  We study the minimization of a rank-one quadratic with indicators and show
that the underlying set function obtained by projecting out the continuous
variables is supermodular. Although supermodular minimization is, in general,
difficult, the specific set function for the rank-one quadratic can be
minimized in linear time. We show that the convex hull of the epigraph of the
quadratic can be obtaining from inequalities for the underlying supermodular
set function by lifting them into nonlinear inequalities in the original space
of variables. Explicit forms of the convex-hull description are given, both in
the original space of variables and in an extended formulation via conic
quadratic-representable inequalities, along with a polynomial separation
algorithm. Computational experiments indicate that the lifted supermodular
inequalities in conic quadratic form are quite effective in reducing the
integrality gap for quadratic optimization with indicators.
