  Quadratic invariance is a condition which has been shown to allow for optimal
decentralized control problems to be cast as convex optimization problems. The
condition relates the constraints that the decentralization imposes on the
controller to the structure of the plant. In this paper, we consider the
problem of finding the closest subset and superset of the decentralization
constraint which are quadratically invariant when the original problem is not.
We show that this can itself be cast as a convex problem for the case where the
controller is subject to delay constraints between subsystems, but that this
fails when we only consider sparsity constraints on the controller. For that
case, we develop an algorithm that finds the closest superset in a fixed number
of steps, and discuss methods of finding a close subset.
