Numerical Study of Semidefinite Bounds for the k-cluster Problem

This paper deals with semidefinite bounds for the k-cluster problem, a classical NP-hard problem in combinatorial optimization. We present numerical experiments to compare the standard semidefinite bound with the new semidefinite bound of [J. Malick and F. Roupin. Solving k-cluster problems to optimality using adjustable semidefinite programming bounds. Submitted, 2009], regarding the trade-off between tightness and computing time. We show that the formulation of the semidefinite bounds has an impact on the efficiency of the numerical solvers, and that the choice of the solver depends on what we expect to get: good accuracy, cheap computational time, or a balance of both.