Conditions for certain Higgs potentials to be bounded below

Necessary and sufficient conditions are obtained for Higgs potentials to be bounded below when they are constructed from: 1) two doublets, and also two doublets and a singlet of $SU(2)$, 2) the adjoint and vector representations of $SO(n)$. For a potential constructed from the adjoint and fundamental multiplets of $SU(n)$, the problem of necessary and sufficient conditions has been only partly solved.