Zinchenko, YuriyWeppler, Sarah2015-09-252015-11-202015-09-252015http://hdl.handle.net/11023/2506The goal of this thesis is to examine tight computable bounds on a probability measure generated by its moments. We study measures supported on a real line and propose an extension of the classical moment problem to the so-called rational moments. Specifically, we examine semidefinite and linear optimization formulations for solving the univariate rational Hausdorff moment problem given a vector of moments. We further investigate shifted moments to reduce the distance between probability bounds, and propose a numerical method to better position such shifts. In addition, when only a few raw and shifted moments are known, we derive novel extensions of the Markov and Chebyshev bounds. Motivated by the problem of optimal radiotherapy treatment design, we present a novel and first in its class cutting plane method to be included within the mixed integer branch and cut scheme. Implementing these cuts results in as much as a 40% runtime reduction.engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.MathematicsOptimizationprobability boundsshifted momentsmaster thesis10.11575/PRISM/28449