  We identify the deviation from the straight line error (DSLE) -- i.e., the
spurious non-linearity of the total energy as a function of fractional particle
number -- as the main source for the discrepancy between experimental vertical
ionization energies and theoretical quasiparticle energies, as obtained from
the $GW$ and $GW$+SOSEX approximations to many-body perturbation theory (MBPT).
For self-consistent calculations, we show that $GW$ suffers from a small DSLE.
Conversely, for perturbative $G_0W_0$ and $G_0W_0$+SOSEX calculations the DSLE
depends on the starting point. We exploit this starting-point dependence to
reduce (or completely eliminate) the DSLE. We find that the agreement with
experiment increases as the DSLE reduces. DSLE-minimized schemes, thus, emerge
as promising avenues for future developments in MBPT.
