1) The document proposes a cardinality-constrained k-means clustering approach to address practical challenges with standard k-means, such as skewed clustering and sensitivity to outliers. 2) It formulates the problem as a mixed integer nonlinear program (MINLP) and provides a convex relaxation to the problem using semidefinite programming (SDP). 3) The approach provides optimality guarantees and a rounding algorithm to recover an integer feasible solution. Numerical experiments demonstrate competitive performance versus heuristics.