Clipping is a necessary signal processing operation in many real-time audio applications, yet it often reduces the sound quality of the signal. The recently proposed perception-based clipping algorithm has been shown to significantly outperform other clipping techniques in terms of objective sound quality scores. However, the real-time solution of the optimization problems that form the core of this algorithm, poses a challenge. In this paper, a fast gradient projection optimization method is proposed and incorporated into the perception-based clipping algorithm. The optimization method will be shown to have an extremely low computational complexity per iteration, allowing the perception-based clipping algorithm to be applied in real-time for a broad range of clipping factors.