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Abstract

This thesis studies globally optimal solution methods for three classical centroid-based clustering problems: K-means, K-centers, and K-medoids. The work has two closely related objectives. First, it presents a unified mathematical treatment of these models, including mixed-integer formulations and a common branch-and-bound viewpoint that clarifies the roles of lower bounds, upper bounds, domain reduction, and branching. Second, it implements this viewpoint in a Julia package, GO_Clustering.jl, that brings together exact and gap-certifying methods from the literature within a consistent computational framework. The package provides common data structures, solver interfaces, MPI-based parallel execution, structured logging, and experiment drivers, so that one software environment can be used to solve and compare K-means, K-centers, and K-medoids under a unified workflow. By integrating these methods into a single package, the thesis supports systematic computational study and makes it easier to report certified lower and upper bounds and global optimality certificates. Overall, the thesis provides a reproducible software foundation for research on globally optimal centroid-based clustering.