K-means clustering (${\min }_{C_1,\dots ,C_K}{\sum }_{k=1}^K{\sum }_{i\in C_k}\Vert x_i-m_k{\Vert }^2$)

For a prescribed number of groups, minimize within-cluster squared Euclidean distances. At fixed assignments, differentiating the objective gives the mean as the optimal centroid. At fixed centroids, assigning each point to a nearest centroid minimizes its contribution. Alternating these steps cannot increase the objective but can converge to a local optimum, so multiple starts are useful. Empty clusters and ties need an explicit handling rule. The method favors roughly spherical groups and depends on variable scaling.