In this contribution, we derive an optimal solution to an ℋ_-/ℋ_∞ fault detection (FD) problem for linear time-invariant systems. Firstly, a novel ℋ_∞-based performance index is formulated to minimize the relative sensitivity of the residual to disturbances with respect to the sensitivity to faults. Then a class of optimal filters for the ℋ_-/ℋ_∞ FD problem is obtained by solving a Linear Matrix Inequality (LMI) optimization. Further specifications such as fault isolation can be achieved by using the remaining degrees of freedom in the optimal FD filter design. Moreover, weighting filters are considered to improve the results. Finally, two examples demonstrate the effectiveness of the proposed scheme.