Estimating the parameters of max-stable parametric models poses significant challenges, particularly when some parameters lie on the boundary of the parameter space. This situation arises when a subset of variables exhibits extreme values simultaneously, while the remaining variables do not -- a phenomenon commonly referred to as an extreme direction. A novel estimator is proposed for the parameters of a general parametric mixture model, incorporating a threshold exceedances approach based on a pseudo-norm penalization. The latter plays a crucial role in accurately identifying parameters at the boundary of the parameter space. Additionally, the estimator comes with a data-driven algorithm to detect groups of variables corresponding to extreme directions. The performance of the estimator is assessed in terms of both parameter estimation and the identification of extreme directions through extensive simulation studies. Finally, the method is applied to two real-world datasets: discharge measurements at stations along the Danube river, and financial portfolio losses from stocks listed on the NYSE, AMEX, and NASDAQ. In both applications, the sets of variables that can become large simultaneously are identified.

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