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Clipping is an essential signal processing operation in many real-time audio applications, yet the use of existing clipping techniques generally has a detrimental effect on the perceived audio signal quality. In this paper, we present a novel multidisciplinary approach to clipping which aims to explicitly minimize the perceptible clipping-induced distortion by embedding a convex optimization criterion and a psychoacoustic model into a frame-based algorithm. The core of this perception-based clipping algorithm consists in solving a convex optimization problem for each time frame in a fast and reliable way. To this end, three different structure-exploiting optimization methods are derived in the common mathematical framework of convex optimization, and corresponding theoretical complexity bounds are provided. From comparative audio quality evaluation experiments, it is concluded that the perception-based clipping algorithm results in significantly higher objective audio quality scores than existing clipping techniques. Moreover, the algorithm is shown to be capable to adhere to real-time deadlines without making a sacrifice in terms of audio quality.