We consider the problem of designing feedback control laws when a complete set of state variables is not available. For linear autonomous control systems with quadratic performance criterion, the design problem consists of choosing an appropriate static output feedback (SOF) gain matrix according to a certain objective function. The corresponding non-linear matrix optimization problem can be interpreted as an equality constrained minimization problem. For solving this problem, we propose a constrained trust region (CTR) algorithm, which presents a new and efficient numerical approach for this problem class. On the other hand, based on the formulation used in the past, the SOF problem can be also interpreted as an unconstrained programming problem. Thus, based on this interpretation, we also develop an unconstrained trust region (UTR) method. Finally, several numerical examples for optimal SOF problems demonstrate the applicability of the considered algorithm.