How to generate weakly infeasible semidefinite programs via Lasserre’s relaxations for polynomial optimization

Examples of weakly infeasible semidefinite programs (SDP) are useful to test whether SDP solvers can detect infeasibility. However, finding non trivial such examples is notoriously difficult. This note shows how to use Lasserre’s semidefinite programming relaxations for polynomial optimization in order to generate examples of weakly infeasible SDP. Such examples could be used to test whether a SDP solver can detect weak infeasibility. In addition, in this note, we generate weakly infeasible SDP from an instance of polynomial optimization with nonempty feasible region and solve them by SDP solvers. Although all semidefinite programming relaxation problems are infeasible, we observe that SDP solvers do not detect the infeasibility and that values returned by SDP solvers are equal to the optimal value of the instance due to numerical round‐off errors.