As a noninvasive technique, electroencephalography (EEG) is commonly used to monitor the brain signals of patients with epilepsy such as the interictal epileptic spikes. However, the recorded data are often corrupted by artifacts originating, for example, from muscle activities, which may have much higher amplitudes than the interictal epileptic signals of interest. To remove these artifacts, a number of independent component analysis (ICA) techniques were successfully applied. In this paper, we propose a new deflation ICA algorithm, called penalized semialgebraic unitary deflation (P-SAUD) algorithm, that improves upon classical ICA methods by leading to a considerably reduced computational complexity at equivalent performance. This is achieved by employing a penalized semialgebraic extraction scheme, which permits us to identify the epileptic components of interest (interictal spikes) first and obviates the need of extracting subsequent components. The proposed method is evaluated on physiologically plausible simulated EEG data and actual measurements of three patients. The results are compared to those of several popular ICA algorithms as well as second-order blind source separation methods, demonstrating that P-SAUD extracts the epileptic spikes with the same accuracy as the best ICA methods, but reduces the computational complexity by a factor of 10 for 32-channel recordings. This superior computational efficiency is of particular interest considering the increasing use of high-resolution EEG recordings, whose analysis requires algorithms with low computational cost.