We consider the problem of constructing decentralized control systems. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. We develop necessary and sufficient conditions under which the constraint set is quadratically invariant, and show that many examples of decentralized synthesis which have been proven to be solvable in the literature are quadratically invariant. As an example, we show that a controller which minimizes the norm of the closed-loop map may be efficiently computed in the case where distributed controllers can communicate faster than the propagation delay of the plant dynamics.