This paper introduces a new method to maximize kurtosis-based contrast functions. Such contrast functions appear in the problem of blind source separation of convolutively mixed sources: the corresponding methods recover the sources one by one using a deflation approach. The proposed maximization algorithm is based on the particular nature of the criterion. The method is similar in spirit to a gradient ascent method, but differs in the fact that a "reference" contrast function is considered at each line search. The convergence of the method to a stationary point of the criterion can be proved. The theoretical result is illustrated by simulation