  Masses of classical Cepheids of 3 to 11 M$\odot$ are predicted by theory but
those measured, clump between 3.6 and 5 M$\odot$. As a result, their
mass-luminosity relation is poorly constrained, impeding our understanding of
basic stellar physics and the Leavitt Law. All Cepheid masses come from the
analysis of 11 binary systems, including only 5 double-lined and well-suited
for accurate dynamical mass determination. We present a project to analyze a
new, numerous group of Cepheids in double-lined binary (SB2) systems to provide
mass determinations in a wide mass interval and study their evolution. We
analyze a sample of 41 candidate binary LMC Cepheids spread along the P-L
relation, that are likely accompanied by luminous red giants, and present
indirect and direct indicators of their binarity. In a spectroscopic study of a
subsample of 18 brightest candidates, for 16 we detected lines of two
components in the spectra, already quadrupling the number of Cepheids in SB2
systems. Observations of the whole sample may thus lead to quadrupling all the
Cepheid mass estimates available now. For the majority of our candidates,
erratic intrinsic period changes dominate over the light travel-time effect due
to binarity. However, the latter may explain the periodic phase modulation for
4 Cepheids. Our project paves the way for future accurate dynamical mass
determinations of Cepheids in the LMC, Milky Way, and other galaxies, which
will potentially increase the number of known Cepheid masses even 10-fold,
hugely improving our knowledge about these important stars.
