We present a new method to compute solutions to the general multiblock l1 control problem. The method is based on solving a standard H2 problem and a finite-dimensional semidefinite quadratic programming problem of appropriate dimension. The new method has most of the properties that separately characterize many existing approaches. In particular, as the dimension of the quadratic programming problem increases, this method provides converging upper and lower bounds on the optimal l1 norm and, for well posed multiblock problems, ensures the convergence in norm of the suboptimal solutions to an optimal l1 solution. The new method does not require the computation of the interpolation conditions, and it allows the direct computation of the suboptimal controller.