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MATRIX REDUCTION IN NUMERICAL OPTIMIZATION

dc.contributor.advisorO'Leary, Dianne Pen_US
dc.contributor.authorPark, Sungwooen_US
dc.date.accessioned2011-07-08T05:30:14Z
dc.date.available2011-07-08T05:30:14Z
dc.date.issued2011en_US
dc.identifier.urihttp://hdl.handle.net/1903/11751
dc.description.abstractMatrix reduction by eliminating some terms in the expansion of a matrix has been applied to a variety of numerical problems in many different areas. Since matrix reduction has different purposes for particular problems, the reduced matrices also have different meanings. In regression problems in statistics, the reduced parts of the matrix are considered to be noise or observation error, so the given raw data are purified by the matrix reduction. In factor analysis and principal component analysis (PCA), the reduced parts are regarded as idiosyncratic (unsystematic) factors, which are not shared by multiple variables in common. In solving constrained convex optimization problems, the reduced terms correspond to unnecessary (inactive) constraints which do not help in the search for an optimal solution. In using matrix reduction, it is both critical and difficult to determine how and how much we will reduce the matrix. This decision is very important since it determines the quality of the reduced matrix and the final solution. If we reduce too much, fundamental properties will be lost. On the other hand, if we reduce too little, we cannot expect enough benefit from the reduction. It is also a difficult decision because the criteria for the reduction must be based on the particular type of problem. In this study, we investigatematrix reduction for three numerical optimization problems. First, the total least squares problem uses matrix reduction to remove noise in observed data which follow an underlying linear model. We propose a new method to make the matrix reduction successful under relaxed noise assumptions. Second, we apply matrix reduction to the problem of estimating a covariance matrix of stock returns, used in financial portfolio optimization problem. We summarize all the previously proposed estimation methods in a common framework and present a new and effective Tikhonov method. Third, we present a new algorithm to solve semidefinite programming problems, adaptively reducing inactive constraints. In the constraint reduction, the Schur complement matrix for the Newton equations is the object of the matrix reduction. For all three problems, we propose appropriate criteria to determine the intensity of the matrix reduction. In addition, we verify the correctness of our criteria by experimental results and mathematical proof.en_US
dc.titleMATRIX REDUCTION IN NUMERICAL OPTIMIZATIONen_US
dc.typeDissertationen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.contributor.departmentComputer Scienceen_US
dc.subject.pqcontrolledComputer Scienceen_US
dc.subject.pqcontrolledStatisticsen_US
dc.subject.pqcontrolledApplied Mathematicsen_US
dc.subject.pquncontrolledconstraint reductionen_US
dc.subject.pquncontrolledcovariance matrixen_US
dc.subject.pquncontrollederrors in variablesen_US
dc.subject.pquncontrolledmatrix reductionen_US
dc.subject.pquncontrolledsemidefinite programmingen_US
dc.subject.pquncontrolledtotal least squaresen_US


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