  We present a comparative full-potential study of generalized Kohn-Sham
schemes (gKS) with explicit focus on their suitability as starting point for
the solution of the quasiparticle equation. We compare $G_0W_0$ quasiparticle
band structures calculated upon LDA, sX, HSE03, PBE0, and HF functionals for
exchange and correlation (XC) for Si, InN and ZnO. Furthermore, the HSE03
functional is studied and compared to the GGA for 15 non-metallic materials for
its use as a starting point in the calculation of quasiparticle excitation
energies. For this case, also the effects of selfconsistency in the $GW$
self-energy are analysed. It is shown that the use of a gKS scheme as a
starting point for a perturbative QP correction can improve upon the
deficiencies found for LDA or GGA staring points for compounds with shallow $d$
bands. For these solids, the order of the valence and conduction bands is often
inverted using local or semi-local approximations for XC, which makes
perturbative $G_0W_0$ calculations unreliable. The use of a gKS starting point
allows for the calculation of fairly accurate band gaps even in these difficult
cases, and generally single-shot $G_0W_0$ calculations following calculations
using the HSE03 functional are very close to experiment.
