Strong feasibility (a.k.a. strict feasibility) of the dual problem of a given linear matrix inequality (LMI) is an important property to guarantee the existence of an optimal solution of the LMI problem. In particular, the LMI problem may not have any optimal solutions if the dual is not strongly feasible. This implies that the computed solutions by SDP solvers may be meaningless and useless for designing the controllers of H∞ output feedback control problems. The facial reduction is a tool to analyze and reduce such non-strongly feasible problems. We introduce the strong feasibility of the dual and facial reduction and provide the necessary and sufficient condition on the strong feasibility. Furthermore, we reveal that the condition is closely related to invariant zeros in the plant.