  We examine the possible extension of the parameter space of the minimal
supersymmetric extension of the Standard Model (MSSM), as expressed via the
renormalization-group equations in terms of universal soft
supersymmetry-breaking terms at the unification scale, to include tachyonic
input scalar masses. Many models with negative masses-squared for scalars at
the unification scale may be viable because the small sizes of the
masses-squared allow them to change signs during the renormalization-group
evolution to the electroweak scale. However, in many cases, there is, in
addition to the electroweak vacuum, a much deeper high-scale vacuum located
along some F- and D-flat direction in the effective potential for the MSSM. We
perform a numerical search for such vacua in both the CMSSM and the NUHM. We
discuss the circumstances under which the existence of such a deep charge- and
color-breaking vacuum is consistent with standard cosmology. A crucial role is
played by the inflation--induced scalar masses, whereas thermal effects are
often irrelevant.
