This paper uses the framework of a Mass Spectrometry application to introduce a new method of peak picking as well as two active set methods for the minimization of the elastic-net-functional. The application of peak picking is essential in mass spectrometry and is often based on mean spectra. In contrast our procedure uses a set of spectra obtained from a basis learning method. Our procedure utilizes the well known l1-minimization and corresponding active set algorithms but comprises ill conditioned operators such that regularization is required. We show, that the elastic-net gives a natural justification for Tikhonov-Philipsregularization in the used algorithms. Therefore we introduce adaptions of known active set algorithms for l1-minimization to the elastic net. Furthermore, we emphasize the differences of the algorithms for `1 and the elastic-net in numerical examples.