In real hyperspectral images, there exist variations within spectra of materials. The inherent spectral variability is one of the major obstacles for the successful hyperspectral image target detection. Although several hyperspectral image target detection algorithms have been proposed, there are few algorithms considering the spectral variability. Under such circumstances, in this paper, we propose a hyperspectral image target detection algorithm that is robust to the target spectral variability. The proposed algorithm utilizes an inequality constraint to guarantee that the outputs of target spectra, which vary in a certain set, are larger than one, so that these target spectra could be detected. The proposed algorithm transforms the target detection to a convex optimization problem and uses a kind of interior point method named barrier method to solve the formulated optimization problem effectively. Two synthetic hyperspectral images and two real hyperspectral images are used to conduct experiments. The experimental results demonstrate the proposed algorithm is robust to the target spectral variability and performs better than other classical algorithms.