Consider the classical state feedback design in the linear system $\dot x = Ax+Bu$ subject to performance specifications with an additional requirement that the control input vector $u=Kx$ has as many zero entries as possible. The corresponding gain $K$ is referred to as a row-sparse controller. We propose an approach to approximate solution of this kind of nonconvex problems by formulating the proper convex surrogate,---the minimization of a certain matrix norm subject to LMI constraints. The novelty of the paper is the problem formulation itself and the construction of the surrogate. The two main contributions are the design of low-dimensional output to be used in static output feedback, and suboptimal design illustrated via LQR. The results of preliminary numerical experiments are twofold. First, in many test problems, the number of controls was considerably reduced without significant loss in performance. Second, the number of nonzero entries obtained by our method is either very close to or coincide with the minimum possible amount. The approach can be further extended to handle numerous problems of optimal and robust control in sparse formulation.