We consider the problem of finding optimal feedback gains in the presence of structural constraints and/or sparsity-promoting penalty functions. Such problems are known to be difficult due to their lack of convexity. We provide an equivalent reformulation of the optimization problem such that its source of nonconvexity is isolated in one nonconvex matrix inequality of the form Y X -1. Furthermore, we preserve the feedback gain as an optimization variable in the reformulated problem. Via linearizations of the nonconvex constraint, we introduce an iterative algorithm that solves a semidefinite program at every stage and for which the nonconvex constraint is satisfied upon convergence. We elaborate on the modular nature of the proposed scheme and show that it can be used in a wide range of network control problems.