Second-order cone programming models for the unitary weighted Weber problem and for the minimum sum of the squares clustering problem

In this work, new mixed integer nonlinear optimization models are proposed for two clustering problems: the unitary weighted Weber problem and the minimum sum of squares clustering. The proposed formulations are convex quadratic models with linear and second-order cone constraints that can be efficiently solved by interior point algorithms. Their continuous relaxation is convex and differentiable. The numerical experiments show the proposed models are more efficient than some classical models for these problems known in the literature.