Abstract: In this paper convexity constraints are derived for a background model of electron energy loss spectra (EELS) that is linear in the fitting parameters. The model outperforms a power-law both on experimental and simulated backgrounds, especially for wide energy ranges, and thus improves elemental quantification results. Owing to the model’s linearity, the constraints can be imposed through fitting by quadratic programming. This has important advantages over conventional nonlinear power-law fitting such as high speed and a guaranteed unique solution without need for initial parameters. As such, the need for user input is significantly reduced, which is essential for unsupervised treatment of large datasets. This is demonstrated on a demanding spectrum image of a semiconductor device sample with a high number of elements over a wide energy range.