  In this paper, we present the novel analytical expressions for the
bounded-from-below or the vacuum stability conditions of scalar potential for a
general CP violating two-Higgs-doublet model by using the concepts of
co-positivity and the gauge orbit spaces. More precisely, several analyticial
sufficient conditions and necessary conditions are established for the vacuum
stability of the general 2HDM potential, respectively. We also give an
equivalent condition of the vacuum stability of the general 2HDM potential in
theory, and then, apply it to derive the analyticial necessary conditions of
the general 2HDM potential. Meanwhile, the semi-positive definiteness is proved
for a class of 4th-order 2-dimensional complex tensor.
