Many current deterministic solvers for NP-hard

combinatorial optimization problems are based on nonlinear

relaxation techniques that use floating point arithmetic.

Occasionally, due to solving these relaxations, rounding errors

may produce erroneous results, although the deterministic

algorithm should compute the exact solution in a finite number of

steps. This may occur especially if the relaxations are

ill-conditioned or ill-posed, and if Slater's constraint

qualifications fail. We show how exact solutions can be obtained

by rigorously bounding the optimal value of semidefinite

relaxations, even in the ill-posed case. All rounding errors due

to floating point arithmetic are taken into account.