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dc.contributor.advisorKuhlmann, Salmaen_US
dc.contributor.advisorMarshall, Murrayen_US
dc.creatorGhasemi, Mehdien_US
dc.date.accessioned2013-01-03T22:33:41Z
dc.date.available2013-01-03T22:33:41Z
dc.date.created2012-08en_US
dc.date.issued2012-09-14en_US
dc.date.submittedAugust 2012en_US
dc.identifier.urihttp://hdl.handle.net/10388/ETD-2012-08-590en_US
dc.description.abstractThere are a wide variety of mathematical problems in different areas which are classified under the title of Moment Problem. We are interested in the moment problem with polynomial data and its relation to real algebra and real algebraic geometry. In this direction, we consider two different variants of moment problem. The first variant is the global polynomial optimization problem, i.e., finding the minimum of a polynomial f in R[X1,...,Xn] on R^n. It is known that this problem is NP-hard. One of the most successful approaches to this problem is semidefinite programming (SDP), which gives a polynomial time method to approximate the largest real number $r$ such that f-r is a sum of squares of polynomials. But current implementations of SDP are not able to minimize polynomials in rather small number of variables (>6) of degree >8. We make use of a result due to Hurwitz, to give a criterion in terms of coefficients of a polynomial to be a sum of squares. Then, using this criterion, we introduce a much faster method to approximate a lower bound, in this case using geometric programming. The second variant is where we wish to determine when a given multi-sequence of reals comes from a Borel measure supported on a given subset K of R^n. This is the so called K-Moment Problem. Using Jacobi's Theorem for archimedean quadratic modules, we generalize a result of Berg et al. which states that the closure of the cone sums of squares in a weighted l1-topology consists of nonnegative polynomials on a hypercube. Then we substitute l1-topology with a locally convex topology tau, sums of squares by a cone C and the hypercube by a closed set K, and study the relation between closure of C in tau and nonnegative polynomials on K to solve the moment problem for a tau-continuous linear functional on R[X1,...,Xn]. We investigate the moment problem for weighted lp-continuous psd functionals and coefficientwise convergence topology on R[X1,...,Xn]. Then, we fix a set K, and find an appropriate locally convex topology tau, such that sums of 2d powers, solves the K-moment problem for tau-continuous linear functionals. In other words, we prove that a tau-continuous linear functional nonnegative on sums of 2d powers is representable by a Borel measure on K.en_US
dc.language.isoengen_US
dc.subjectOptimizationen_US
dc.subjectMoment Problemen_US
dc.subjectsums of squaresen_US
dc.subjectlinear functionalen_US
dc.titlePolynomial Optimization and the Moment Problemen_US
thesis.degree.departmentMathematics and Statisticsen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Saskatchewanen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophy (Ph.D.)en_US
dc.type.materialtexten_US
dc.type.genreThesisen_US
dc.contributor.committeeMemberBremner, Murrayen_US
dc.contributor.committeeMemberSamei, Ebrahimen_US
dc.contributor.committeeMemberSarty, Gordonen_US
dc.contributor.committeeMemberSoteros, Chrisen_US


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