The variational inequality problem arises in numerous contexts. In this dissertation, we consider solving a semi-infinite variational inequality problem, which is a variational inequality problem defined on a domain described by infinitely many constraints. We present characterization and the solution analysis for semi-infinite variational inequalities. After introducing the solution analysis, three solution methodologies, namely a discretizationbased smoothing method, an exchange method and an entropic analytic center cutting plane method are proposed. A comprehensive computational results with the comparison of the algorithms is provided.