It is estimated that Periodontal Diseases effects up to 90% of the adult population. Given the complexity of the host environment, many factors contribute to expression of the disease. Age, Gender, Socioeconomic Status, Smoking Status, and Race/Ethnicity are all known risk factors, as well as a handful of known comorbidities. Certain vitamins and minerals have been shown to be protective for the disease, while some toxins and chemicals have been associated with an increased prevalence. The role of toxins, chemicals, vitamins, and minerals in relation to disease is believed to be complex and potentially modified by known risk factors. A large comprehensive dataset from 1999-2003 from the National Health and Nutrition Examination Survey (NHANES) contains full and partial mouth examinations on subjects for measurement of periodontal diseases as well as patient demographic information and approximately 150 environmental variables. In this dissertation, a Feasible Solution Algorithm (FSA) will be used to investigate statistical interactions of these various chemical and environmental variables related to periodontal disease. This sequential algorithm can be used on traditional statistical modeling methods to explore two and three way interactions related to the outcome of interest. FSA can also be used to identify unique subgroups of patients where periodontitis is most (or least) prevalent. In this dissertation, FSA is used to explore the NHANES data and suggest interesting relationships between the toxins, chemicals, vitamins, minerals and known risk factors that have not been previously identified.

Linear programming (LP) stands for an optimization of a linear objective function, subject to linear and non-negativity constraints. For this purpose many methods for LP emerged. The best known is Simplex Method. Another group of methods for LP is represented by Interior Point Methods (IPM). These methods are based on interior points of feasible region of a problem, while Simplex Method uses basic feasible solution of a problem. This thesis focuses on theoretical background of IPM and brings it into relation with algorithms based on IPM. KKT system and its significance are included and the algorithm solving Linear Complementarity Problem is discussed as well. In this thesis, two algorithms based on IPM are introduced and used for solving a sample LP problem.