Why does K-Means converge and what is it optimizing mathematically?

K-Means converges because it monotonically minimizes the within-cluster sum of squares (WCSS) objective function.

K-Means optimizes the objective: J = Σ ||x_i - μ_k||². Each iteration has two steps: assignment (fix centroids, assign points) and update (fix assignments, recompute centroids). Both steps reduce or maintain the objective value, ensuring convergence to a local minimum, not necessarily global.