  The regularization and renormalization of the radiative mass-type quadrupole
moment of inspiralling compact binaries (without spins) is investigated at the
fourth post-Newtonian (4PN) approximation of general relativity. As clear from
the conservative 4PN equations of motion, a dimensional regularization has to
be implemented in order to properly treat the non-linear interactions
experienced by gravitational waves during their propagation toward future null
infinity. By implementing such procedure, we show that the poles coming from
the source moment (computed in a companion paper) are exactly cancelled in the
radiative moment, as expected for a physical quantity. We thus define and
obtain a "renormalized" source quadrupole, three-dimensional by nature, which
is an important step towards the computation of the gravitational-wave flux
with 4PN accuracy. Furthermore, we explicitly prove the equivalence between the
dimensional regularization and the previously used Hadamard partie finie scheme
up to the 3PN order.
