier.eissn1872-6771


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