We consider the problem of designing a suboptimal H-2/H-infinity feedback control law for a linear time-invariant control system when a complete set of state variables is not available. This problem can be necessarily restated as a nonconvex optimization problem with a bilinear, multiobjective functional under suitably chosen linear matrix inequality (LMI) constraints. To solve such a problem, we propose an LMI-based procedure which is a sequential linearization programming approach. The properties and the convergence of the algorithm are discussed in detail. Finally, several numerical examples for static H-2/H-infinity output feedback problems demonstrate the applicability of the considered algorithm and also verify the theoretical results numerically.