Dimension reduction techniques and cluster analysis play an important role in efficiently managing and extracting insights from data. This thesis focuses on developing structural properties and mathematical programming algorithms to solve problems related to dimension reduction and clustering.The first part of this thesis concerns correspondence analysis, a dimension reduction technique typically applied to contingency tables. Specifically, we investigate the inverse correspondence analysis problem, where we use a correspondence analysis solution to retrieve the original contingency table. Although experiments in the literature suggest that correspondence analysis solutions uniquely correspond to a table, we establish the existence of distinct contingency tables that have the same correspondence analysis solution.In the second part, we aim to generate a uniform distribution of contingency tables satisfying given marginals and levels of association. To tackle this problem, we first develop an efficient method to generate a uniform distribution of vectors satisfying linear and nonlinear constraints. Afterwards, we present optimisation-based and heuristic methods to generate contingency tables with given properties.In the last part, we address clustering and location problems, specifically, we consider the hierarchical clustering and p-median problem. To solve large-scale instances, we develop exact mathematical programming methods that leverage decomposition techniques. Our methods are validated on real-world instances, demonstrating the ability to solve larger instances to optimality compared to existing approaches in the literature.