Objective: The electroencephalographic (EEG) inverse problem is ill-posed owing to the electromagnetism Helmholtz theorem and since there are fewer observations than the unknown variables. Apart from the strong background activities (ongoing EEG), evoked EEG is also inevitably contaminated by strong outliers caused by head movements or ocular movements during recordings. Methods: Considering the sparse activations during high cognitive processing, we propose a novel robust EEG source imaging algorithm, least absolute l-P (0