  In this paper, we consider convex quadratic optimization problems with
indicators on the continuous variables. In particular, we assume that the
Hessian of the quadratic term is a Stieltjes matrix, which naturally appears in
sparse graphical inference problems and others. We describe an explicit convex
formulation for the problem by studying the Stieltjes polyhedron arising as
part of an extended formulation and exploiting the supermodularity of a set
function defined on its extreme points. Our computational results confirm that
the proposed convex relaxation provides an exact optimal solution and may be an
effective alternative, especially for instances with large integrality gaps
that are challenging with the standard approaches.
