  We show in all dimensions that minimizers of variational problems with a
convexity constraint, which arise from the Rochet-Chon\'e model with a
quadratic cost in the monopolist's problem in economics, can be approximated in
the uniform norm by solutions of singular Abreu equations. The difficulty of
our Abreu equations consists of having singularities that occur only in a
proper subdomain and they cannot be completely removed by any transformations.
To solve them, we rely on a new tool which we establish here: a Harnack
inequality for singular linearized Monge-Amp\`ere type equations that satisfy
certain twisted conditions.
