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dc.contributor.advisorPablo A. Parrilo.en_US
dc.contributor.authorYuan, Chenyangen_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.en_US
dc.date.accessioned2019-02-14T15:47:51Z
dc.date.available2019-02-14T15:47:51Z
dc.date.copyright2018en_US
dc.date.issued2018en_US
dc.identifier.urihttp://hdl.handle.net/1721.1/120396
dc.descriptionThesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.en_US
dc.descriptionCataloged from PDF version of thesis.en_US
dc.descriptionIncludes bibliographical references (pages 73-75).en_US
dc.description.abstractIn this thesis, we study approximation algorithms for polynomial optimization over the sphere, concentrating on classes of polynomials whose optimum on the sphere can be efficiently approximated to a factor that only depends on the degree of the polynomial, and not on the dimension of the problem. We extend and generalize an existing class of polynomials known as focused polynomials which exhibit this property. These polynomials can be well approximated by a random projection, reducing the problem to optimization over a sphere of a much smaller dimension. We then introduce polynomials generated from a focused cone, which generalizes focused polynomials, and show that the dimension required for the projection is related to a geometrical property of the focused cone, its Gaussian width. Next we study the behavior of the maximum of quadratic polynomials under a random projection, and show that if the dimension of the random projection is at least the stable rank of the matrix representation of this polynomial, its maximum over the sphere is preserved within a constant factor. We then show that the stable rank of matrices representing quadratic focused polynomials is also small, and how some properties of focused polynomials generalizes statements about matrices. Finally we apply sum of squares optimization to focused polynomials, and show that given a polynomial generated from a focused cone one can devise a rounding algorithm that finds a vector close to this cone, which gives a good approximation to the optimum.en_US
dc.description.statementofresponsibilityby Chenyang Yuan.en_US
dc.format.extent75 pagesen_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsMIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectElectrical Engineering and Computer Science.en_US
dc.titleFocused polynomials, random projections and approximation algorithms for polynomial optimization over the sphereen_US
dc.typeThesisen_US
dc.description.degreeS.M.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
dc.identifier.oclc1083761637en_US


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